Free Flow Region Research Articles

A method for modeling the lateral distributions of depth-averaged velocities behind an emergent vegetation patch

Aquatic vegetation provides ecological, hydrological, and aesthetic functions for rivers, and measuring velocity in vegetated channels is essential for river management. The paper presents a method for modeling the lateral distributions of depth-averaged velocities behind an emergent vegetation patch. Based on SKM, this approach divides the channel behind an emergent vegetation patch into pseudo-vegetation and free-flow regions, offering analytical solutions for the depth-averaged velocities in these two regions. The model incorporates several critical parameters, including the Darcy–Weisbach coefficient, lateral dimensionless eddy viscosity, drag force coefficient, and secondary flow coefficients. These coefficients are associated with bed friction, vegetation-induced resistance, secondary flow effects, and lateral momentum exchange, respectively, affecting the depth-averaged velocities. A comparison with published experimental data validates that the proposed model can predict the lateral distributions of depth-averaged velocities behind an emergent vegetation patch. A steady wake section exists behind an emergent vegetation patch for low-flow blockage. In the steady wake section, the secondary flow coefficients in pseudo-vegetation and free-flow regions remain relatively constant. Upon exiting the stable wake section, the secondary flow coefficient in the pseudo-vegetation region decreases with increasing distance from an emergent vegetation patch, and it in the free-flow region increases with increasing distance from an emergent vegetation patch. A sensitivity analysis of drag coefficient and secondary flow coefficients suggests that secondary flow coefficient in the free-flow region has a more significant effect on the lateral distributions of depth-averaged velocities compared to drag coefficient and secondary flow coefficient in the pseudo-vegetation region.

Turbulence and sediment deposition in a channel with floating vegetation

Floating canopies of vegetation alter the vertical profile of flow velocity within and below canopies, which in turn impacts sediment deposition below canopies. This study explored the impacts of channel-average velocity (U0), vegetation density (a), and relative flow depth (hg/H, where hg is the height of the free flow region below the floating canopy and H is the entire flow depth) on the longitudinal distributions of near-bed turbulent kinetic energy (TKE) and sediment deposition in a channel with a floating vegetation canopy. Below a floating canopy, sediment deposition was correlated with the near-bed TKE. For the low-density and low-velocity cases, sediment deposition was spatially uniform across the entire channel because the near-bed TKE was lower than the critical value for sediment resuspension. As the vegetation density and channel-average velocity increased, the near-bed TKE increased and surpassed the critical TKE, which caused resuspension and reduced deposition below the canopy relative to the upstream reference. For the same vegetation density and channel-average velocity, a lower relative flow depth (hg/H) produced a higher below-canopy depth-averaged velocity and near-bed TKE, resulting in less deposition below the canopy. A model was proposed for predicting the longitudinal evolution of near-bed TKE based on the predicted flow velocity below the canopy throughout the channel. The longitudinal evolution of deposition below a floating canopy was predicted by combining the predicted near-bed TKE with the deposition probability. The predicted near-bed TKE and deposition matched laboratory measurements. Finally, the proposed model can be used to predict the longitudinal profile of deposition below a real Eichhornia crassipes canopy.

Pressure-robust finite element scheme for the time-dependent fully coupled Stokes–Darcy-transport problem

We present a pressure-robust mixed finite element scheme for the time-dependent fully coupled Stokes–Darcy-transport problem. The P1c⊕RT0 discretization and Raviart–Thomas mixed elements are used to discretize the velocity in the free flow region and the porous medium region, respectively. We approximate the pressure with piecewise constant finite elements and the concentration with piecewise linear finite elements. We perform a divergence-free reconstruction of the velocity in the Stokes domain. Our scheme preserves local mass conservation in the sense of projection and has pressure-robust property. The stability and error analysis of the method are obtained. The errors of velocity and concentration are independent of pressure, and our method works well even with small viscosity. The theoretical results are validated with numerical examples.

Controlling electrothermal behavior of Metal–Carbon hybrid wire in free molecular flow region

We investigated the heat loss of the metal wire to the gases in the free molecular region, whose pressure is below 10−3 Pa. The heat transfer characteristics at the interface between the metal surface and the gases are only accessible when the conductional heat loss along the wire is comparable to or less than 100 nW/K. Unfortunately, such an infinitesimal heat flow is not controllable from typical metal wire. For this reason, we have synthesized a new material, metal-carbon hybrid wire (MCHW). In the free molecular flow region, the resistance of MCHW is inversely proportional to the pressure (P), R∼1/P, regardless of gas species, which contradicts the gas-dependent heat loss theory. At a pressure <1 × 10−4 Pa, we observe a deviation from reciprocal linearity attributed to a growing radiational heat loss. Our results realize the thermal conductive sensing of the pressure below 10−5 Pa, which has been unprecedented.

Porous structures impact on particle dynamics of non-Brownian and noncolloidal suspensions

This study aims to provide valuable insights into the impact of porous structures on particle dynamics in non-Brownian, non-colloidal suspension flows at very low Reynolds numbers. Two experimental approaches, Particle Image Velocimetry (PIV) with refractive index matching and Optical Flow Tracking Velocimetry (OFTV) were employed to analyze very dilute suspensions over various porous media models. The study considered three different porous structures with permeabilities ranging from 0.7 to 0.9 and three different thicknesses ranging from 0.2 cm to 0.5 cm, while the suspension bulk volume fraction was maintained at 3 %. In the PIV analysis, we observed that decreasing the porous permeability resulted in the maximum velocity location within the free flow region moving closer towards the interface between the flow and the porous media. We further quantified the effect of the porous structure on the suspension by characterizing interface properties, such as dimensionless slip velocity, shear rate, and slip length. These interface properties were found to be influenced by both the thickness and permeability of the porous media. Next, we analyzed particle migration due to the presence of porous structures using OFTV for very dilute suspensions of 1 %, 2 %, and 3 %, considering a porous medium with known physical properties and thickness. The study revealed two local concentration maxima: one within the free flow region on top of the rod arrays used to create the porous structure and a second along the rods' centerline inside the porous media model.

Stability of regional traffic networks employing maximum throughput demand management

This paper considers the stability and optimality properties of traffic demand management schemes, motivated by the integration of smart monitoring and control technologies in traffic networks. First, a suitable optimization problem is formulated that aims to obtain demand input values that maximize the throughput within traffic networks adhering to regional traffic dynamics with triangular macroscopic fundamental diagrams. We show that optimal solutions to this problem may lead to unstable behaviour, revealing a trade-off between stability and optimality. To address this issue, we analytically study the stability properties of traffic networks at the presence of constant demand input and provide suitable local conditions that guarantee stability when the system’s equilibrium densities are strictly within the free-flow region, but not at the critical density. The latter case is significant, since the maximum throughput behaviour coincides in many cases with the local critical density. We resolve this by proposing a decentralized proportional demand control scheme and suitable local design conditions such that stability is guaranteed. Our analytic results are validated with numerical simulations in a 3-region system that demonstrate the effectiveness and practicality of the proposed approach.

Effect of speed humps on instantaneous traffic emissions in a microscopic model with limited deceleration capacity

As a common transportation facility, speed humps can control the speed of vehicles on special road sections to reduce traffic risks. At the same time, they also cause instantaneous traffic emissions. Based on the classic instantaneous traffic emission model and the limited deceleration capacity microscopic traffic flow model with slow-to-start rules, this paper has investigated the impact of speed humps on traffic flow and the instantaneous emissions of vehicle pollutants in a single lane situation. The numerical simulation results have shown that speed humps have significant effects on traffic flow and traffic emissions. In a free-flow region, the increase of speed humps leads to the continuous rise of CO2, NO X and PM emissions. Within some density ranges, one finds that these pollutant emissions can evolve into some higher values under some random seeds. Under other random seeds, they can evolve into some lower values. In a wide moving jam region, the emission values of these pollutants sometimes appear as continuous or intermittent phenomenon. Compared to the refined NaSch model, the present model has lower instantaneous emissions such as CO2, NO X and PM and higher volatile organic components (VOC) emissions. Compared to the limited deceleration capacity model without slow-to-start rules, the present model also has lower instantaneous emissions such as CO2, NO X and PM and higher VOC emissions in a wide moving jam region. These results can also be confirmed or explained by the statistical values of vehicle velocity and acceleration.

Influence of thickness on porous lining when the inclined channel walls are bounded by a deformable porous layer

This paper deals with the viscous fluid flow in an inclined channel bounded by a deformable porous layer. The coupled phenomenon of fluid movement and solid deformation in the porous layer was considered. The governing equations were obtained for the fluid velocity, solid displacement and temperature distribution in the deformable porous medium. Analytical solutions were obtained for solid displacement, fluid velocity, velocity in the free flow region and temperature distributions. The results were analysed for different physical parameters using graphs. The results revealed that the effect of the thickness of the porous layer was significant on solid displacement and fluid velocity. The solid displacement decreased by increasing the volume fraction of the fluid and the thickness of the porous layer. Since the drag coefficient was the ratio of fluid viscosity to the permeability of the porous material, it follows that the enhancement in drag reduces the permeability of the bounding porous layer. The flow in the blood vessels (capillaries) is surrounded by a tissue layer which is modelled as a deformable porous medium.

Flow velocity adjustment in a channel with a floating vegetation canopy

Floating vegetation with unanchored and connected roots can completely cover the water surface of rivers, ditches, and streams. Eichhornia crassipes is a typical macrophyte with a stable network of roots that is present in aquatic environments, forming a complicated flow velocity adjustment. This study constructed model floating canopies and real E. crassipes canopies to investigate flow velocity adjustment and verify a new proposed velocity model. Laboratory experiments showed that flow velocity was modified in both the streamwise (x) and vertical (z) directions and impacted by the mean channel velocity, canopy density, and relative flow depth. The penetration distance δe of the Kelvin–Helmholtz vortices penetrating into the floating canopy increased to its maximum beyond the flow adjustment distance XD, where XD was the distance from the canopy leading edge to the position at which the flow was fully developed. The thickness of the bed boundary layer δb was related to the vegetation drag (FD(x)), the bed shear stress (τb(x)), and the height of the free-flow region below the canopy hg. New estimators for δe and δb were proposed and validated for floating canopies. Combined with the new estimators (δe and δb), flow continuity equation, momentum equation and four-layer divided method, a new analytical model was proposed to predict vertical profiles of velocity within and below a floating canopy in both the flow adjustment region (XD > x > 0) and fully developed flow region (x > XD). Fourteen groups of velocity data from this study and published literature were used to verify the proposed model. The predicted vertical profiles of velocity in flow adjustment and fully developed flow regions agreed with measurements under a wide range of vegetation and flow conditions, suggesting that the proposed model was capable of accurately predicting velocity profiles in a channel with a floating canopy. Finally, the model was validated in a channel with a real E. crassipes canopy. A comparison between predicted and measured velocities confirmed that the new model is valid for application in a channel with real floating macrophytes.

Pore-Scale Simulation of Fracture Propagation by CO2 Flow Induced in Deep Shale Based on Hydro-Mechanical Coupled Model

Summary The depletion of conventional reservoirs has led to increased interest in deep shale gas. Hydraulic fracturing addresses the challenge of developing low-permeability shale, involving hydro-mechanical coupling fracture propagation mechanics. Supercritical CO2 (SC-CO2) has become a promising alternative to fracturing fluids due to its ability to be buried underground after use. The high temperature, pressure, and stress of deep shale lead to the flow of fracturing fluid to plastic deformation of rock, resulting in microfractures. In this paper, we simulate the fracture propagation process of deep shale fractured by SC-CO2 based on the coupling of the Darcy-Brinkman-Biot method, which adopts the Navier-Stokes-like equation to solve the free flow region, and the Darcy equation with Biot’s theory to solve flow in the matrix. To clearly probe the mechanism of deep fracturing from a microscopic perspective, the plastic rock property is taken into consideration. We investigate the effects of injection velocity, rock plastic yield stress, formation pressure, and gas slippage effect on fluid saturation and fracture morphology, and find that increasing the injection rate of fracturing fluid can form better extended fractures and complex fracture networks, improving the fracturing effect. Furthermore, we find that it is more appropriate to adopt SC-CO2 as a fracturing fluid alternative in deep shale with higher plastic yield stress due to higher CO2 saturation in the matrix, indicating greater carbon sequestration potential. High confining pressure promotes the growth of shear fractures, which are capable of more complex fracture profiles. The gas slip effect has a significant impact on the stress field while ignoring the flow field. This study sheds light on which deep shale gas reservoirs are appropriate for the use of SC-CO2 as a fracturing fluid and offers recommendations for how to enhance the fracturing effect at the pore scale.

Hydrodynamic pressure and effective mixing length in vegetated free-flow region

Submerged vegetation acts as large-scale roughness element that changes flow structure and promotes flow subdivision, thereby affecting scalar transport and energy transfer in the river. This article explores vertical distributions of both the hydrodynamic pressure P D and effective mixing length L in vegetated free- flow region through flume experiments. Firstly, formulas have been proposed in the present study to calculate the P D and L values by some theoretical analyses. Then, we discuss the P D and L profiles under different vegetation density λ, submergence degree H/h v and mean bulk velocity U m conditions. The P D value is positive and attains its vertical maximum around the middle position of this region, indicating that shear turbulence promotes the upwards release of the hydro pressure. The vertical variation coefficient for P D is nearly 1.15 and independent of λ and U m in this study. The dimensionless L value varies little in the lower part of this region and increases linearly upwards in the upper part, reflecting vertical variation of shear turbulence movement. Since the vegetation density λ and submergence H/h v control turbulence coherent motion, the P D and L/(h o - h v) values change monotonically with them in this study. The high-velocity flow accelerates turbulence waves and promotes an increase in P D.

Global sensitivity analysis using multi-resolution polynomial chaos expansion for coupled Stokes–Darcy flow problems

Determination of relevant model parameters is crucial for accurate mathematical modelling and efficient numerical simulation of a wide spectrum of applications in geosciences. The conventional method of choice is the global sensitivity analysis (GSA). Unfortunately, at least the classical Monte-Carlo based GSA requires a high number of model runs. Response surfaces based techniques, e.g. arbitrary Polynomial Chaos (aPC) expansion, can reduce computational effort, however, they suffer from the Gibbs phenomena and high hardware requirements for higher accuracy. We introduce GSA for arbitrary Multi-Resolution Polynomial Chaos (aMR-PC) which is a localized aPC based data-driven polynomial discretization. The aMR-PC allows to reduce the Gibbs phenomena by construction and to achieve higher accuracy by means of localization also for lower polynomial degrees. We apply these techniques to perform the sensitivity analysis for the Stokes–Darcy problem which describes fluid flow in coupled free-flow and porous-medium systems. We consider the Stokes equations in the free-flow region, Darcy’s law in the porous-medium domain and the classical interface conditions across the fluid–porous interface including the conservation of mass, the balance of normal forces and the Beavers–Joseph condition for the tangential velocity. This coupled problem formulation contains four uncertain parameters: the exact location of the interface, the permeability, the Beavers–Joseph slip coefficient and the uncertainty in the boundary conditions. We carry out the sensitivity analysis of the coupled model with respect to these parameters using the Sobol indices on the aMR-PC expansion and conduct the corresponding numerical simulations.

Fully decoupled energy-stable numerical schemes for two-phase coupled porous media and free flow with different densities and viscosities

In this article, we consider a phase field model with different densities and viscosities for the coupled two-phase porous media flow and two-phase free flow, as well as the corresponding numerical simulation. This model consists of three parts: a Cahn–Hilliard–Darcy system with different densities/viscosities describing the porous media flow in matrix, a Cahn–Hilliard–Navier–Stokes system with different densities/viscosities describing the free fluid in conduit, and seven interface conditions coupling the flows in the matrix and the conduit. Based on the separate Cahn–Hilliard equations in the porous media region and the free flow region, a weak formulation is proposed to incorporate the two-phase systems of the two regions and the seven interface conditions between them, and the corresponding energy law is proved for the model. A fully decoupled numerical scheme, including the novel decoupling of the Cahn–Hilliard equations through the four phase interface conditions, is developed to solve this coupled nonlinear phase field model. An energy-law preservation is analyzed for the temporal semi-discretization scheme. Furthermore, a fully discretized Galerkin finite element method is proposed. Six numerical examples are provided to demonstrate the accuracy, discrete energy law, and applicability of the proposed fully decoupled scheme.

New hyperbolic tangent formula for mixing layer in vegetated flow

Flow velocity profile in the vegetal mixing layer was investigated using data collected in the present experiments and those reported in the literature, which indicates that the inflection point of the velocity profile does not coincide with the middle position of the vegetal mixing layer. Aiming to solve this problem, the hyperbolic tangent formula that is applicable to the plane mixing layer is modified by revising the inflection point and applying different characteristic scales in the free flow and penetration regions. For the problem of predicting flow velocity on top of vegetation, we constructed a new calculation method for flow velocity on top of vegetation based on correlation analysis. The proposed formula yields significant improvements in representing measured velocity profiles in the vegetated open channel flows. In comparison with previous ones, the modified formula performs better in exhibiting the similarity of measured velocity profiles over a wide range of flow and vegetation conditions. The research results provide technical support for the high precision calculation of the vegetated mixing layer.

Turbulent channel flow of suspensions of neutrally buoyant particles over porous media

This study discusses turbulent suspension flows of non-Brownian, non-colloidal, neutrally buoyant and rigid spherical particles in a Newtonian fluid over porous media with particles too large to penetrate and move through the porous layer. We consider suspension flows with the solid volume fraction ${{\varPhi _b}}$ ranging from 0 to 0.2, and different wall permeabilities, while porosity is constant at 0.6. Direct numerical simulations with an immersed boundary method are employed to resolve the particles and flow phase, with the volume-averaged Navier–Stokes equations modelling the flow within the porous layer. The results show that in the presence of particles in the free-flow region, the mean velocity and the concentration profiles are altered with increasing porous layer permeability because of the variations in the slip velocity and wall-normal fluctuations at the suspension-porous interface. Furthermore, we show that variations in the stress condition at the interface significantly affect the particle near-wall dynamics and migration toward the channel core, thereby inducing large modulations of the overall flow drag. At the highest volume fraction investigated here, ${{\varPhi _b}}= 0.2$ , the velocity fluctuations and the Reynolds shear stress are found to decrease, and the overall drag increases due to the increase in the particle-induced stresses.

Flow velocity evolution through a floating rigid cylinder array under unidirectional flow

Floating canopies with unanchored roots are often distributed in streams and wetlands, which alters the evolution of the flow velocity inside the canopy interiors. Based on the flow continuity equation and layer-averaged momentum equations, an analytical model was proposed to predict the velocity evolution through floating canopies under unidirectional flow. Laboratory experiments showed that the velocity evolution within the canopy interior was impacted by the mean channel velocity, canopy density and ratio of the height of the free flow region below the floating canopy to the flow depth. Inside a floating canopy, the flow adjustment distance, XD, was defined as the distance from the canopy leading edge to the position at which the velocity decreased to a constant value. A new estimator of XD was proposed and validated for floating canopies. The shear parameter (C = 0.076 ± 0.025) was determined and used to characterize the vertical momentum exchange between the in-canopy and below-canopy flows. The velocities of the proposed model predicted using the new estimator (XD) and shear parameter (C) were in good agreement with fourteen groups of measurements from different sources. The vegetation rigidity and flexibility had a weak influence on the flow velocity evolution inside the floating canopy. This indicated that the proposed model was capable of accurately predicting the velocity evolution through floating canopies of vegetation. In addition, the proposed model can be applied to predict the velocity evolution inside the root zones of floating vegetation islands (FVIs). Compared to that of the drag of the root zones, the influence of the drag of the floating island on the velocity evolution inside the root zones was negligible.

Численный анализ динамики молекулярного газа при импульсной лазерной абляции

The evaporation of a molecular gas with rotational degrees of freedom caused by the action of a pulsed nanosecond laser (pulsed laser ablation) of moderate intensity is considered. Model kinetic equations that account for the exchange of translational and rotational energies using a two-temperature model are used to account for the influence of internal energy on the vapor-gas cloud dynamics. Rykov equations (R-model) are integrated when modeling the flow of a diatomic gas, and a generalization of the Bhatnagar-Gross-Krook (BGK) equation for a molecular gas is used to calculate the flow of nonlinear molecules with a number of rotational degrees of freedom equal to three. Taking into account the influence of internal energy on the gas flow is realized by relaxation terms approximating the collision integral as a sum of elastic and inelastic collisions. Since in the model kinetic equations the collision frequency does not depend on the velocities, and the influence of internal energy is taken into account only due to temperature changes, it is necessary to compare with more realistic approaches, which include, for example, the method of direct simulation Monte Carlo (DSMC). The comparison shows that the change in the average temperature over time and the vapor-gas cloud parameters (density and temperature), obtained by the solution of the kinetic equations and the DSMC method, are close. Calculations of the problem of pulsed laser ablation by direct integration of kinetic equations are carried out by the method of discrete ordinates and are a difficult computational problem, which is associated with discontinuous boundary conditions, the presence of both continuum regions and free molecular flow regions in the solution, as well as the need to calculate the vapor-gas cloud dynamics for a large time interval. To reduce computational costs, grid adaptation in physical and velocity spaces is used.

Experimental and numerical study on the development process and flow characteristics of powder fuel jet in the powder fuel scramjet

The injection process of powder fuel directly affects the scramjet operating performance. This study analyzes gas-particle two-phase flow during powder fuel injection. The macroscopic behavior of powder fuel jets under different injection pressure conditions is characterized using the high-speed shadow imaging technology. Simultaneously, the coupled model of computational fluid dynamics-discrete element method is employed to determine the experimental conditions, and the flow field structure and gas-particle interactions are analyzed. The results show that the development of powder fuel jets mainly occurs in three stages: the low-speed development stage, the high-speed development stage, and the stable stages. A long transition period exists between different stages, and the transition time decreases with increasing injection pressure. In the low-speed development stage, the powder fuel jet initially develops slowly and then develops rapidly. In the high-speed development stage, the jet initially develops rapidly and then develops slowly. The jet in the stable stage is classified into the powder particle, fluidized gas, and free flow regions. The distribution range of the particles decreases with increasing injection pressure. The fluidized gas velocity and drag force curve on the axis experience many fluctuations, and the particle velocity is almost always accelerated. Fluidized gas velocity, drag force, particle velocity, and particle acceleration distance increase with the injection pressure.

Effect of porous media models on rheological properties of suspensions

This work reports an experimental investigation of the flow of non-colloidal, non-Brownian suspensions via rheological tests over and through well-designed porous media models. We performed a series of simple shear flow experiments using a rheometer with a parallel-plate geometry where different porous structures were placed on the lower plate. The impact of various bulk particle volume fractions of suspensions φb, ranging from 0 to 0.4, was studied by considering four different porous microstructures with porosities of 0.7 and 0.9. We designed and built porous media models with known permeability and porosity, where the particles were either allowed to move or prevented from moving inside the porous structures. We examined the impact of viscosity by the existence of porous media. We found that, when particles move inside the porous layer, the slip velocity at the fluid-porous interface increases and varies with the porosity compared to when they do not flow into the porous layer. Additionally, with the increase in volume fraction of suspensions, the slip velocity decreases at a constant stress while the slip length rises. We then compared these data with the volume-averaged Navier-Stokes (VANS) equations for flow in the porous medium coupled with the Stokes equations in the free-flow region to further examine the slip velocity.

Chaotic jam and phase transitions in a lattice model with density dependent passing

In real traffic dynamics, passing has a significant impact on the traffic flow. Passing/Overtaking is primarily influenced by the traffic density in the surroundings, therefore considering passing as constant is impractical. In this paper, we proposed a lattice hydrodynamic model with the consideration of density dependent passing for a unidirectional single lane highway to examine the traffic system more realistically. Due to various experimental investigations, the passing behavior is considered similar to the flow-density curve. The passing increases with the density and after achieving a maximum at a critical value, it decreases. Thus, implementing the idea to model density dependent passing similar to the optimal velocity function. The impact of density dependent passing on the lattice model is investigated through linear stability analysis and it is shown that with an increase in passing, the stability region reduces significantly. Using nonlinear analysis, the kink-antikink solution of the mKdV equation is obtained to describe the propagating behavior of the density wave near the critical point. For the small values of passing, there is a phase transition from the kink jam region to the free flow region, with decreasing sensitivity. On the other hand, for large values of passing, the phase transition occurs from the uniform to the kink jam region through the chaotic jam region, with increasing delay time. The influence of parameters involved in density dependent passing is investigated theoretically as well as numerically.