Wave Flow Research Articles

The Effect of Contraction–Expansion Nozzle on High-Temperature Shock Tube Flow

To achieve higher enthalpy and pressure, the technique of variable cross-section drive is effectively combined with the heating of light gas to enhance the intensity of the incident shock wave. A study was conducted to predict the impact of variable cross-sections on the performance of high-temperature shock tube flow using a shock tube with a 2.6:1 diameter ratio between the driver and driven sections. The driver section was filled with a helium–argon gas mixture (mass ratio of 1:9), while the driven section contained dry air. Under total pressure conditions of 14.5 MPa and total temperature of 3404 K, as well as total pressure of 45 MPa and total temperature of 4845 K in the driver section, corresponding to driven section pressures of 10 kPa and 80 kPa, the results of chemical non-equilibrium numerical simulations were compared to experimental measurements of the incident shock Mach number and total pressure. The results indicated the following: First, after adding the contraction–expansion nozzle, the incident shock accelerated through the contraction section and reflected within the contraction section. Strong oscillations occurred during the flow, with increasing intensity as the throat size decreased. Second, without the nozzle, the shock velocity increased and then decreased. However, with the nozzle, the Mach number was highest near the nozzle exit and gradually decreased thereafter. Third, the presence of the nozzle led to the formation of a distinct fan-shaped wavefront, accompanied by significant variations in flow variables such as pressure, temperature, and Mach number in the region. This phenomenon was attributed to the interaction between the shock wave and the nozzle geometry, which altered the flow dynamics. Finally, as the throat size decreased, the intensity of the incident shock also decreased. After reflecting at the end of the shock tube, the total pressure in the driven section also decreased. The numerical simulations employed a multi-component, multi-temperature chemical non-equilibrium model, validated against experimental data, to accurately capture the complex flow behavior and wave interactions within the shock tube.

East Asian Gene flow bridged by northern coastal populations over past 6000 years

Coastal areas of northern East Asia in the ShanDong region, which show complex cultural transitions in the last 10,000 years, have helped to facilitate population interactions between more inland regions of mainland East Asia and islands such as those in the Japanese archipelago. To examine how ShanDong populations changed over time and interacted with island and inland East Asian populations, we sequenced 85 individuals from 11 ancient sites in the ShanDong region dating to ~6000-1500 BP. We found that ancestry related to ShanDong populations likely explains the mainland East Asian ancestry observed in post-Yayoi populations from the Japanese archipelago, particularly recent populations who lived in the Ryukyu Islands after ~2800 BP. In the ShanDong region, we observed gene flow from populations to the north and south of this region by at least ~7700 BP, and two waves of gene flow associated with the inland Yellow River populations into the ShanDong region during the DaWenKou cultural period (6000-4600 BP) and in the early dynastic period (3500-1500 BP). Reconstructing the genetic history of the Neolithic, Bronze, and Iron Age populations of coastal northern East Asia shows gene flow on both a north-south and an east-west (inland-coastal-island) scale.

Point-of-care ultrasound of the common carotid arteries for detection of large vessel occlusion stroke: Results of the POCUS-LVO study.

Distal arterial occlusions can cause measurable changes in the flow wave profile in proximal segments of the feeding artery. Our objective was to study the diagnostic accuracy of point-of-care ultrasound (POCUS) of the common carotid arteries (CCA) for detection of anterior circulation large vessel occlusion (ac-LVO) in patients with suspected stroke. We conducted a prospective, single-center, observational study of adult patients with suspected stroke admitted in the emergency department. Flow wave profiles of both CCAs were generated by non-specialists using POCUS as soon as possible after admission. ac-LVO was defined as an internal carotid artery or M1 occlusion in CT- or MR-angiography. The diagnostic performances for detection of ac-LVO using flow wave parameters were calculated. Among 283 patients recruited during a 10-month period, 257 patients (91%) had CCA ultrasound images of sufficient quality and were included for analysis. The mean age was 75 years (IQR 62-83), 131 were female (51.0%), median baseline NIHSS was 2 (IQR 0-5). The most frequent final diagnosis was ischemic stroke (49.4%), ac-LVO was present in 30 patients (11.9%). The median duration of POCUS was 3 min (IQR 2-5). Among all flow wave parameters, the highest diagnostic accuracy for ac-LVO detection was found for end-diastolic velocity difference between sides (AUC = 0.90, 95%CI = 0.85-0.93), with a specificity of 83% (95%CI = 78-88%) at a predefined sensitivity threshold of 80%. POCUS of the CCA in patients with suspected stroke can predict the presence of ac-LVO. These results need to be replicated in a prehospital setting.

Direct detection method for cusp thickness in wavy thin-film flow using an optical waveguide film

The wave crest (cusp) of the disturbance wave in thin liquid film flow is an important factor contributing to heat/mass transfer, e.g., fuel rods in boiling water reactors, stator/rotor blades in steam turbines, and cleaning/drying wafer processes in semiconductor manufacturing. We developed a new technique for directly detecting the thickness of wave cusps using array-based sensing with an optical waveguide film (OWF). This technique, based on geometrical optics assumptions, simultaneously obtains information on liquid films’ thickness and their interfacial shape, i.e., whether or not the local interface is convex upward. We first performed a pseudo-film flow measurement using a metal specimen to confirm the basic principle. According to the results, a meaningful signal indicating the wave-cusp passage, along with a thickness signal, was detected simultaneously. The OWF signal processing for cusp thickness detection was newly established based on this fact. We then applied this technique to a wavy liquid film flow formed on a flat plate in the entry region. A series of experiments were performed over a wide range of air speeds (jG = 20–70 m/s). As a result, the cusp thicknesses of relatively large waves on the wavy interface were successfully extracted from the OWF output signal. Further, the major thickness variables (i.e., base film thickness, median film thickness, and cusp thickness) were compared with those of conventional thickness estimation methods, which showed reasonable agreement. This paper provides a framework for wavy thin-film flow measurements via OWF that is specialized for directly detecting local thickness profiles.Graphical abstract

Sex-specific ultrasound imaging biomarkers of neurodegeneration in a mouse model.

Early detection of neurodegeneration is essential for optimizing interventions. The highly reproducible progression of neurodegeneration in the decrepit (dcr) mouse allows investigation of early biomarkers and mechanisms of brain injury. Using high-frequency ultrasound, the common carotid arteries of female and male dcr and control mice were imaged longitudinally at time points bracketing the disease progression (50, 75, and 125 days of age) (n = 6 mice/group/sex). Over the disease time course, the female dcr mice demonstrated increased carotid artery blood flow and pulse wave velocity while the male dcr mice had a decrease in heart rate and no change in carotid artery ultrasound parameters. Early imaging biomarkers were sex-specific, with decreased carotid artery blood flow in female dcr mice and increased carotid artery diameter and decreased pulse wave velocity in males. Carotid artery and wave reflection ultrasound is a promising screening tool for early detection of neurodegeneration.

Marginal microleakage at the tooth-material interface with different types of sealants. Application of the ultrasonic sealant technique

Introduction: Pit and fissure sealants are the most effective method of prevention, since they act as a physical barrier preventing the entry of microorganisms into these hard-to-reach areas during brushing and self-cleaning carried out by saliva. The effectiveness of sealants depends mainly on retention on the tooth surface and microleakage is the factor most linked to their failure. Microleakage is defined as the passage of molecules, bacteria or fluids between the cavity walls and the restorative material and is the result of an inadequate application protocol, tooth preparation techniques, the chemical composition of the material and shrinkage due to polymerization. The most commonly used sealant materials are resin-based, since in theory they present less microleakage than ionomers. The most recent ones have incorporated nanometric inorganic filler particles (0.005 - 0.01 microns) in their composition in order to improve their physical-mechanical properties and reduce polymerization shrinkage.Objectives: To evaluate the degree of microleakage of the tooth-material interface in premolars and third molars using the ultrasound technique with three different materials. Sealer, Vitreous Ionomer and Fluid Resin.Materials and methods: An in vitro, comparative and transversal study was carried out on 30 dental pieces (premolars and third molars), randomly divided into 3 study groups of 10 pieces each, sealed with different types of materials. The groups were divided into A (sealant), B (vitreous ionomer) and C (flow composite) in order to evaluate the degree of microleakage of the different materials used. For the cavity preparations, the enamel surface was etched with 37% phosphoric acid propelled with a micromotor and a dental pneumatic cavitator tip with a small rubber band and a k 10 file for 15 seconds. Subsequently, they were sealed with different types of materials to evaluate the degree of microleakage, analyzed by direct observation.Group A: Sealer (Conseal f and Conseal)Group B: Vitreous Ionomer (Riva Light Cure)Group C: Flow Composite (Flow Wave)The materials used during the investigation are SDI brand.Results: The material that showed the highest degree of microfiltration was the one used in group B (vitreous ionomer) with 2 out of 10 pieces filtered. Group A (sealant) and group C (flow composite) obtained better results: 1 out of 10 pieces filtered in both cases. Direct observation showed that all three groups presented marginal microleakage, to a lesser or greater degree. It could not be determined that the permanence of the materials depended on the morphology of the dental pieces, since it was mixed, that is to say, they were detached in premolars as well as in third molars.Conclusion: In comparison with groups A and C, the marginal microleakage observed was significantly higher in group B (vitreous ionomer). However, marginal microleakage was present in all three groups, to a lesser or greater degree. Therefore, we estimate that there will always be a percentage probability of this occurring

Influence mechanisms of the submerged sand engine on coastal protection as the extension of beach nourishment

As an innovative practice of beach nourishment, a submerged sand engine was constructed for coastal restoration in Jinsha Bay, China. The submerged sand engine is composed of a nourished beach, submerged sand groins and submerged sandbars. To investigate hydrodynamic and geomorphological responses to the system, a multi-coupled model is established for numerical simulations. The numerical results indicate that the submerged sand engine serves structural functions in wave attenuation and flow regulation. The system reduces significant wave heights of storm waves by 79% at most, and mitigates alongshore current speed in embayed areas by a 10% decrease in the embayed area. In a typical typhoon event, the system reduces erosion spots with erosion depth over 0.09 m by 100% along embayed beaches. Moreover, submerged sandbars serve similar contributions on cross-shore and alongshore sediment supplements, while alongshore contributions of submerged sand groins are over 75%. Hence the cross-shore profiles generally present shoreward movement of submerged sandbars and head scour of submerged sand groins. Furthermore, the submerged sand engine saves 90% expenses of the Sand Engine in terms of the sand volume cost per unit length of the target shore. The submerged sand engine is of flexible and economic advantages in coastal protection.

Positivity-preserving pseudo arc-length method for solving extreme explosive shock wave flow field problems in hyperbolic conservation law equations

Positivity-preserving pseudo arc-length method for solving extreme explosive shock wave flow field problems in hyperbolic conservation law equations

Tuning photocatalytic activity of Ce-doped BaTiO3 nanoparticles by encountering acoustic shock wave flow exposure

Tuning photocatalytic activity of Ce-doped BaTiO3 nanoparticles by encountering acoustic shock wave flow exposure

Abnormal Venous Flow in Pregnant Women with Mild Right Ventricular Dysfunction in Repaired Tetralogy of Fallot: A Clinical Model for Organ Dysfunction in Preeclampsia.

Background: Pregnant women with congenital heart disease carry a high risk of complications, especially when cardiac function is suboptimal. Increasing evidence suggests that impaired right ventricular (RV) function has a negative effect on placental function, possibly through venous congestion. We report a case series of hepatic and renal venous flow patterns in pregnant women with right ventricular dysfunction after repaired Tetralogy of Fallot (ToF), relative to those observed in normal pregnancy and preeclampsia. Methods: At 20-24 weeks pregnancy, RV function was measured by echocardiography and by cardiovascular magnetic resonance in women with repaired ToF. Combined Doppler-ECG of the hepatic and renal interlobular veins were performed in three women with asymptomatic right ventricular dysfunction. Venous impedance index and pulse transit time were measured and classified as abnormal at >75th and <25th reference percentile, respectively. Results: All three women showed dilated RV and mildly impaired RV function. Both hepatic and intrarenal Doppler flow waves were abnormal and very much resembled the patterns seen in preeclampsia. One of the three women had complications including ventricular tachycardia, intrauterine growth restriction, antenatal bleeding, emergency cesarean section and acute heart failure 2 days postpartum. Conclusions: Pregnant women with mild right ventricular dysfunction after repaired ToF show abnormal venous Doppler flow waves in the liver and kidneys, similar to those observed in preeclampsia. These findings are in line with reported observations on the association between impaired RV function, abnormal return of venous blood, venous congestion and organ dysfunction. The parallel with venous Doppler flow observations in preeclampsia suggest that the venous compartment might play an important role in the etiology of preeclampsia-induced organ dysfunction. Whether this phenomenon directly affects the uteroplacental circulation is to be assessed in future research.

Smooth Muscle Mechanosensitivity Generates and Maintains Pressure Gradients Across the Intestine.

The gut, the ureter, or the Fallopian tube all transport biological fluids by generating trains of propagating smooth muscle constrictions collectively known as peristalsis. These tubes connect body compartments at different pressures. We extend here Poiseuille's experiments on liquid flow in inert tubes to an active, mechanosensitive tube: the intestine. We use as a miniature myogenic peristaltic pump model, the fetal chicken gut, and measured the flow and contractile wave propagation as a function of the initially applied pressures and pressure gradients. We dissect the molecular pathways of smooth muscle mechanosensitivity by measuring the force generated by gut rings in different pharmacological conditions. We demonstrate that smooth muscle contractions in response to stretch or pressure is mediated by L-type Ca2+ channels and IP3 receptors. We show that this positive-feedback mechanosensitive behavior can spontaneously generate pressure gradients across gut segments initially subject to equal pressure; this same mechanism tends to stabilize initially applied pressure gradients; it can act jointly or compete with the pressure gradient induced by directional peristaltic waves. We demonstrate that high pressure differentials can reverse the physiological propagation direction of contractile waves imparted by interstitial cell of Cajal pacemaker activity. We find that flow rate increases with tube length, but that the maximum pressure differential generated depends solely on smooth muscle contractile force and on the initial resting pressure applied inside the organ. We provide fundamental mechanical and hydrodynamic insight into the myogenic mechanisms of transport in the gastrointestinal tract. We scale up our results to other human peristaltic organs and discuss their implications for pathophysiology of intestinal obstruction, vesicoureteral reflux and endometriosis.

Sonar Image Augmentation for Underwater Bridge Piers Via Cyclic GANS

Underwater bridge pier structures are frequently exposed to stressors like water flow scour, ship collisions, and wave forces, leading to defects such as cracks, exposed rebar, and holes. Timely detection of these defects is critical to preventing catastrophic failures. Recent advancements in sonar imaging have improved underwater inspections, but the resulting images often suffer from noise and lower quality compared to optical images, complicating defect detection. Additionally, the scarcity of publicly available sonar datasets presents challenges in training accurate detection models.This paper addresses these challenges by proposing a novel data augmentation method using CycleGAN, which converts optical images of underwater piers into sonar images. This approach leverages the power of Generative Adversarial Networks (GANs) for effective sample augmentation, overcoming the limitations of traditional data augmentation techniques. This method significantly improves the robustness and accuracy of deep learning models for detecting defects in underwater bridge piers, providing a novel solution for augmenting limited sonar image datasets and advancing automated defect detection.

Towards the modern theory of parturition

A system analysis of the results of modern scientific research has shown the most complex organization of the parturition, during which the unfolding and activation of the genetic birth program initially embedded in the mother and child occurs. 2–3 weeks before birth, desympatization and formation of an acupuncture network begin in the uterus. Along these acupuncture channels the wave flows of biologically active substances with both contractile and inhibitory properties move. These substances are delivered to the uterus by the bloodstream and blood cells. Some of them also have psychotropic properties, thereby enhancing the effect on the brain and causing a state of altered consciousness in the woman and her child. As labor approaches, the prenate revealed activation of the locus on chromosome 2, which allowed the researchers to assert that the prenate is initiator of the birth beginning. The totality of the data presented in the article served as a prerequisite for the formulation of a scientifically substantiated Theory of the parturition, according to which childbirth is a genetic-psychosomatic phenomenon, free of pain.

Protective performance of helmets based on the shock wave from rocket launcher

Abstract Traumatic brain injury(TBI) is one of the most common cause of major casualties in the battlefield. Soldiers may be endangering their brains when they operate certain shoulder-fired weapons or large-caliber artillery for long periods of time. It is necessary to pay attention to protection and prevention of brain injury in daily training. In this paper, the characteristic of shock wave propagation on the head with/without helmet protection when soldiers operated shoulder-fired rocket-propelled grenade launcher. The head-helmet surrogate was constructed in combination with pressure sensors. The pressure sensors were installed on the forehead, temples, top, and back of the head surrogate respectively and the sensors kept a plane with the surface of the head surrogate. The pressure variation of the shock wave flow field under different working conditions was analyzed in three types such as with/without combat helmet and integrated helmet. The results show that the peak overpressure of the shock wave from the back of the head can reach 80∼105kPa without combat helmet. The combat helmet can effectively reduce the shock wave overpressure. The integrated helmet has the best protection and can reduce the overpressure on the forehead, temple, top, and back of the head by 73%, 58%, 80%, and 83% respectively. Meanwhile the shock wave is prone to reflection and diffraction as it transmits inside the helmet. This significantly increase the overpressure of the shock wave on the forehead and the duration of the shock wave in the helmet, which has a negative effect on the protection provided by the shock wave.

Research and application of flying-wing aircraft bow wave modeling method based on convolutional neural network

With the development of aviation technology, aircraft voyage improvement has become an urgent problem to be addressed. Aerial refueling technology has been widely used as an important method for improving endurance. In probe-and-drogue refueling (PDR), the tanker's hose-drogue system is offset by the bow wave generated by the receiver's head. By probe-and-drogue refueling, we mean soft aerial refueling, which is a method that transfers fuel by connecting the drogue of tanker with the probe near the oil receiver's head to transfer fuel. The ability to accurately calculate flow field changes caused by the bow wave and the hose-drogue system offset position have become key factors for successful docking during PDR. However, existing bow wave models based on the potential flow method cannot accurately calculate the bow wave flow field of a flying-wing aircraft. Therefore, this paper proposes a variety of bow wave models based on a convolutional neural network (CNN), which aims to accurately solve the bow wave flow field of a flying-wing unmanned aerial vehicle in the process of autonomous aerial refueling. In addition, a method to calculate the hose-drogue system position based on the dynamics of a multi-body system is proposed, which calculates the final position of the hose-drogue system after docking from the coupled flow field of the tanker and receiver. Finally, by applying the bow wave model to the position calculation of the hose-drogue system, it was verified that the CNN model can be accurately applied to research related to aerial refueling.

Deep phenotyping the right ventricle to establish translational MRI biomarkers for characterization of adaptive and maladaptive states in pulmonary hypertension

Deep phenotyping the right ventricle (RV) is essential for understanding the mechanisms of adaptive and maladaptive RV responses to pulmonary hypertension (PH). In this study, feature selection coupled with machine learning classification/ranking of specific cardiac magnetic resonance imaging (MRI) features from cine-MRI, flow-sensitized, and extracellular-volume techniques were used to assess RV remodelling in monocrotaline (MCT) and Sugen hypoxia (SuHx) PH rats. Early physiological changes associated with RV adaptation were detected along with prediction of RV maladaptive outcomes. Key adaptation features included haemodynamic alterations of pulmonary blood flow ejection and wave reflection, mild RV dilatation, progressive RV hypertrophy with subtle extracellular volume growth of RV wall. A dominant component of maladaptation was the extracellular matrix increase at RV insertion points and septum, observations compatible with histopathologic and RNA-sequencing results. The upregulation of mammalian target of rapamycin (mTOR) paralleled by AMP-activated protein kinase (AMPK) deactivation was seen at 4-week MCT and 8-week SuHx, along with reduced sarcoplasmic/endoplasmic reticulum Ca2+ATPase (SERCA2) expression, strongly associated with the RV systolic malfunction seen at this stage in vivo. The here established MRI features can serve as potential imaging biomarkers to evaluate PH treatment efficacy in preclinical studies and build up translational markers for the PH clinic.

A novel flow channel inspired by classical mathematical function: Enhancing output performance and low-grade heat recovery efficiency of thermal regeneration ammonia-based flow battery

Thermal Regeneration Ammonia-based Flow Battery (TRAFB) faces significant challenges in enhancing its output performance and low-grade waste heat recovery efficiency due to limitations in mass transfer and an incomplete understanding of the mass transfer mechanism. To address these issues, this study innovatively introduces a novel sinusoidal wave flow channel (SWFC), which is inspired by the classic mathematical function. The impact of the SWFC on TRAFB performance is thoroughly discussed and compared in detail with the conventional straight flow channel (SFC). Most importantly, this work unveils for the first time the distribution mechanism of the three mass transfer modes in TRAFB, which are convection, diffusion, and electrophoretic mass transfer. The main findings reveal that the mass transfer performance of TRAFB is primarily governed by convection and diffusion, with electrophoretic mass transfer playing a minimal role, and there is a threshold of current density beyond which the dominant mass transfer mechanism transitions from convection to diffusion. The unique design of the SWFC, which induces the Venturi effect, significantly enhances the overall mass transfer performance of the TRAFB due to the distinct local acceleration, achieving a remarkable enhancement in flux of up to 118.48 times. By optimizing the amplitude and period, the battery's output performance can be further enhanced, with the maximum peak power achieved by the SWFC with an amplitude of 0.8 mm and a period of 0.5π, which is 95.63 % over the SFC. Notably, the SWFC also excels in improving the thermoelectric conversion efficiency, particularly at high current densities, achieving an enhancement of up to 9.81 times.

Localization of internal gravity waves in stratified channel flow

Linear and nonlinear contributions to the localization and dynamics of internal gravity waves in a stably stratified turbulent channel flow are investigated using data from direct numerical simulations (DNS). The classification into linear and nonlinear mechanisms is based on the resolvent formulation of the Navier–Stokes equations, which interprets velocity and temperature fluctuations (flow response) as the result of a linear operator (resolvent) acting on the nonlinear advection terms (forcing). Spatial and spatio-temporal power spectral densities computed from DNS data demonstrate that the stratified flow response is localized in spectral space and in the channel core, while the nonlinear forcing is broadband and spans up to the entire channel height. The localization of the velocity and temperature fluctuations in wavenumber and frequency is captured by the leading singular value of the resolvent operator. The wall-normal localization on the other hand results from a combination of linear dynamics and nonlinear forcing, and the latter is further examined using the cross-spectral density (CSD) tensor. Wall-normal subsets of the forcing CSD lead to flow responses that reveal a three-layer structure. The middle one hosts the critical layer of the gravity wave, and is termed the outer layer since it is flanked by an inner layer at the wall and the core region at the channel centre. Forcing within this outer layer generates the majority of the flow response in the channel core. Furthermore, a decomposition of the forcing CSD into velocity and temperature demonstrates that each imprints distinct phase relations on their associated responses, which lead to destructive interference and localization of the gravity waves in the channel core.

Lie group of similarity analysis of shock waves in viscous flow under magnetic field

This paper investigates the propagation of planar shock waves in an ideal gas under the viscous stress and magnetic field using the Lie group of similarity analysis. The ambient density and magnetic field both vary with shock radius as power and exponential law. Newton’s law of viscosity has been used and shock jump conditions have been derived for viscous flow by introducing shock Reynolds number Res. The system of governing partial differential equations is reduced into a system of ordinary differential equations using the Lie group of invariance method and numerical solutions have been obtained for power and exponential law both. The effects of shock Reynolds number Res=10 (highly viscous flow), 50, 200, 1000 (slightly viscous flow), and Res→∞ (non-viscous flow), Alfven-Mach number, and ratio of specific heats have been discussed on the shock strength, piston position and flow variables behind the shock front. The magnetic field enhances the effect of viscous stress in exponential law but reduces in power law. All flow variables except viscous stress increase for power law and decrease for exponential law in viscous flow in comparison to non-viscous flow. Comparison of results of highly non-viscous flow with the corresponding results of inviscid flow establishes the validity of the model presented in this work. The results of this paper show the significant effect of viscosity on shock propagation contrary to the negligible effect in earlier studies.

Roll waves on laminar sheet flow of Newtonian fluid with negligible surface tension

We conduct direct numerical simulations (DNS) to study the temporal and spatial developments of the roll waves on a laminar sheet flow of Newtonian fluid. The DNS unveil the physics of the wavefront and show the limitation of the widely used shallow-layer approximations. The most prominent wave, the front runner, is determined by the DNS for the first time in studying the spatial development of the laminar sheet flow with negligible surface tension. Depending on the Froude and Reynolds numbers, the front runner can be a multi-peaked undular bore or a single-peaked non-breaking or breaking wave. The simulation has uncovered an extended region behind the wavefront, where the bed-friction stress is much higher than the corresponding friction in the undisturbed uniform flow. It also produces an uplift velocity needed in the description of wave breaking. For comparison, we also examine the nonlinear development of the instability using two-equation and four-equation shallow-layer models. The two-equation shallow-layer model has produced the bulk of the wave profile but is deficient because it fails to predict the uplift velocity and the substantial increase in bed friction in the frontal region. The four-equation shallow-layer model correctly predicts the bed friction but cannot produce the breaking wave. The simulations also determine the celerity and amplitude of the front runner to follow a linear relationship, qualitatively similar to the roll waves in a turbulent flow.