Nonlinear Fluid Flow Research Articles

Study on flow regimes and seepage models through open rough-walled rock joints under high hydraulic gradient

Little attention has been paid to non-Darcy fluid flow through rough-walled rock joints under high hydraulic gradient. Water fluid flow through splitting granite joints was tested beyond the range of previously tested conditions by using an experimental apparatus designed for this study. This apparatus facilitated the quantification of the effects of joint surface roughness and aperture on fluid flow properties. Experimental results showed that Forchheimer’s law could provide an excellent description of the nonlinear relationship between hydraulic gradient and flow velocity, and the variations of Forchheimer coefficients with joint surface roughness and aperture were further explored. In this work, the Reynolds numbers varied between 2,881 and 290,338, and greatly exceeded the critical value of 2,300. Moreover, the fluid flow entered the turbulent flow regime. In accordance with the discriminant factor α = 0.9 on the basis of Forchheimer’s law, the fluid flow regimes were further divided into two categories: weak turbulence and fully developed turbulence. The corresponding critical Reynolds numbers were obtained and ranged from 17,169 to 94,385. Finally, a new seepage calculation formula for fluid flow through open rough-walled rock joints was established and verified by experimental observations and other formulae. Through a comprehensive consideration of the joint surface roughness and the uneven aperture distribution, the proposed formula could reflect the real fluid flow situation. The findings may prove beneficial for improving understanding of the nonlinear fluid flow in jointed rocks.

MHD slip flow of chemically reacting UCM fluid through a dilating channel with heat source/sink

Abstract The present article explores the effects of uniform heat source and first order destructive chemical reaction on an upper convected Maxwell fluid through an expanding or contracting channel with the porous slip condition at the upper plate. It is assumed that the fluid is sucked or injected through the upper plate. The temperature and concentration at the plates is maintained constant. Using suitable similarity transformations, nonlinear coupled ODEs are developed from the governing PDEs. The subsequent ODEs are converted into a first order system and integrated via shooting method. The effect of various prominent parameters on heat, flow and mass transfer characteristics are studied in detail through graphs and tables. The present results suggest that the presence of chemical reaction and heat source yields in the reduction of concentration and of the enhancement of temperature the fluid. It is also observed that the wall expansion shows an increasing effect on the radial velocity component, but the slip parameter exhibits an opposing effect. The viscous case has been studied as a special case where the present results are found to be close to the earlier ones. The flow of such nonlinear viscoelastic fluids has important applications in separation processes like petroleum and medical industries.

Hydromechanical modelling of CO2 sequestration using a component-based multiphysics code

Geologic carbon dioxide sequestration (GCS) is a complex process with coupled multiphysics mechanisms, including non-linear solid deformations and fluid flow. When applying a finite-element method to solve such a complex problem, implementing and solving non-linear partial differential equations with sophisticated material models pose great challenges. In this work, a generic component-based multiphysics analysis code – Albany – is introduced for the modelling and analysis of GCS problems. The component-based approach allows the application code development effort to focus on writing physics models. A notable feature of Albany is the powerful automatic differentiation utilities that can be used to calculate automatically system Jacobians and consistent tangents. Two numerical problems, namely, wellbore injection and carbon dioxide transport in a confined underground channel, are developed to demonstrate the performance of the Albany analysis framework. The results are verified against and matched well with those from Comsol – a robust commercial multiphysics finite-element package. The numerical results are also thoroughly analysed to gain insights into distribution profiles and temporal evolutions of pore pressure and displacement fields during a GCS process.

Transported snapshot model order reduction approach for parametric, steady‐state fluid flows containing parameter‐dependent shocks

SummaryA new model order reduction approach is proposed for parametric steady‐state nonlinear fluid flows characterized by shocks and discontinuities whose spatial locations and orientations are strongly parameter dependent. In this method, solutions in the predictive regime are approximated using a linear superposition of parameter‐dependent basis. The sought‐after parametric reduced bases are obtained by transporting the snapshots in a spatially and parametrically dependent transport field. Key to the proposed approach is the observation that the transport fields are typically smooth and continuous, despite the solution themselves not being so. As a result, the transport fields can be accurately expressed using a low‐order polynomial expansion. Similar to traditional projection‐based model order reduction approaches, the proposed method is formulated mathematically as a residual minimization problem for the generalized coordinates. The proposed approach is also integrated with well‐known hyper‐reduction strategies to obtain significant computational speedups. The method is successfully applied to the reduction of a parametric one‐dimensional flow in a converging‐diverging nozzle, a parametric two‐dimensional supersonic flow over a forward‐facing step, and a parametric two‐dimensional jet diffusion flame in a combustor.

Influences of connectivity and conductivity on nonlinear flow behaviours through three-dimension discrete fracture networks

A numerical procedure about nonlinear flow in three-dimension discrete fracture networks was developed by solving the Reynolds equation and Forchheimer equation using the Galerkin method. The numerical simulation results of three intersecting models were compared with experimental data and a good agreement was observed. Based on further applications of the model to fracture networks with the consideration of Barton’s and Chen’s empirical model, it was found that nonlinear fluid flows in fracture networks could be appropriately described by the Forchheimer equation. The smoother the fracture surfaces and the greater the connectivity of the fracture network are, the lower the hydraulic gradient for the onset of nonlinear flows is.

Dissipative shock waves generated by a quantum-mechanical piston

The piston shock problem is a prototypical example of strongly nonlinear fluid flow that enables the experimental exploration of fluid dynamics in extreme regimes. Here we investigate this problem for a nominally dissipationless, superfluid Bose-Einstein condensate and observe rich dynamics including the formation of a plateau region, a non-expanding shock front, and rarefaction waves. Many aspects of the observed dynamics follow predictions of classical dissipative—rather than superfluid dispersive—shock theory. The emergence of dissipative-like dynamics is attributed to the decay of large amplitude excitations at the shock front into turbulent vortex excitations, which allow us to invoke an eddy viscosity hypothesis. Our experimental observations are accompanied by numerical simulations of the mean-field, Gross-Pitaevskii equation that exhibit quantitative agreement with no fitting parameters. This work provides an avenue for the investigation of quantum shock waves and turbulence in channel geometries, which are currently the focus of intense research efforts.

Retracted: A simple algorithm for reducing the operation frequency of residential water pumps during peak hours of power consumption

AbstractIn developing countries where water supply pressure is low, or frequent water outages and electricity shortages happen, domestic end‐users are forced to install water pump‐storage systems, consisting of water pumps and ground‐level and rooftop tanks to satisfy their daily water demands. This contributes to increasing electrical energy consumption, particularly during peak electricity demand hours. This study presents a simple, practical, computational, and cost‐effective shifting water level control algorithm to manage water‐energy nexus, by reducing pump‐storage system electric energy consumption during peak hours. The proposed algorithm requires a simple modification to the existing water level control scheme, by installing an additional float switch in the rooftop water tank below the currently available float switch that is usually adjusted to trigger the pump when the level in the tank drops by 5‐10% from the maximum level. Based on the simulation results, the algorithm preserves the domestic end‐users’ comfortable daily water demand and reduces water pump energy consumption during peak hours by 90%. During off‐peak hours, the control algorithm triggers the pump to refill the rooftop tank based on the upper float switch when water level drops by 5%, while during peak hours, the pump is triggered only when the water drops by 30%. The performance of the algorithm is found to be comparable to the performance of the model predictive control (MPC) approach developed for the same purpose, but MPC needs a high computation capacity and a complex analog feedback level sensor. The algorithm succeeds in reducing and shifting pump energy consumption under various possible operation scenarios and water demand disturbances. A mathematical model is developed for the domestic water pump‐storage system using Matlab/Simscape to cope with the complexity of solving nonlinear fluid flow equations and measure the data required to develop the control algorithm. The performance of the algorithm is tested based on a real case study.

Modeling three-dimensional flow of the polymer melts in a converging channel

Temperature and volume rate impact of the polymer melt on the sizes of the vortex area has been considered. The vortex area appears during the flow at the inlet of the slotted channel. A modified Vinogradov and Pokrovskii rheological model is used for the mathematical modeling of three-dimensional flow of the melt in the convergent plane-parallel channel. This model was extended to take account of the nature of non-monotonic gradient dependence of elongation viscosity. On the solid wall the sticking conditions were used for velocity. The temperature dependence of the initial shear viscosity of the polymer melt has Arrhenius form. The initial relaxation time was rated by comparison with experimental data for agradient based stationary viscosity in uniaxial tension, and on the basis of the molecular-kinetic approach. Discrete analogues of a system of equations of the dynamics for the polymer liquid were obtained via the method of controlling the volume with the division due to physical processes. The numerical algorithm was implemented so that the technology of parallel computing based on GPU was conveniently used. Calculations of hydrodynamic characteristics of polymer melt flows at different temperatures in a convergent channel with a rectangular cross section have shown that the sudden narrowing of the channel leads to the appearance of eddy flows. The sizes of these flows pass through a maximum with increasing melt temperature which is detected in the experiments. It also confirms that the rheological model can be used for the flows of polymer melts in areas with complex geometry. The results are proving the effectiveness of CUDA technology parallel computing for unsteady calculations of three-dimensional flows of nonlinear viscoelastic fluids with rheological law of behavior in differential form. The above calculations show that some convergent flow of polymermelts may show substantial three-dimensional picture, which manifests itself in a velocity component in a neutral flow direction. This should be taken into account when arranging experiments, as there are methods that cannot measure all velocity components. Also note that the velocity profile in the slit of the channel is set at a considerable distance from the entrance to the channel.

Simulations of variable concentration aspects in a fractional nonlinear viscoelastic fluid flow

This article elucidates viscoelastic fractional fluid flow along with homogeneous-heterogeneous reactions. Non-integer derivatives are important for the systems having hereditary behavior as they depend on the past conditions along with the local conditions. Viscoelastic fluids keep memory of old deformations and their behavior is related to these deformations. The fractional derivatives are more adequate in predicting the characteristics of viscoelastic fluids than the ordinary derivatives. In this paper we have examined characteristics of fractional homogeneous-heterogeneous reactions with variable concentrations of both the reactant species. Flow is induced by the variable moving surface. Diffusion coefficients of reactants and auto catalyst are of comparable size. For better understanding of hereditary properties, non integer Caputo time derivatives have been incorporated in mathematical formulation of the flow regime. Unsteady motion of a magnetohydrodynamic incompressible viscoelastic fluid is governed by highly nonlinear fractional partial differential equations. Flux conditions are imposed on the chemical species at the boundary. Governing flow equations with suitable initial and boundary conditions have been discretized using finite difference-finite element scheme. Proposed numerical scheme is flexible for the solution of non-linear flow problems. Skin friction coefficients are calculated for fractional viscoelastic model. Flow field is demonstrated for different values of involved parameters. The acquired results revealed that with the increase of fractional exponent α concentration of chemical specie decreases for the final time t=2. It is also noted that concentration gradients of chemical specie increase with the increase of α at both the boundaries of flow configuration. In literature no such result exits with non integer Caputo fractional derivatives. Various viscoelastic flows particularly in chemical, plastics and polymer industries can be modeled in a similar manner.

Experimental Study on Stress-Dependent Nonlinear Flow Behavior and Normalized Transmissivity of Real Rock Fracture Networks

The mechanism and quantitative descriptions of nonlinear fluid flow through rock fractures are difficult issues of high concern in underground engineering fields. In order to study the effects of fracture geometry and loading conditions on nonlinear flow properties and normalized transmissivity through fracture networks, stress-dependent fluid flow tests were conducted on real rock fracture networks with different number of intersections (1, 4, 7, and 12) and subjected to various applied boundary loads (7, 14, 21, 28, and 35 kN). For all cases, the inlet hydraulic pressures ranged from 0 to 0.6 MPa. The test results show that Forchheimer’s law provides an excellent description of the nonlinear fluid flow in fracture networks. The linear coefficient a and nonlinear coefficient b in Forchheimer’s law J=aQ+bQ2 generally decrease with the number of intersections but increase with the boundary load. The relationships between a and b can be well fitted with a power function. A nonlinear effect factor E=bQ2/(aQ+bQ2) was used to quantitatively characterize the nonlinear behaviors of fluid flow through fracture networks. By defining a critical value of E = 10%, the critical hydraulic gradient was calculated. The critical hydraulic gradient decreases with the number of intersections due to richer flowing paths but increases with the boundary load due to fracture closure. The transmissivity of fracture networks decreases with the hydraulic gradient, and the variation process can be estimated using an exponential function. A mathematical expression T/T0=1-exp⁡(-αJ-0.45) for decreased normalized transmissivity T/T0 against the hydraulic gradient J was established. When the hydraulic gradient is small, T/T0 holds a constant value of 1.0. With increasing hydraulic gradient, the reduction rate of T/T0 first increases and then decreases. The equivalent permeability of fracture networks decreases with the applied boundary load, and permeability changes at low load levels are more sensitive.

Full-Annulus Simulation of Nonsynchronous Blade Vibration Excitation of an Axial Compressor

A high speed 1–1/2 axial compressor stage is simulated in this paper using an unsteady Reynolds-averaged Navier–Stokes (URANS) solver for a full-annulus configuration to capture its nonsynchronous vibration (NSV) flow excitation with rigid blades. A third-order weighted essentially nonoscillatory scheme for the inviscid flux and a second-order central differencing for the viscous terms are used to resolve nonlinear unsteady fluid flows. A fully conservative rotor/stator sliding boundary condition (BC) is employed with multiple-processor capability for rotor/stator sliding interface that accurately captures unsteady wake propagation between the rotor and stator blades while conserving fluxes across the rotor/stator interfaces. The predicted dominant frequencies using the blade tip response signals are not harmonic to the engine order, which is the NSV excitation. The simulation is based on a rotor blade with a 1.1% tip-chord clearance. Comparison with the previous 1/7 annulus simulations show that the time-shifted phase-lag BCs used in the 1/7 annulus are accurate. For most of the blades, the NSV excitation frequency is 6.2% lower than the measurement in the rig test, although some blades displayed slightly different NSV excitation frequencies. The simulation confirms that the NSV is a full annulus phenomenon. The instability of the circumferential traveling vortices in the vicinity of the rotor tip due to the strong interaction of incoming flow is the main cause of the NSV excitation. This instability is present in all blades of the rotor annulus. For circumferentially averaged parameters like total pressure ratio, NSV is observed to have an effect on the radial profile, particularly at radial locations above 70% span. A design with a lower loading of the upper blade span and a higher loading of the midblade spans is recommended to mitigate or remove NSV.

Von Neumann stability analysis of first-order accurate discretization schemes for one-dimensional (1D) and two-dimensional (2D) fluid flow equations

In literature, the von Neumann stability analysis of simplified model equations, such as the wave equation, is typically used to determine stability conditions for the non-linear partial differential fluid flow equations (Navier–Stokes and Euler). However, practical experience suggests that such simplistic stability conditions are grossly inadequate for computations involving the system of coupled flow equations. The goal of this paper is to determine stability conditions for the full system of fluid flow equations – the Euler equations are examined, as any conditions derived for the Euler equations will apply to the Navier–Stokes (NS) equations in the limit of convection-dominated flows. A von Neumann stability analysis is conducted for the one-dimensional (1D) and two-dimensional (2D) Euler equations. The system of equations is discretized on a staggered grid using finite-difference discretization techniques; the use of a staggered grid allows equivalence to finite-volume discretization. By combining the different discretization techniques, ten solution schemes are formulated – eight solution schemes are considered for the 1D Euler equations, and two schemes for the 2D Euler equations. For each scheme, error amplification matrices are determined from the stability analysis, stable and unstable regimes are identified, and practical stability limits are predicted in terms of the maximum-allowable CFL (Courant–Friedrichs–Lewy) number as a function of Mach number. The predictions are verified for selected schemes using the Riemann problem at incompressible and compressible Mach numbers. Very good agreement is obtained between the analytically predicted and the “experimentally” observed CFL values. The successfully tested stability limits are presented in graphical form, which offer a viable alternative to complicated mathematical expressions often reported in published literature, and should benefit everyday CFD (Computational Fluid Dynamics) users. The stability regions are used to discuss the effect of time integration (explicit vs. implicit), density bias in continuity equation and momentum convection term linearization on stability. A comparison of the predicted stability limits for 1D and 2D Euler equations with commonly-used stability conditions arising from the wave equation shows that the stability thresholds for the Euler equations lie well below those predicted by the wave equation analysis; in addition, the 2D Euler stability limits are more restrictive as compared to 1D Euler limits. Since the present analysis accounts for the full system of fluid flow (Euler) equations, the derived stability conditions can be used by CFD practitioners to estimate a timestep or CFL number to guide the stability of their computations.

Fully Coupled Nonlinear Fluid Flow and Poroelasticity in Arbitrarily Fractured Porous Media: A Hybrid‐Dimensional Computational Model

AbstractWe formulate the problem of fully coupled transient fluid flow and quasi‐static poroelasticity in arbitrarily fractured, deformable porous media saturated with a single‐phase compressible fluid. The fractures we consider are hydraulically highly conductive, allowing discontinuous fluid flux across them; mechanically, they act as finite‐thickness shear deformation zones prior to failure (i.e., nonslipping and nonpropagating), leading to “apparent discontinuity” in strain and stress across them. Local nonlinearity arising from pressure‐dependent permeability of fractures is also included. Taking advantage of typically high aspect ratio of a fracture, we do not resolve transversal variations and instead assume uniform flow velocity and simple shear strain within each fracture, rendering the coupled problem numerically more tractable. Fractures are discretized as lower dimensional zero‐thickness elements tangentially conforming to unstructured matrix elements. A hybrid‐dimensional, equal‐low‐order, two‐field mixed finite element method is developed, which is free from stability issues for a drained coupled system. The fully implicit backward Euler scheme is employed for advancing the fully coupled solution in time, and the Newton‐Raphson scheme is implemented for linearization. We show that the fully discretized system retains a canonical form of a fracture‐free poromechanical problem; the effect of fractures is translated to the modification of some existing terms as well as the addition of several terms to the capacity, conductivity, and stiffness matrices therefore allowing the development of independent subroutines for treating fractures within a standard computational framework. Our computational model provides more realistic inputs for some fracture‐dominated poromechanical problems like fluid‐induced seismicity.

A Review of Critical Conditions for the Onset of Nonlinear Fluid Flow in Rock Fractures

Selecting appropriate governing equations for fluid flow in fractured rock masses is of special importance for estimating the permeability of rock fracture networks. When the flow velocity is small, the flow is in the linear regime and obeys the cubic law, whereas when the flow velocity is large, the flow is in the nonlinear regime and should be simulated by solving the complex Navier-Stokes equations. The critical conditions such as critical Reynolds number and critical hydraulic gradient are commonly defined in the previous works to quantify the onset of nonlinear fluid flow. This study reviews the simplifications of governing equations from the Navier-Stokes equations, Stokes equation, and Reynold equation to the cubic law and reviews the evolutions of critical Reynolds number and critical hydraulic gradient for fluid flow in rock fractures and fracture networks, considering the influences of shear displacement, normal stress and/or confining pressure, fracture surface roughness, aperture, and number of intersections. This review provides a reference for the engineers and hydrogeologists especially the beginners to thoroughly understand the nonlinear flow regimes/mechanisms within complex fractured rock masses.

Mesoscopic Rhelogical Model for Polymeric Media Flows

The paper compares hydrodynamic properties of three-dimensional flows of polymer melts. A modified Vinogradov and Pokrovskii rheological model is used for the mathematical description of nonlinear viscoelastic fluid flows in a planeparallel channel with a sudden convergence. Discrete analogs for partial differential equations were obtained via the control volume method separating physical processes. The numerical implementation is carried out using the GPU-based parallel computing technology. Velocity and pressure fields have been calculated for two samples of polyethylene melts and the circulating flow at the entrance of the slit channel is noticeable. It is shown that the size of the vortex zone depends significantly on melt rheology.

Influence of Geometry and Velocity of Rotating Solids on Hydrodynamics of a Confined Volume

Three cylinder-based geometries were evaluated at five different rotating speeds (ω = 20.94, 62.83, 94.25, 125.66, and 157.08 rad·s−1) to obtain the fluid flow pattern in nonsteady conditions. Two of the models were modified at the lower region, also known as tip section, by means of inverted and right truncated cone geometries, respectively. The experimental technique used a visualization cell and a Particle Imaging Velocimetry installation to obtain the vector field at the central plane of the volume. The Line Integral Convolution Method was used to obtain the fluid motion at the plane. In addition, the scalar kinetic energy and the time series were calculated to perform the normal probability plot. This procedure was used to determine the nonlinear fluid flow pattern. It was also used to identify two different flow regimens in physical and numerical results. As the rotation speed increased, the turbulent regions were placed together and moved. The process makes experimental observation difficult. The biphasic and turbulence constitutive equations were solved with the Computational Fluid Dynamics technique. Numerical results were compared with physical experiments for validation. The model with the inverted truncated cone tip presented better stability in the fluid flow pattern along the rotation speed range.

Quantitative Estimates of Normalized Transmissivity and the Onset of Nonlinear Fluid Flow Through Rough Rock Fractures

Quantitative Estimates of Normalized Transmissivity and the Onset of Nonlinear Fluid Flow Through Rough Rock Fractures

Experimental Investigation of Nonlinear Flow Characteristics of Real Rock Joints under Different Contact Conditions

AbstractLaboratory experiments were conducted on three artificial rock joints with natural characteristics to study the nonlinear fluid flow characteristics in a single joint under different contact conditions. A noncontact, three-dimensional stereotopometric measurement instrument was used to measure the three-dimensional surface morphology of rock joints before they were tested. Subsequently, the composite morphology parameters of rock joints under different contact conditions were calculated by using self-programmed software. The experimental results showed that both the Forchheimer’s equation (emphatically discussed) and Izbash’s equation could provide an excellent description for the nonlinear fluid flow in a single joint. The linear coefficient a and nonlinear coefficient b of the Forchheimer’s law both generally decreased, and such rate of decrease was gradually reduced along with increasing offset distance. The value of linear coefficient a approached zero and the value of coefficient n was close ...

Analytical solution of the Graetz problem for non-linear viscoelastic fluids in tubes of arbitrary cross-section

Graetz problem is solved analytically for steady laminar flow of non-linear viscoelastic fluids in straight tubes of arbitrary cross-section. The one-to-one mapping method used to model the cross-sectional geometry satisfies the no-slip and thermal boundary conditions for a wide range of arbitrary tube contours. Field variables are expanded in asymptotic series in terms of the Weissenberg number Wi, leading to a set of linear hierarchical equations which are solved successively up to and including the third order in Wi. Exact analytical solution for the Graetz problem for Newtonian fluids in round as well as arbitrary cross-sectional tubes is recovered at the lowest order. Secondary flows arise at the third order of the analysis as shown previously by Siginer and Letelier [1]. Their effect on the convection field is fully taken into account. The temperature distributions for triangular and square cross-sectional tubes are computed as particular cases of the general analysis together with the variation of the asymptotic Nusselt number for the triangular cross-section as a function of Wi to demonstrate the substantial enhancement of the heat transfer rates due to the non-linear viscoelastic constitutive structure of the fluid. The analytical algorithm presented is very versatile and can be easily applied to a wide spectrum of non-circular tube contours.

Influence of surface roughness on nonlinear flow behaviors in 3D self-affine rough fractures: Lattice Boltzmann simulations

This study investigates the impacts of surface roughness on the nonlinear fluid flow through three-dimensional (3D) self-affine rock fractures, whose original surface roughness is decomposed into primary roughness (i.e. the large-scale waviness of the fracture morphology) and secondary roughness (i.e. the small-scale unevenness) with a wavelet analysis technique. A 3D Lattice Boltzmann method (LBM) is adopted to predict the flow physics in rock fractures numerically created with and without consideration of the secondary roughness, respectively. The simulation results show that the primary roughness mostly controls the pressure distribution and fracture flow paths at a large scale, whereas the secondary roughness determines the nonlinear properties of the fluid flow at a local scale. As the pressure gradient increases, the secondary roughness enhances the local complexity of velocity distribution by generating and expanding the eddy flow and back flow regions in the vicinity of asperities. It was found that the Forchheimer's law characterizes well the nonlinear flow behavior in fractures of varying roughness. The inertial effects induced by the primary roughness differ only marginally in fractures with the roughness exponent varying from 0.5 to 0.8, and it is the secondary roughness that significantly enhances the nonlinear flow and leads to earlier onset of nonlinearity. Further examined were the effects of surface roughness on the transmissivity, hydraulic aperture and the tortuosity of flow paths, demonstrating again the dominant role of the secondary roughness, especially for the apparent transmissivity and the equivalent hydraulic aperture at high pressure gradient or high Reynolds number. The results may enhance our understanding of the role of surface roughness in the nonlinear flow behaviors in natural rock fractures.