Unsteady Surface Research Articles

Swale, and: Ghost Forest, and: "I Have Written Myself into a Tropical Glow", and: Scurvy, and: You Like to Think the Whales Are Listening

Swale, and: Ghost Forest, and: "I Have Written Myself into a Tropical Glow", and: Scurvy, and: You Like to Think the Whales Are Listening Allison Hutchcraft (bio) Swale In my winter by the sea, I fashioneda new habit: each day walking to Crowley Creek through mudand leafless alder, their branches cupped by the plush green of mosses and rollingbeds of sword fern, whose serrated edges thrust extravagantly into cold and humid air.The creek fed the estuary, which in turn fed the sea—and I liked to see how far upthe tide had reached, or how far it had receded, the marshy banks transformedby that lunar clockwork on which my hours turned.Water called slack, like the grip on a rope loosened, at which point the river would swelland still, the brackish tide having expanded the limits of the creek, submergedgrasses swaying like the drowned hair of a doll. Cold and hard and clear,the water looked like the creek I felt in me. [End Page 29] Day after day I watched gulls float like wooden toys,rocking on the unsteady surface, and studied barnacles clasped to rocks, theshell-white skeletons of small shoreline animals, discarded limbsof driftwood. Swale also meaning a depression, a low placein the land, the sour smell once the water has drawn back,unmasking river sludge and battered sea debris:luminous blue Velella with their fan-like sails, hollow carapaces of crabs,picked at and cleaned— When I swale, I cannottell border from border, land from water. I feel the loamof day crumble. Washed up, what's left?An accumulation of silt. Or sand, sifted, rubbery tendrils of seaweed dotted with notcheslike taste buds inflamed— Sometimes I think love is swale, andsometimes sadness, how each comes in like a tide, how eachalters the bodies beneath. [End Page 30] Heart, be complete, come out of your grave-light—It was decades before I was alive when the estuary was diked to make moreland for pasture, the water no longer water then but fields of sown grassesfor the cows to eat. How they, too, must have tasted it—the memory of waterburied in the new green shoots, the verdant nourishment, still tasting, faintly,of brine. [End Page 31] Ghost Forest As if resurrected, the trees have returned: what's leftof their once buried trunks now jagging through beach sand. We go at low tide so we can see them,and it is easy to walk to those farthest up shore, to peer into their hollows made by wind and by water, followingthe ancient rings, the swollen wood, dark and soft, something to flake with a tool or fingernail.In all the available spaces, blue-gray stones, the surprising white of shattered shells,as if someone had lodged them there deliberately. Delicate, bone-like, barnacles encase trunk after trunk,each exoskeleton as small as a stud earring and open at the top, like an erupted volcano or extracted molar.Farther north, in the outskirts of Alaska's archipelagos, [End Page 32] if the earth ruptures we will feel it. The seawill swell with tsunami waves, covering roads and towns, breaking apart the shore as two thousand years agothese spruce broke from high cliff shelves. And when they were buried it was instantly—flanks of land collapsing, swallowed, and the animals, too, their pelty sinking. Called ghost, but lookhow what's left stubbornly remains, not ghost at all but solid, stunted branches, radial and smooth,the polished wood now green with a fine algae or limp strands of seaweed, which, when the tide returns, will riselike candles taking flame. Look closer: there is always something ready to bury us.Slabs of damp sand break off into the creek that cuts this beach in two— in the Arctic, whole glaciersfracture this way, the oceans everywhere rising. [End Page 33] A woman wades across the silty creek, carryingher small dog in the cleft of her jacket— her legs more and more submerged,nearly up to her hips now— the look on her face— [End Page 34...

Numerical investigation of low-noise airfoils inspired by the down coat of owls

Numerical analysis of airfoil geometries inspired by the down coat of the night owl is presented. The bioinspired geometry consists of an array of ‘finlet fences’, which is placed near the trailing edge of the baseline (NACA 0012) airfoil. Two fences with maximum nondimensional heights, and are investigated, where is the displacement thickness at 2.9% chord upstream of the airfoil trailing edge. Wall-resolved large eddy simulations are performed at chord-based Reynolds number, , flow Mach number, , and angle of attack, . The simulation results show significant reductions in unsteady surface pressure and farfield radiated noise with the fences, in agreement with the measurements available in the literature. Analysis of the results reveals that the fences increase the distance between the boundary layer turbulence (source) and the airfoil trailing (scattering) edge, which is identified to be the mechanism behind high-frequency noise reduction. These reductions are larger for the taller fence as the source-scattering edge separation is greater. Two-point correlations show that the fences reduce the spanwise coherence at low frequencies for separation distances greater than a fence pitch (distance between two adjacent fences) and increase the coherence for smaller distances, the increase being higher for the taller fence. This increase in coherence and the reduced obliqueness of the leading edge of the fence are hypothesized to be responsible for the small increase in farfield noise at low frequencies observed in the simulations with the taller fence.

Unsteady Gerstner waves

We present an analytical description of the class of unsteady vortex surface waves generated by non-uniformly distributed, time-harmonic pressure. The fluid motion is described by an exact solution of the equations of hydrodynamics generalizing the Gerstner solution. The trajectories of the fluid particles are circumferences. The particles on a free surface rotate around circumferences of the same radii, with the centers of the circumferences lying on different horizons. A family of waves has been found in which a variable pressure acts on a limited section of the free surface. The law of external pressure distribution includes an arbitrary function. An example of the evolution of a non-uniform wave packet is considered. The wave and pressure profiles, as well as vorticity distribution are studied. It is shown that, in the case of a uniform traveling wave of external pressure, the Gerstner solution is valid but with a different form of the dispersion relation. A possibility of observing the studied waves in laboratory and in the real ocean is discussed.

An investigation of the mechanisms causing large-amplitude wind-induced vibrations in stay cables using unsteady surface pressure measurements

Dry galloping has been observed on in-service bridges and has been reproduced in several wind tunnel experiments of inclined stay cables. In certain cases, the large-amplitude vibrations caused by dry galloping could not be mitigated with damping levels specified by the Post Tensioning Institute (PTI). Wind tunnel investigations were conducted for IHI Corporation on a 1:1 scale sectional model of an inclined bridge stay cable in the 3 m × 6 m Wind Tunnel at the National Research Council Canada in 2015. The purpose of the investigation was to reproduce large amplitude cable galloping, which had been observed at an existing cable-stayed bridge in Japan. The experiment was designed to investigate the influence of different damping levels and the ability of a 5 mm-diameter helical fillet to mitigate dry cable vibrations. The current work identified several important factors that contributed to dry galloping and large-amplitude cable motions. It was shown that the physical mechanisms leading to the onset of large-amplitude motion in the critical and supercritical Reynolds number regimes are distinct. In the critical Reynolds number regime, the drop in drag, increase in lift, and fluctuations in the laminar separation bubbles along the cable destabilize the low pressure lobes, allowing the changes in pressure along the cable to become synchronized with the cable motion in a highly-correlated manner. This type of galloping could not be mitigated with damping greater than that required by the PTI. At high Reynolds numbers, the boundary layer state changed and was associated with the upstream movement of the separation point and an asymmetric pressure distribution, Kármán vortex shedding re-emerged and appeared to combine with low-frequency variations to induce large-amplitude motion. The large-amplitude motion at high Reynolds numbers could be mitigated with additional damping.

Unsteady Flow Physics of Airfoil Dynamic Stall

A series of wind-tunnel experiments was conducted on a dynamically pitching NACA 0012 airfoil in order to understand the unsteady flow physics associated with dynamic stall at a transitional Reynolds number of . The airfoil was pitched about the quarter-chord axis following a linear ramp maneuver at . A series of high-frequency unsteady surface pressure measurements were acquired in order to investigate the time-dependent behavior of the flow in the immediate vicinity of the airfoil. These surface pressure measurements actively displayed the evolution of the airfoil boundary layer, leading-edge laminar separation bubble, and dynamic stall vortex. A detailed time-frequency analysis of surface pressure measurements was performed in order to identify the dominant frequencies associated with the unsteady laminar separation bubble. This analysis was also used to study the evolution of the length scales associated with flow structures near the airfoil leading edge, which emerged before the formation of the dynamic stall vortex. Finally, time-resolved particle image velocimetry measurements were acquired to study the evolution of the off-body flowfield during the development of dynamic stall.

Three non-Newtonian fluids flow considering thin film over an unsteady stretching surface with variable fluid properties

This research examines the features of liquid film of non-Newtonian fluids under the influence of thermophoresis. For this study, we proposed a mathematical model for Jeffrey, Maxwell, and Oldroyd-B fluids and concluded the unsteady stretched surface in the existence of a magnetic field and also the thermal conductivity was measured which is directly related to the temperature whereas the viscosity inversely related to the temperature. Inserting the thermophoretic effect which improved the thermal conductivity of Jeffrey fluid over the Oldroyd-B and Maxwell fluids. The model is helpful for the liquid flow of Jeffrey, Maxwell, and Oldroyd-B fluid including the Brownian motion parameter effect. The results have been obtained through optimal approach compared with numerical (ND-Solve) method. Study mainly focused to understand the physical appearance of the embedded parameters based on the characteristic length of the liquid flow. The behavior of skin friction, local Nusselt number, and Sherwood number has been described numerically for the dynamic constraints of the problem. The obtained results are drafted graphically and discussed.

Single-shot two-dimensional measurement of nanoparticles in turbulent jet-diffusion flames using phase-selective laser-induced breakdown spectroscopy

In this work, we demonstrate a single-shot two-dimensional measurement of vanadic oxide (V2O5) nanoparticles in a turbulent jet-diffusion flame based on phase-selective laser-induced breakdown spectroscopy (PS-LIBS). By collecting the atomic spectra of vanadium near 439 nm from nano-sized plasmas within the ∼30 ns lifetime, particle information can be revealed without interference from elastic scattering or Bremsstrahlung emissions. As the laser intensity increases, the signal intensity first increases and then saturates when the laser intensity reaches 0.5 GW/cm2. In the saturation regime, a proportional correlation is established between the signal intensity and the particle volume fraction. Based on the parametric study in the laminar condition, we image the instantaneous distribution of particle volume fraction in the turbulent condition using single-shot PS-LIBS measurements. The snapshots show that vanadic oxide nanoparticles concentrate near the diffusion flame surface, which may be caused by the rapid formation of particles on the oxidizer/precursor side and quick dilution across the reaction layer. The Proper Orthogonal Decomposition (POD) analysis further indicates that the fluctuations of the particle volume fraction originate from the unsteady flame surface at the upstream positions and large-scale mixing at the downstream positions. The single-shot PS-LIBS measurement shows promising potential for resolving complex processes of particle formation in turbulent flame synthesis.

Trailing-edge flow and noise control using porous treatments

This paper is concerned with the application of porous treatments as a means of flow and aerodynamic noise reduction. An extensive experimental investigation is undertaken to study the effects of flow interaction with porous media, in particular in the context of the manipulation of flow over blunt trailing edges and attenuation of vortex shedding. Comprehensive boundary layer and wake measurements have been carried out for a long flat plate with solid and porous blunt trailing edges. Unsteady velocity and surface pressure measurements have also been performed to gain an in-depth understanding of the changes to the energy–frequency content and coherence of the boundary layer and wake structures as a result of the flow interaction with a porous treatment. Results have shown that permeable treatments can effectively delay the vortex shedding and stabilize the flow over the blunt edge via mechanisms involving flow penetration into the porous medium and discharge into the near-wake region. It has also been shown that the porous treatment can effectively destroy the spanwise coherence of the boundary layer structures and suppress the velocity and pressure coherence, particularly at the vortex shedding frequency. The flow–porous scrubbing and its effects on the near-wall and large coherent structures have also been studied. The emergence of a quasi-periodic recirculating flow field inside highly permeable surface treatments has also been investigated. Finally, the paper has identified several important mechanisms concerning the application of porous treatments for aerodynamic and aeroacoustic purposes, which can help more effective and tailored designs for specific applications.

Unsteady surface pressure characteristics of asymmetrically beveled trailing edges

The characteristics of unsteady surface pressure (USP) created by turbulent flow over a family of asymmetrically beveled trailing edges were studied experimentally. The geometries had a trailing edge angle $$\theta =25^\circ$$ with a flat lower surface and a rounded upper surface with radii of curvature between zero and ten times the airfoil thickness. The Reynolds number was $$Re=2.1\times 10^6$$ based on chord. A detailed description of the USP and flow field around the trailing edge was obtained using remote microphone probes (RMP) and particle image velocimetry (PIV), respectively. The lower surface exhibited USP auto-spectral density magnitudes that were similar to those of a zero-pressure-gradient turbulent boundary layer at higher frequency. The low-frequency pressure fluctuations were influenced by the turbulent wake, leading to large increases in magnitude closer to the trailing edge. An empirical model of these results is proposed. The beveled upper surface was characterized by a region of favorable pressure gradient, followed by a strong adverse pressure gradient. The cases with smaller radius of curvature were found to exhibit separated flow over the trailing edge. The spectral magnitudes were largest in these regions, and significant attention is given to the proper scaling of these results. The PIV measurements provided the length and velocity scales for this purpose.

Comparison of two low-order models for the prediction of fan broadband noise

Two low-order methods for predicting the broadband noise downstream of a fan exit guide vane (FEGV) due to interaction with rotor wake turbulence are compared. The noise is predicted with a two-step approach in each method. The methods share the exact same first step. The unsteady vane surface pressure is computed using a strip approach. At each strip, the 2D flat-plate cascade response to a gust in the wave-number, frequency domain is calculated. An integral approach is used for this calculation. The two methods then differ in the second step. One method integrates the unsteady surface pressure together with the Green's function for a cylindrical annulus to obtain the downstream duct acoustic pressure modes. The other method computes rectilinear acoustic modes relevant to the cascade at each strip. In the second method, the total power is obtained by adding the acoustic power downstream of each strip. Both methods neglect mean flow swirl effects on acoustic propagation. The fan broadband noise workshop RC1 and FC2 cases are used as a platform to compare the two methods. When the FEGV inflow turbulent parameters are taken from different sources (e.g. hot-wire, CFD) the predictions from both methods respond similarly. The trends predicted with both methods based on turbulence level or rotor speed are similar. The trends with vane count predicted by the two methods do not match and it is unclear which prediction is correct. The actual spectrum predicted by the Green's function method for all of the cases is shown to be slightly closer to experimental results.

Active Flow Control via Discrete Sweeping and Steady Jets on a Simple-Hinged Flap

Active flow control (AFC) in the form of sweeping jet (SWJ) excitation and discrete steady jet excitation is used to control the flow separation on a National Advisory Council for Aeronautics (NACA) 0015 semispan wing with a deflected, simple-hinged, trailing edge flap. This geometry has been the focus of several recent publications that investigated methods to improve the efficiency of SWJ actuators. In the current study, the interaction of the AFC excitation with the separated flowfields of several flap deflection angles was examined. Previous studies with this model have been limited to a maximum flap deflection angle of 40 deg. The flap deflection range was extended to 60 deg because systems studies have indicated that a high-lift system with simple-hinged flaps may require larger flap deflections than the Fowler flaps found on most high-lift systems. The results obtained at flap deflection angles of 20, 40, and 60 deg are presented and compared. Force and moment data, particle image velocimetry (PIV) data, and steady and unsteady surface pressure data are used to describe the flowfield with and without AFC. With a flap deflection of 60 deg, increasing the SWJ actuator momentum at the flap shoulder increased lift due to an increase in circulation but did not completely eliminate the recirculation region above the flap surface. AFC using the discrete steady jet actuators of this study increased lift as well but required more mass flow than the SWJ actuators and had a detrimental effect on lift at the highest mass flow level tested. PIV results showed that the angle between the excitation and the flap surface was not optimal for attaching the separated shear layer.

Wall Pressure and Blade Surface Pressure in a Side Channel Blower

In side channel blowers, the pressure field is the result of complex, inner flow mechanisms. While there are already experimental investigations on the wall pressure distributions, little is known about the pressure in the rotating system, i.e., on the blade surface. In this work, we present an experimental setup for measuring the unsteady blade surface pressure in several positions. The acquired data will be complemented by and compared to the additionally measured wall pressure on the side channel housing. Miniature pressure sensors are integrated into the impeller. It is modified to ensure flush mounted membranes of the sensors and to avoid impacting on the flow field. A telemetric system is used for a wireless transfer of the data from the rotating system to the data recorder. As a result, we show the time-resolved pressure distribution as well as its phase-locked ensemble average. The variations of the pressure field are related to the integral pressure difference across the turbomachine and to its rotational speed. Due to the high temporal resolution of the measurement data, an exact spatial localization of crucial flow phenomena is achieved. Low integral pressure differences show a nearly linear increase of the pressure in circumferential direction, while greater integral pressure differences evolve exponentially over the azimuth. The results confirm the circulatory flow theory. Different rotational speeds elicit a comparable behavior. The stripper is a dominant source for pressure fluctuations. Its individual geometric discontinuities are correlated to the flow field. Our results provide a deeper understanding of the flow phenomena in side channel blowers and the theory of pressure generation. Although the measurements were performed for only one type of side channel blower with a double-flow configuration and open blades, the energy transfer mechanism is the same for other modifications like single flow or closed blade versions.

Investigation into Boundary Layer Transition on the MEXICO Blade

Boundary layer transition studies have been carried out using the unsteady surface pressure data from the EU project MEXICO and the later New-MEXICO experiment at the DNW wind tunnel, which have been subject of investigation in IEA Wind Task 29 Mexnext. The experiments were conducted on a specially designed 4.5 m diameter wind turbine with and without zig-zag roughness strips applied to the outboard part of the blades. Standstill tests with the MEXICO blades were also conducted at the TU Delft LTT wind tunnel including oil flow and stethoscope tests.Transition is determined by detecting and observing the growth of Tollmien-Schlichting (TS) packets through a Power Spectral Density plot in the frequency domain. The expected frequency range of these waves is determined using a database developed by Van Ingen. The transition results are compared to those determined by RFOIL, a program for the analysis of aerofoils which uses the eN method for transition detection. Generally a good agreement was observed between measurements and predictions. In addition to that the effectiveness of the roughness strips was studied. The result is a unique database in controlled conditions to be used for validation and improvement of transition modelling of wind turbine blades.

Unsteady free surface flow above a moving circular cylinder

A problem on non-stationary free surface flow of infinitely deep ideal fluid generated by the motion of a submerged body is considered. The water wave problem is reduced to the integral–differential system of equations for the functions defining free surface shape, normal, and tangential velocity components on the free boundary. Small-time asymptotic solution is constructed for the case of circular cylinder that moves with constant acceleration from rest. The role of non-linearity is clarified by analysis of this approximate solution which describes the formation of added mass layers, splash jets, and finite amplitude surface waves.

An aeroacoustic study of a leading-edge slat: Beamforming and far field estimation using near field quantities

The leading edge slat of a high-lift system is a large contributor to the overall radiated acoustic field from an aircraft during the approach phase of the flight path. This is due to the unsteady flow field generated in the slat-cove and near the leading edge of the main element. In an effort to understand the noise-source mechanisms, a suite of experimental measurements has been performed on a two-dimensional multi-element MD-30P30N airfoil, including PIV, steady and unsteady surface pressure, and phased microphone array measurements. Acoustic trends are given with angle of attack and flow speed which corresponds to a stowed chord Reynolds number range of 1.2 × 106 ≤ Rec ≤ 1.71 × 106. Spatially integrated beamforming is used to isolate the slat noise, while Kevlar sidewalls are utilized to minimize environmental influence. In addition, temporally-resolved estimates of a low-dimensional representation of the velocity vector fields are obtained through the use of proper orthogonal decomposition and spectral linear stochastic estimation and show good agreement with independent phase-locked PIV measurements. A time-resolved estimate of the pressure field is then computed via solution of the pressure Poisson equation. From this, Curle's acoustic analogy scaled for the frequency-dependent spanwise coherence length projects the time-resolved pressure forces in the slat region to the acoustic far field for comparison to the array measurements. The favorable results establish the connection between the dominant unsteady flow structures most responsible for the radiated noise and provide insight for future noise reduction efforts.

Adaptation of kinematic synergy and postural control to mechanical ankle constraint on an unsteady stance surface

Joint constraint interferes with the coordinative structure in joint movements used to optimize postural stability. This study aimed to investigate changes in postural synergy when the ankle joints were bilaterally braced during a stabilometer stance. Twenty-four young adults stood on a stabilometer plate while wearing a pair of ankle–foot orthoses, which were either unlocked or locked to restrict ankle motion (the ankle constraint (AC) and non-constraint (NC) conditions). Although ankle constraint did not significantly affect the dynamics of the stabilometer movements, the size and regularity of the first principal component (PC1), which explained more than 80% of the variance of joint movements in the lower limb, were increased. In addition, PC1 exhibited higher communalities with angular movements of the knee and hip joints in the AC condition than in the NC condition. Those subjects who exhibited a constraint-induced increase in postural sway (the I group) showed greater increases in the size and regularity of PC1 than did those who exhibited reduced postural sway during ankle constraint (the D group). Constraint-induced changes in postural synergy were group-dependent. Only the I group exhibited an increase of communality of PC1 with the hip angular movement following bilateral ankle constraint. In summary, bilateral ankle constraint altered the coordination solution, with increasing reliance on compensatory knee movement to maintain a balanced posture on the stabilometer. However, accessory hip movement due to ankle constraint was not economical and was disadvantageous to stance stability.

بررسی تجربی اثر آشفتگی جریان ورودی بر میدان فشار ناپایا و نویز جریان پیرامون استوانه دایرهای

Surface pressure fluctuations on a circular cylinder model are the main sources of far- field noise emitted and have complex physical behavior. So far, limited studies have been devoted to identifying the unsteady surface pressure behavior, especially for turbulence incident flow. In the present study to measure the surface pressure fluctuations under both smooth and turbulence incoming flows, a circular cylinder with an outer diameter of 22mm has been used. In order to change the turbulence characteristics of free stream incoming flow, different biplane grids with square meshes were designed. Power spectral density, coherence, autocorrelation and cross-correlation, spanwise length scale, and convection velocity were different interesting parameters which were calculated using measured unsteady pressures in order to better clarify the flow structure and flow noise around the model. The results revealed that the energy level of both tonal and broadband noises is increased when incident flow changes from smooth to turbulent. Moreover, the tonal noise frequency in the turbulence flow ( f ' = 88 Hz ) shifts to low frequencies compared to that of smooth flow ( f = 98 Hz ). Furthermore, results showed that the physical size of eddies and their convection velocity in turbulence flow are greater than corresponding values for smooth flow.

Unsteady Pressure Measurement on Oscillating Blade in Transonic Flow Using Fast-Response Pressure-Sensitive Paint

A fast-response pressure-sensitive paint (PSP) technique was applied to the measurement of unsteady surface pressure of an oscillating cascade blade in a transonic flow. A linear cascade was used, and its central blade was oscillated in a translational manner. The unsteady pressure distributions of the oscillating blade and two stationary neighbors were measured using the fast-response PSP technique, and the unsteady aerodynamic force on the blade was obtained by integrating the data obtained on the pressures. The measurements made with the PSP technique were compared with those obtained by conventional methods for the purpose of validation. From the results, the PSP technique was revealed to be capable of measuring the unsteady surface pressure, which is used for flutter analysis in transonic conditions.

Standing balance on unsteady surfaces in children on the autism spectrum: The effects of IQ

BackgroundPostural stability difficulties are commonly reported in people on the autism spectrum. However, it is unclear whether unsteady surfaces may exacerbate postural stability difficulties in children and adolescents with autism spectrum disorder (ASD). Understanding balance on unsteady surfaces is important because uneven surfaces are commonly encountered in daily life. MethodsTwenty-one youth on the autism spectrum and 16 youth with typical development (ages 6–16 years, IQ ≥ 79) stood on both a fixed and unsteady (tiltable) platform, and center of pressure was measured. ResultsThe group with ASD exhibited differentially more postural sway on the unsteady surface compared to the group with typical development. However, there was substantial variability within the ASD group. Follow-up analyses suggested that much of the variability in postural sway in the ASD group was accounted for by IQ. ConclusionsClinically, these findings suggest that not all individuals with ASD struggle more with postural stability on unsteady surfaces. Instead children and adolescents with ASD and below-average IQ may have particular difficulty on unsteady surfaces and may require accommodations. Further, these findings lay the groundwork for future research to investigate the underlying mechanisms of poorer balance across the autism spectrum.

Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face

This note examines the two-dimensional unsteady isothermal free surface flow of an incompressible fluid in a non-deformable, homogeneous, isotropic, and saturated porous medium (with zero recharge and neglecting capillary effects). Coupling a Boussinesq-type model for nonlinear water waves with Darcy's law, the two-dimensional flow problem is solved using one-dimensional model equations including vertical effects and seepage face. In order to take into account the seepage face development, the system equations (given by the continuity and momentum equations) are completed by an integral relation (deduced from the Cauchy theorem). After testing the model against data sets available in the literature, some numerical simulations, concerning the unsteady flow through a rectangular dam (with an impermeable horizontal bottom), are presented and discussed.