Flow Law Research Articles

Threshold pressure gradient in fractured reservoirs: Experimental simulation and mathematical modeling

Aiming to solve the problems of unclear fluid flow laws and controlling factors in fractured reservoirs, as well as the lack of a mathematical model of threshold pressure gradient (TPG) considering fracture structural parameters, research on the structural characterization of fractured reservoirs, physical simulation of fluid flow, and TPG prediction models has been carried out. Physical simulations demonstrate that fluid flow in fractured reservoirs under stress is nonlinear and exhibits TPG. The TPG displays a significant positive exponential relationship with confining pressure and water saturation. As the confining pressure or water saturation increases, the effective opening of fractures decreases. The TPG will increase dramatically when the fracture opening decreases to the critical thickness for the rapid increase in TPG. On the basis of the fluid flow law study, the TPG fractal prediction model is further developed based on fractal dimension of fracture length distribution (Df), fractal dimension of tortuosity (DT), water saturation (Sw), maximum fracture width (wmax), limit shear stress (η0), and sample characteristic length (L0). The results predicted by the model show good agreement with physical simulations (with average relative error of 7.56%). Experimental simulation and mathematical modeling of TPG in fractured reservoirs can reveal the fluid flow mechanism and improve the accurate evaluation of production capacity during different development processes.

A modified viscous flow law for natural glacier ice: Scaling from laboratories to ice sheets

Glacier flow modulates sea level and is governed largely by the viscous deformation of ice. Multiple molecular-scale mechanisms facilitate viscous deformation, but it remains unclear how each contributes to glacier-scale deformation. Here, we present a model of ice deformation that bridges laboratory and glacier scales, unifies existing estimates of the viscous parameters, and provides a framework for estimating the parameters from observations and incorporating flow laws derived from laboratory observations into glacier-flow models. Our results yield a map of the dominant deformation mechanisms in the Antarctic Ice Sheet, showing that, contrary to long-standing assumptions, dislocation creep, characterized by a value of the stress exponent [Formula: see text], likely dominates in all fast-flowing areas. This increase from the canonical value of [Formula: see text] dramatically alters the climate conditions under which marine ice sheets may become unstable and drive rapid rates of sea-level rise.

Application of Delaunay adaptive mesh refinement in flood risk assessment of multi-bridge system with short distance

To ensure bridge safety, the flood risk analysis is significant important. However, due to the small size and large number of piers in the short-distance multi-bridge system, the extremely long calculation time and low efficiency of the numerical model are induced by the small mesh size and large mesh number. In this paper, a flood risk assessment model of the multi-bridge system with short distance was established to improve the calculation efficiency based on the finite volume method combined with the Delaunay mesh adaptive refinement method. The calculated water level with refined and non-refined mesh was compared with the experimental data of a partial failure dam break test case and Shukry experiment of open channel bend flow. The calculated water level results are in good agreement with the experimental data. In addition, the mesh refinement model improved the calculation efficiency by more than 73% with ensuring the calculation accuracy. Finally, the flood risk of a real multi-bridge system with short distance was evaluated by using the numerical model. The calculated results shown that, different from the general flow law, the water level in the upstream and downstream channel of Bridge 2 rose with a maximum difference value of 0.326 m while the water level in the far downstream channel of Bridge 2 dropped result from the construction of Bridge 2 on the basis of the Bridges 1, 3 and 4. The construction of Bridge 2 also increased the flow velocity around Bridge 3 with maximum 0.013 m/s. This study provides a new tool and technical reference for flood risk analyses of similar multi-bridge system with short distance.

Size dependent scaling law of plastic flow in FCC nanolattices: A dislocation dynamics study

Micro- and nanolattices can achieve simultaneously both lightweight and ultra-high strength due to a combination of resilient architecture and enhanced mechanical properties of small-scale unit constituents. However, understanding the fundamental deformation mechanism for these novel materials remains intriguing due to the intricate interplay of various geometric characteristics and intrinsic microscopic defects. Here, we investigate the fundamental mechanism of plastic deformation in terms of dynamic evolution of defects and corresponding mechanical responses in aluminum nanolattices using a mesoscale defect dynamics model which couples three-dimensional dislocation dynamics and finite element method. Our concurrently coupled model could capture detailed dislocation motion under complex loading conditions and predict the plastic flow stress of the nanolattices. We demonstrate that the size of individual constituent beam plays a critical role in the properties of nanolattices through small-scale strengthening, showing the strength of the nanolattices increases with decreasing beam size with the scaling law of σ∝db−0.89. As a result, the scaling law of plastic yielding with material density drastically increases from the continuum prediction. In addition, our modeling could allow us to study the geometric effect with various nanolattices. These findings establish a foundation for the fundamental understanding of the governing mechanism of plastic deformation, allowing access to a new regime in designing optimal structures using nanolattices.

A Comparative Study on Coupled Fluid–Thermal Field of a Large Nuclear Turbine Generator with Radial and Composited Radial–Axial–Radial Ventilation Systems

With the continuous growth of energy demand, the advantages of nuclear power, such as high energy density, low emissions, and cleanliness, are gradually highlighted. However, the increasing capacity of the turbine generator in nuclear power plants has led to greater losses and critical heating issues. Designing an effective cooling system plays an important role in improving the rotor’s heat dissipation ability, especially under the condition of limited rotor space. In this study, the cooling effects of the rotor using a radial straight-type cooling structure and a composited radial–axial–radial cooling structure are compared and analyzed for a 1555 MVA hydrogen-cooled nuclear turbine generator. Three-dimensional fluid thermal coupled models of the rotor with both cooling structures are established, and corresponding boundary conditions are provided. The models are solved using the finite volume method. The flow law of cooling hydrogen gas inside the rotor and the temperature distribution of various parts of the rotor are studied in detail. Compared with the radial straight-type cooling structure, adopting the composited radial–axial–radial cooling structure can reduce the average temperature of the rotor field windings by 4.5 °C. The research results provide a reference for the design and optimization of the rotor cooling system for large-capacity nuclear turbine generators.

Experimental study on the pseudo threshold pressure gradient of supported fractures in shale reservoirs

Pseudo threshold pressure gradient (PTPG) exists in the propped fractured reservoir, but its nonlinear flow law remains unclear. The effects of the mineral composition of shale and microstructure of fracturing fluid on PTPG were analyzed by X-ray diffraction and liquid nitrogen quick-freezing method. The results demonstrate that a proppant with a large particle size is more likely to form an effective flow channel and reduce liquid flow resistance, thus decreasing PTPG and increasing conductivity. The polymer fracturing fluid with rectangular microstructures significantly increased the PTPG supporting the fractured core. Experimental results show that the PTPG of the resin-coated sand-supported core in the fracturing fluid with a concentration of 1.2% is 245 times higher than that in the fracturing fluid with a concentration of 0.1% when the confining pressure is 5 MPa. Wetting hysteresis and the Jamin effect are responsible for the rise of PTPG in two-phase flow. The equivalent fracture width shows a good power function relationship with the PTPG. Thus, this study further explains the nonlinear flow behavior of reservoirs with fully propped fractures.

The influence of viscous slab rheology on numerical models of subduction

Abstract. Numerical models of subduction commonly use diffusion and dislocation creep laws from laboratory deformation experiments to determine the rheology of the lithosphere. The specific implementation of these laws varies from study to study, and the impacts of this variation on model behavior have not been thoroughly explored. We run simplified 2D numerical models of free subduction in SULEC, with viscoplastic slabs following (1) a diffusion creep law, (2) a dislocation creep law, and (3) both simultaneously, as well as several variations of model 3 with reduced resistance to bending. We compare the results of these models to a model with a constant-viscosity slab to determine the impact of the implementation of different lithospheric flow laws on subduction dynamics. In creep-governed models, higher subduction velocity causes a longer effective slab length, increasing slab pull and asthenospheric drag, which, in turn, affect subduction velocity. Numerical and analogue models implementing constant-viscosity slabs lack this feedback but still capture morphological patterns observed in more complex models. Dislocation creep is the primary deformation mechanism throughout the subducting lithosphere in our models. However, both diffusion creep and dislocation creep predict very high viscosities in the cold core of the slab. At the trench, the effective viscosity is lowered by plastic failure, rendering effective slab thickness the primary control on bending resistance and subduction velocity. However, at depth, plastic failure is not active, and the viscosity cap is reached in significant portions of the slab. The resulting high slab stiffness causes the subducting plate to curl under itself at the mantle transition zone, affecting patterns in subduction velocity, slab dip, and trench migration over time. Peierls creep and localized grain size reduction likely limit the stress and viscosity in the cores of real slabs. Numerical models implementing only power-law creep and neglecting Peierls creep are likely to overestimate the stiffness of subducting lithosphere, which may impact model results in a variety of respects.

A tensorial energy-release-rate based anisotropic damage-plasticity model for concrete

This paper proposes an anisotropic damage model for concrete to capture its plasticity and damage behavior effectively. Utilizing the framework of continuum damage mechanics, the model considers the influence of plastic Helmholtz free energy on damage evolution. This approach explicitly deduces damage-related thermodynamic forces without resorting to assumptions such as strain equivalence or energy equivalence. The proposed model employs tensile and compressive damage tensors to describe the development of anisotropic damage in concrete. The solution for plastic strain utilizes the empirical plasticity model. Furthermore, a unified evolutionary criterion for isotropic and anisotropic damage is also established, incorporating damage yield functions and orthogonal flow laws. Numerical simulations are conducted to validate the model’s ability to reproduce typical nonlinear behaviors of concrete structures under various load conditions.

Computational simulation of the effects of blood flow velocity on atherosclerosis progression in a human carotid artery

Background Atherosclerosis is a build-up of low-density lipoproteins (LDL) in the channels of blood vessels. This occludes the vessels and, occurring in the carotid arteries, portends conditions that favour stroke. This work is an attempt to mathematically represent the physiological process of atherosclerosis caused by plaques on the walls of the human arteries. Aim Provide insight into the effect of blood flow velocity on wall shear stress and its implications on atherosclerosis progression in a human carotid artery via computational simulation. Methods The effect of blood velocity on plaque growth and progression is simulated using COMSOL multi-physics. The human carotid was modeled in 2-D with Stokes law for model flow. The simulation began with a plaque-free vessel with velocities of 30 m/s – 125 m/s. Results Results showed that the rate of plaque initiation dropped as the blood velocity increased from 30 m/s to 125 m/s; higher inlet velocities gave lower plaque growth; the highest degree of 30% stenosis was recorded at a blood velocity of 30 m/s. Plaque height significantly affects the Plaque wall Stress, PWS, and its distribution around the plaque and arterial wall; higher plaque heights experience higher velocity distribution around the plaque, causing a higher force associated with blood flow around the plaque, resulting in higher compression stress. More compressional stresses are localized around the root, which would encourage growth as well as possible rupture at higher velocities. These ruptured plaques potentially narrow or block the arteries and prevent blood flow. This is atherosclerosis and can lead to a heart attack. Conclusion Results from this study can find significant use in the understanding, management, and treatment of atherosclerosis since the regulation of blood velocity and pressure plays a major role in the progress of atherosclerosis in the carotid artery which raises the risk of stroke.

Complexity upgrade and triggering mechanism of mixed-wettability: Comparative study of CO2 displacement in different phases

Complex pore structures and wettability affect CO2 storage in underground reservoirs. Revealing the flow law of CO2 in complex reservoirs is crucial for evaluating storage stability and security. Considering the complexity of the pore structure and wettability of rock, a complex mixed-wettability characterization method based on a complex pore structure is proposed. Matrix numerical models for two-dimensional heterogeneous complex mixed-wettability rock were developed to simulate CO2 displacement in different phases. A comparative study found that, under single-phase wettability, the weaker the wettability, the better the effect of gaseous CO2 displacement. Moreover, the stronger the wettability, the better the displacement effect of ScCO2. The wetting phase of CO2 was transformed from the non-wetting phase. At the late stage of gaseous CO2 displacement, when the contact angles were 30° and 60°, the displaced water phase was absorbed back into the matrix owing to imbibition, resulting in the formation of a peak in the displacement efficiency curve. Compared to that at a contact angle of 60°, the displacement efficiency at a contact angle of 30° decreased more significantly after the peaks, while the suck-back phenomenon did not occur when the contact angle was 90°. Under mixed-wettability conditions, the flow law of CO2 in different phases is directly affected by the contact angle of the pore walls. The displacement efficiencies of gaseous CO2 and ScCO2 increased by 9.58% and 17.96%, respectively, owing to the dual driving force caused by imbibition. The complexity of mixed-wettability is upgraded by providing pore walls with different proportions of contact angles (six types). The flow law of different phases of CO2 showed that, the higher the proportion of walls with small contact angles (3:2:1), the worse the displacement effect of gaseous CO2 and the better the displacement effect of ScCO2. Compared with single-phase wettability and mixed-wettability, the displacement efficiency of gaseous CO2 and ScCO2 under complex mixed-wettability decreased by 4.26%, 1.00% and 20.41%, 3.37%, respectively. Therefore, ignoring complex mixed-wettability often leads to a higher displacement efficiency.

A little mica goes a long way: Impact of phyllosilicates on quartz deformation fabrics in naturally deformed rocks

Quartz deformation fabrics reflect stress and strain conditions in mylonites, and their interpretation has become a mainstay of kinematic and structural analysis. Quantification of grain size and shape and interpretation of textures reflecting deformation mechanisms can provide estimates of flow stress, strain rate, kinematic vorticity, and deformation temperatures. Empirical calibration and determination of quartz flow laws is based on laboratory experiments of pure samples; however, pure quartzite mylonites are relatively uncommon. In particular, phyllosilicates may localize and partition strain that can inhibit or enhance different deformation mechanisms. Experimental results demonstrate that even minor phyllosilicate content (<15 vol%) can dramatically alter the strain behavior of quartz; however, few field studies have demonstrated these effects in a natural setting. To investigate the role of phyllosilicates on quartz strain fabrics, we quantify phyllosilicate content and distribution in quartzite mylonites from the Miocene Raft River detachment shear zone (NW Utah, USA). We use microstructural analysis and electron backscatter diffraction to quantify quartz deformation fabrics and muscovite spatial distribution, and X-ray computed tomography to quantify muscovite content in samples with varying amounts of muscovite collected across the detachment shear zone. Phyllosilicate content has a direct control on quartz deformation mechanisms, and application of piezometers and flow laws based on quartz deformation fabrics yield strain rates and flow stresses that vary by up to two orders of magnitude across our samples. These findings have important implications for the application of flow laws in quartzite mylonites and strain localization mechanisms in mid-crustal shear zones.

Three-dimensional elastoplastic post-buckling analysis of porous FG plates resting on Winkler/Pasternak foundation using meshless RRKPM

This paper presents a three-dimensional (3D) analysis of the elastoplastic post-buckling behavior of porous functionally graded (FG) plates resting on Winkler/Pasternak foundations under uniaxial and biaxial in-plane loadings using an enhanced meshless approach. The principle of virtual work is used to derive the governing equations, and 3D nonlinear Green–Lagrange strains are considered. Plastic deformation is modeled employing the Prandtl–Reuss flow law and the isotropic hardening von Mises criterion. A novel meshless radial basis reproducing kernel particle approach is used to obtain the discretized system of equations. The Newton–Raphson method, coupled with the arc-length technique, is used to compute post-buckling paths of porous FG plates. Numerical assessments show that post-buckling paths are significantly influenced by porosity parameter, porosity distribution, foundation parameters, material gradient, plate thickness, loading ratio, and boundary conditions (BCs).

Study on the evolution of permeability properties of limestone under different stress paths

Abstract Stress path change has a great relationship with the effect of deformation and strength of rock. However, the underground rock body is in the engineering environment where the stress field, seepage field, and other fields are coupled to change, the law of fluid flow in the rock body is complex and variable. The change in the stress field has an important effect on the seepage characteristics of rock body; therefore, it is necessary to study the pattern of rock permeability evolution pattern by different stress paths. This study is based on the study of limestone, conducting triaxial unloading seepage mechanics experiments, the evolution of permeability properties of limestone specimens was analyzed based on the test results. The results show that in the conventional triaxial loading seepage test, the permeability of the limestone decreases before the rock stress reaches the peak intensity and increases after that. Increasing axial pressure unloading surrounding pressure compression section, permeability loss rate and effective stress are in line with the changing law of Gaussian distribution function. Under the action of constant axial pressure unloading surrounding pressure, with the increase in unloading amount, the permeability rate of change appears to increase slowly, and in the late stage of unloading section, the permeability rate of change appears to surge. Unloading section permeability change rate and cumulative unloading amount are in line with the law of change of the exponential function. Creep unloading pressure seepage test found that unloading pressure stage strain-time and permeability-time evolution characteristics are in line with the exponential rule of change. The experimental results of this study can provide an important experimental and theoretical basis for the permeability analysis of low-permeability rock body under complex stress conditions in underground engineering.

Research on hydro-thermal coupling model of canal foundation soil based on particle grading curve predicting soil-water characteristic curve

Hydro-thermal coupling is the essence of the freeze-thaw process, and theoretical studies of this coupled process have been hot topics in the field of frozen soil. Darcy's law of unsaturated soil water flow, heat conduction theory, and relative saturation and solid-liquid ratio are based on this paper. According to the principle that the cumulative curve of particle gradation of canal foundation soil is similar to soil-water properties. A soil-water characteristic curve is derived using the cumulative particle gradation curve. VG model is then used to fit soil-water characteristic curves to obtain the canal foundation soil's hydraulic characteristic parameters, and the established hydro-thermal coupling model is modified to reflect canal foundation soil hydro-thermal evolution more objectively. A closed system one-way freezing test method is used to verify the feasibility of the proposed method in this part. The results show that the optimal parameters of the VG model of the subsoil are a = 0.06, n = 1.2, and m = 0.17, and the temperature and water fields obtained from the simulation are in good agreement with the measured data, showing the utility of the hydro-thermal coupling model in predicting hydraulic parameters. Analysis of the multi-field interaction mechanism and dynamic coupling process of the canal foundation soil during freezing and thawing. This has great importance for preventing freezing damage in canals and protecting agricultural safety.

Research on theoretical and experimental investigations of burr formation mechanism in micro-milling process of medical ploy-l-lactide

Medical grade poly-l-lactide (PLLA) has been widely applied for medical devices and consumables, e.g. biodegradable cardiovascular stents and surgical thread, due to its excellent biocompatibility. High-speed micro-milling technology shows great potential in high-precision and high-quality manufacturing of material PLLA, while it usually produces problems such as material burrs and defects. There is few research and analysis on the material removal mechanism and theory in the micro-milling process of polymers, and thus this paper aims to investigate the burr formation mechanism and processing quality in micro-milling process of medical PLLA. Firstly, a novel thermal-mechanical coupled theoretical model was proposed by considering the minimum cutting thickness and cutting-edge radius size effects, which can reveal the plastic flow law during micro-milling process of polymer material. This model was used to calculate the burr width at up and down milling sides of cutting area. Then material characterization and micro-milling experiments on two types of PLLA with different molecular weights were carried out to validate the established theoretical model, and further the micro-milling parameters were optimized to achieve the control of material burr formation and growth. As a result, a cardiovascular stent was fabricated by micro-milling with the optimized parameters (spindle speed of 25,000 rpm, feed rate of 100 mm/min and axial depth of cut of 5 μm), and the stents obtained significantly high surface quality. These results suggest that the proposed theoretical model has great advantages in the high-quality micro-milling of medical material PLLA. This work provides a potential stent manufacture technique in the development and clinical application of bioresorbable polymeric stents.

3D Stresses and Velocities Caused by Continental Plateaus: Scaling Analysis and Numerical Calculations With Application to the Tibetan Plateau

AbstractUnderstanding stresses is crucial for geodynamics since they govern rock deformation and metamorphic reactions. However, the magnitudes and distribution of crustal stresses are still uncertain. Here, we use a 3D numerical model in spherical coordinates to investigate stresses and velocities that result from lateral crustal thickness variations around continental plateaus like those observed for the Tibetan plateau. We do not consider any far‐field deformation so that the plateau deforms by horizontal dilatation and vertical thinning. We assume viscous creep, a simplified plateau geometry, and simplified viscosity and density distributions to couple the numerical results with a scaling analysis. Specifically, we study the impact of the viscosity ratio between crust and lithospheric mantle, a rectangular plateau corner, a stress‐dependent power‐law flow law and Earth's double curvature on the crustal stress field and horizontal velocities. Two orders of magnitude variation in crustal and lithospheric mantle viscosities change the maximum crustal differential stress only by a factor of ≈2. We derive simple analytical estimates for the crustal deviatoric stress and horizontal velocity which agree to first order with 3D numerical calculations. We apply these estimates to calculate the average crustal viscosity in the eastern Tibetan plateau as ≈1022 Pa · s. Furthermore, our results show that a corner strongly affects the stress distribution, particularly the shear stresses, while Earth's curvature has a minor impact on the stresses. We further discuss potential implications of our results to strike‐slip faulting and fast exhumation around the Tibetan plateau's syntaxes.

Invariant scaling laws for plane Couette flow with wall-transpiration

The effect of the wall-transpiration Reynolds number ReV0 on the large-scale rolls of the plane Couette flow with wall-transpiration is investigated using resolvent analysis. We observe streamwise-elongated large-scale rolls, moving closer to the suction wall and becoming narrower in the wall-normal and spanwise direction as well as shortened in the streamwise direction for growing ReV0 due to the decreasing characteristic length of the sheared region. Invariant scaling laws describing the growth of the amplification gain over various parameters and determining the optimal parameters yielding most amplification are found enhancing our understanding of the influence on the coherent structures, which can be used to predict their exact size. In a potential second step, this allows to manipulate and efficiently modify these flow characteristics increasing the functionality for various engineering applications as for mass, momentum and heat transfer, which is dominated by turbulent superstructures. For large enough wall-transpiration, the most amplified structures are oblique at optimal streamwise wavenumbers αmax≠0 growing linearly with ReV0. The optimal spanwise wavenumber βmax yielding most amplified structures shows a polynomial dependence of second order on ReV0. For a constant ratio of the wall-transpiration and streamwise Reynolds numbers γ=ReV0Re, the most amplified structures occur at optimal streamwise Reynolds numbers represented by Remax·γa=C. The amplification gain changes its trend from growing with Re2 to decreasing with Re−1.368 after Remax is reached. The resulting streamwise structures are identical for each set of Remax and γ within this invariant scaling law.

The Transition from Darcy to Nonlinear Flow in Heterogeneous Porous Media: I—Single-Phase Flow

Using extensive numerical simulation of the Navier–Stokes equations, we study the transition from the Darcy’s law for slow flow of fluids through a disordered porous medium to the nonlinear flow regime in which the effect of inertia cannot be neglected. The porous medium is represented by two-dimensional slices of a three-dimensional image of a sandstone. We study the problem over wide ranges of porosity and the Reynolds number, as well as two types of boundary conditions, and compute essential features of fluid flow, namely, the strength of the vorticity, the effective permeability of the pore space, the frictional drag, and the relationship between the macroscopic pressure gradient ∇P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\ abla }}P$$\\end{document} and the fluid velocity v. The results indicate that when the Reynolds number Re is low enough that the Darcy’s law holds, the magnitude ωz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _z$$\\end{document} of the vorticity is nearly zero. As Re increases, however, so also does ωz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _z$$\\end{document}, and its rise from nearly zero begins at the same Re at which the Darcy’s law breaks down. We also show that a nonlinear relation between the macroscopic pressure gradient and the fluid velocity v, given by, -∇P=(μ/Ke)v+βnρ|v|2v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-{\\varvec{\ abla }}P=(\\mu /K_e)\ extbf{v}+\\beta _n\\rho |\ extbf{v}|^2\ extbf{v}$$\\end{document}, provides accurate representation of the numerical data, where μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu$$\\end{document} and ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho$$\\end{document} are the fluid’s viscosity and density, Ke\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_e$$\\end{document} is the effective Darcy permeability in the linear regime, and βn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _n$$\\end{document} is a generalized nonlinear resistance. Theoretical justification for the relation is presented, and its predictions are also compared with those of the Forchheimer’s equation.

A controllable sliding law for thin-film flows over slippery fluid-saturated substrates: theory and experiments

We develop a theoretical and experimental framework for generating slip underneath thin-film flows of viscous fluids in the laboratory, with the ability to control slip as desired. Such a framework is useful for large-scale fluid-mechanical experiments in which basal sliding is important. In particular, we consider the flow of a thin film of viscous fluid spreading over a structured, slippery substrate, involving a sequence of two-dimensional cavities that are prewetted with a fluid of smaller viscosity. By averaging over small-scale inhomogeneities, we demonstrate that such a substrate gives rise to a macroscopic linear sliding law, or Navier slip condition, that is effectively homogeneous on the large scale. The slip length, determining the slipperiness of the substrate, is proportional to the viscosity ratio and width of each cavity. As such, the slipperiness of the substrate can be controlled by altering the viscosity ratio, as desired. Two asymptotic regimes arise, describing flow over very slippery substrates and flow over no-slip substrates. The former regime is valid for early times, when the depth of the overlying fluid is much less than the slip length, and the latter is valid for late times, when the depth is much greater than the slip length. Solutions to the full model approach similarity solutions describing the two regimes for early and late times. We confirm our theoretical predictions by conducting a series of analogue laboratory experiments.

Analysis of vortex characteristics in hump region of reversible pump-turbine based on omega vortex identification method

As the core equipment of pumped storage power stations, reversible pump-turbines will frequently switch between different working modes during operation, and it is easy to appear hump region under pump condition, which will greatly affect the performance of the pumped storage unit. Therefore, in order to explore the causes of the hump region, this paper takes the model reversible pump-turbine as the research object. First, the unit speed–unit flow characteristics are compared with the model test results under different working conditions. Then, based on the omega vortex identification method, the vortex distribution in the flow channel of the unit is analyzed. By analyzing the flow characteristics of the flow components of the unit under different flow rates in the hump region, the internal flow law of the hump region is revealed. It is found that when the flow rate decreases, the rising head cannot offset the head lost by the hydraulic loss, thus forming the hump region, and the unstable flow gradually appears in the guide vane and runner area. There are unstable phenomena, such as flow separation, in the guide vane area. There is a sudden change of radial pressure in the vaneless area between the guide vane and the runner; that is, the unstable vortex almost occupies the whole flow channel due to the change of pressure in this area, resulting in the deterioration of the instability of the unit operation. When the flow rate is large, the radial pressure mutation zone mainly exists between the guide vane cascades, making it easy to form an unstable vortex. The research results can provide a theoretical reference for improving the stability of reversible pump-turbines.