Nonlinear Fluid Flow Research Articles

Quantitative evaluation of fracture geometry influence on nonlinear flow in a single rock fracture

The nonlinear flow behavior in a single rock fracture (SRF) has been studied for decades. However, the existing models to link fracture geometries and nonlinear flow behavior are less accurate due to insufficient characterization of fracture geometries. The primary goal of this study is to propose new parameters to characterize the geometric characteristics of a SRF, and then to evaluate the nonlinear flow behavior in a SRF. A series of high precision hydraulic tests are conducted on 50 mated and unmated artificial fractured samples. The results show that Forchheimer equation adequately describes the nonlinear fluid flow through a SRF with rough walls and variable apertures, and its nonlinear coefficients can be determined by quantifying the fracture geometries. By introducing the concept of a threshold aperture, meaning that only those zones with greater apertures than a threshold value are involved in the flow process, the heterogeneity of aperture distribution is quantified based on the cumulative distribution of individual apertures of sampling points. In addition, the influence of surface roughness on the nonlinear flow in a SRF is quantified with the standard deviation of secondary roughness. Lastly, an equation that relates the nonlinear coefficient to the newly proposed geometrical parameters is arrived, which is further used to link the friction factor, nonlinear flow model, and critical Reynolds number with the fracture geometric characteristics. The newly proposed equations in this study predict and characterize the nonlinear flow characteristics in a SRF very well.

The Experimental Investigation of the Development Potential of Low-Permeability Reservoirs in the Daqing Oilfield

It is very risky and difficult to develop low-permeability reservoirs, but reservoir development can be guided by the development potential of different low-permeability reservoirs. In this study, natural cores of the Daqing Oilfield were used as the research objects. The throat radius distributions of the different low-permeability cores were determined by the constant velocity mercury injection method, the movable fluid distribution characteristics were determined by nuclear magnetic resonance, and the nonlinear fluid flow characteristics were analyzed via fluid flow experimentation. From these data, the development potential for low-permeability reservoirs was determined. The results show that when the permeability is 1×10−3 μm2, the average throat radius is only approximately 0.9 μm and throats with radii less than 0.1 μm account for approximately 30% of the throats. The throats with an average radius less than 1 μm, especially throats with radii less than 0.1 μm, are the main factor restricting the fluid flow in these cores. The movable fluid is only approximately 20% of the fluid in a core, and the threshold pressure gradient reaches 0.15 MPa/m when the permeability is 1×10−3 μm2, indicating that it is more difficult to develop reservoirs with permeabilities less than 1×10−3 μm2.

Electrokinetic effects in helical flow of non-linear viscoelastic fluids

Helical flows of complex fluids through annular regions have been studied extensively for their large spectrum of applications ranging from extrusion to mud drilling. Such flows include an axial flow in the presence of cylinder rotations, and they become coupled for complex fluids because of their shear thinning nature. Flows in annular confinements also find important applications in porous media and many biological and chemical systems, where electrokinetic effects in the form of surface charge and Electrical Double Layers (EDLs) might also play important roles. However, studies on helical flows of complex fluids in the presence of such effects are rather scarce. As such, here, we investigate helical flows of non-linear viscoelastic fluids in the presence of electrokinetic influences. We show that it is possible to write semi-analytical solutions for the velocity and the stress fields, even when the effects of EDL are present. We demonstrate that electro-osmotic effects compound the influence of the cylinder rotation, which can rapidly accelerate the volume throughput, even for moderate viscoelasticity. Our analysis brings out novel scaling regimes between the axial throughput and viscoelasticity. We further analyze to probe into the streaming potential resulting from a purely pressure driven flow and show that similar to the volume throughput, angular rotation also helps the streaming potential grow rapidly, even when the fluid is weakly viscoelastic.

Utilization of modified Darcy's law in peristalsis with a compliant channel: applications to thermal science

This research addresses the combined influence of MHD and slip on the peristaltic flow of non-Newtonian Sisko fluid model in a symmetric compliant characteristic channel. The fluid under investigation fills the porous space in a channel. The flexible walls of the channel surface propagate sinusoidal along the channel. Using long wavelength and low Reynolds number assumptions, the governing non-linear fluid flow equations are derived and tackled using perturbation scheme. The series form of solution are computed for fluid thermal distribution, concentration profile and momentum field along with the rate of heat transfer coefficient. Effects of numerous significant parameters are studied and deliberated through graphs. Excellent mode for regulating flows are obtained for larger values of electrical and magnetic fields and such results are significantly important at the time of surgery operation. It is observed that momentum distribution is decreased by augmenting the values of the Sisko parameter. The blood velocity can be controlled by regulating the magnetic field strength. Shear- thinning and shear- thickening nature of Sisko fluid model is also explained through these graphs.

Data-driven modelling of nonlinear spatio-temporal fluid flows using a deep convolutional generative adversarial network

Deep learning techniques for fluid flow modelling have gained significant attention in recent years. Advanced deep learning techniques achieve great progress in rapidly predicting fluid flows without prior knowledge of the underlying physical relationships. However, most of existing researches focused mainly on either sequence learning or spatial learning, rarely on both spatial and temporal dynamics of fluid flows (Reichstein et al., 2019). In this work, an Artificial Intelligence (AI) fluid model based on a general deep convolutional generative adversarial network (DCGAN) has been developed for predicting spatio-temporal flow distributions. In deep convolutional networks, the high-dimensional flows can be converted into the low-dimensional “latent” representations. The complex features of flow dynamics can be captured by the adversarial networks. The above DCGAN fluid model enables us to provide reasonable predictive accuracy of flow fields while maintaining a high computational efficiency. The performance of the DCGAN is illustrated for two test cases of Hokkaido tsunami with different incoming waves along the coastal line. It is demonstrated that the results from the DCGAN are comparable with those from the original high fidelity model (Fluidity). The spatio-temporal flow features have been represented as the flow evolves, especially, the wave phases and flow peaks can be captured accurately. In addition, the results illustrate that the online CPU cost is reduced by five orders of magnitude compared to the original high fidelity model simulations. The promising results show that the DCGAN can provide rapid and reliable spatio-temporal prediction for nonlinear fluid flows.

Numerical Comparison for Focused Wave Propagation Between the Fully Nonlinear Potential Flow and the Viscous Fluid Flow Models

Numerical simulations on focused wave propagation are carried out by using three types of numerical models, including the linear potential flow, the nonlinear potential flow and the viscous fluid flow models. The wave—wave interaction of the focused wave group with different frequency bands and input wave amplitudes is examined, by which the influence of free surface nonlinearity and fluid viscosity on the related phenomenon of focused wave is investigated. The significant influence of free surface nonlinearity on the characteristics of focused wave can be observed, including the increased focused wave crest, delayed focused time and downstream shift of focused position with the increase of input amplitude. It can plot the evident difference between the results of the nonlinear potential flow and linear potential flow models. However, only a little discrepancy between the nonlinear potential flow and viscous fluid flow models can be observed, implying the insignificant effect of fluid viscosity on focused wave behavior. Therefore, the nonlinear potential flow model is recommended for simulating the non-breaking focused wave problem in this study.

Experimental study of non-linear fluid flow though rough fracture based on fractal theory and 3D printing technique

In this note, the non-linear fluid flow through rough fracture is investigated experimentally. The rough surface of fracture is generated synthetically based on the fractal dimension (D) and standard deviation (σ). 3D printing technique is used to produce the fracture specimen, in which, different components of the specimen are printed and assembled so that the fracture roughness and aperture can be easily controlled. After that, the self-designed apparatus is used to conduct the fluid flow test on the fractured specimens under various pressure gradients. It is found that, the surface roughness imposes an important impact on the nonlinear characteristics of fluid flow through fracture; and the root-mean-square of the first derivative of profile, Z2, is found effective in characterizing the fracture roughness. Based on the test results and the derived parameters in Forchheimer equation, an empirical equation is proposed to relate the hydraulic aperture (eh) to Z2 and mechanical aperture (em). Furthermore, the correlation among the nonlinear coefficient β, em and Z2 is established.

A New Numerical Study Method of Thermal Stress Distribution and Tortuosity Effectiveness in an Anode Porous Electrode for a Planar Solid Oxide Fuel Cell

A fuel cell is an electro-chemical tool capable of converting chemical energy into electricity. High operating temperature of solid oxide fuel cell, between 700oC to 1000oC, causes thermal stress. Thermal stress causes gas escape, structure variability and cease operation of the SOFC before its lifetime.The purpose of the current paper is to present a method that predicts the thermal stress distribution in an anisotropic porous anode of planar SOFC. The coupled governing non-linear differential equations, heat transfer, fluid flow, mass transfer, mass continuity, and momentum are solved numerically. A code based on computational fluid dynamics (CFD), computational structural mechanics and finite element method (FEM) is developed and utilized. The code uses the generated data inside the porous anode in order to detect the temperature and the stress distribution using the Darcy’s law and the Navier-Stokes equations. The numerical results used to govern the areas of high values of stresses were higher than the yield strength of materials. The results show that a highest thermal stress occurs at lower corners of the anode. The concentrated temperature occurs at the middle of the electrolyte-anode whereas the maximum pressure occurs at the middle of the upper and lower section of the anode.

Modeling linear and nonlinear fluid flow through sheared rough-walled joints taking into account boundary stiffness

Abstract This study modeled four hundred fluid flow cases through shear-induced aperture fields of a rough-walled joint with self-affine surface characteristics under different boundary stress and stiffness conditions by solving the Navier-Stokes equations. The results show that as the hydraulic gradient increases from 10−8 to 101, the fluid flow experiences changes characterized by a linear regime, a weak nonlinear regime and a strong nonlinear regime. The permeability is a constant in the linear regime and starts to decline when entering the weak nonlinear regime, and the declining rate gradually converges when reaching the strong nonlinear regime. Such transition is poorly indicated by the global Reynolds number alone; it is instead primarily governed by the magnitudes and distribution characters of local Reynolds numbers. The boundary stiffness has linear/nonlinear relations with the permeability, depending on the magnitude of the joint aperture, which is governed by surface roughness, joint compressive strength and mechanical boundary conditions. With increasing boundary stiffness from 0 to 2.0 GPa/m, the ratio of permeability in y-direction to that in x-direction increases from 2.49 to 26.49 by approximately a factor of 10, suggesting that anomalous flow may prevail in sheared zones with stiff surrounding confinements.

Modelling of Shear Banding in Plane Couette Flow of Non-linear Viscoelastic Fluids

We investigate shear banding phenomena in the plane Couette flow of viscoelastic fluids.The viscoelastic fluids are modelled viathe Giesekus constitutive equations with stress diffusion. The nonlinear and coupled systems of equations, comprising the momentum equation and the constitutive equations, are solved numerically using semi-implicit finite difference methods. Theeffects of the fluid parameters on the flow variables are also investigated. Under certain values of the material parameters, we observe formation of shear bands in plane Couette flow.

NIMFA Software Complex for Solving Problems of Nonlinear Single-Phase Fluid Flows and Heat-and-Mass Transport in Porous Media

NIMFA Software Complex for Solving Problems of Nonlinear Single-Phase Fluid Flows and Heat-and-Mass Transport in Porous Media

Non-linear effects in the oscillating drop method for viscosity measurements

The contribution of non-linear fluid flow effects to the damping of surface oscillations in the oscillation drop method was investigated in a series of experiments in an electromagnetic levitation device installed on the International Space station, ISS-EML. In order to correctly evaluate the damping time constant from measured surface oscillation decays the effect of a modulated signal response on measured surface oscillation decay curves was investigated. It could be shown that various experimentally observed signal patterns could be well represented by a modulated response. The physical origin of such modulations is seen in rotation and precession. Over a temperature range of 220 K covered by different surface oscillation excitation pulses with an initial sample shape deformation of 5 – 10% the amplitude of surface oscillations as a function of time could be very well represented by a Lamb type damping with a temperature dependent viscosity. A direct comparison of surface oscillation decay times measured in the same temperature range but for different oscillation amplitudes showed no non-linear contribution to the damping time constant with a confidence level better 10%.

Modeling hydrogen networks for future energy systems: A comparison of linear and nonlinear approaches

Common energy system models that integrate hydrogen transport in pipelines typically simplify fluid flow models and reduce the network size in order to achieve solutions quickly. This contribution analyzes two different types of pipeline network topologies (namely, star and tree networks) and two different fluid flow models (linear and nonlinear) for a given hydrogen capacity scenario of electrical reconversion in Germany to analyze the impact of these simplifications. For each network topology, robust demand and supply scenarios are generated. The results show that a simplified topology, as well as the consideration of detailed fluid flow, could heavily influence the total pipeline investment costs. For the given capacity scenario, an overall cost reduction of the pipeline costs of 37% is observed for the star network with linear cost compared to the tree network with nonlinear fluid flow. The impact of these improvements regarding the total electricity reconversion costs has led to a cost reduction of 1.4%, which is fairly small. Therefore, the integration of nonlinearities into energy system optimization models is not recommended due to their high computational burden. However, the applied method for generating robust demand and supply scenarios improved the credibility and robustness of the network topology, while the simplified fluid flow consideration can lead to infeasibilities. Thus, we suggest the utilization of the nonlinear model for post-processing to prove the feasibility of the results and strengthen their credibility, while retaining the computational performance of linear modeling.

Generalized decomposition method: Applications to nonlinear oscillator and MHD fluid flow past cone/wedge geometries

In this article, Adomian decomposition method has been modified and two of the recursive relations for nonlinear oscillator and the nonlinear ordinary differential equation that arise from the boundary layer flow problem has been applied. The MHD, electrically conducted, steady, incompressible, and boundary layer flow past a cone and a wedge problem is modeled and reduced into boundary value problem by applying similarity transformations. Four recursive relations have been constructed with some parameters and some additional functions. The rate of convergence of solutions series is enhanced by imposing the conditions of no secular terms in the solutions series. It is also to be noted that the way of choosing linear operator yields the exact solution of nonlinear boundary value problem in the case of flow past a wedge. Comparisons of the numerical solutions obtained from Matlab solver “bvp4c” with the approximate solutions obtained by the recursive relations of the present method are also presented.

Nonintrusive reduced order modeling framework for quasigeostrophic turbulence.

In this study, we present a nonintrusive reduced order modeling (ROM) framework for large-scale quasistationary systems. The framework proposed herein exploits the time series prediction capability of long short-term memory (LSTM) recurrent neural network architecture such that (1) in the training phase, the LSTM model is trained on the modal coefficients extracted from the high-resolution data snapshots using proper orthogonal decomposition (POD) transform, and (2) in the testing phase, the trained model predicts the modal coefficients for the total time recursively based on the initial time history. Hence, no prior information about the underlying governing equations is required to generate the ROM. To illustrate the predictive performance of the proposed framework, the mean flow fields and time series response of the field values are reconstructed from the predicted modal coefficients by using an inverse POD transform. As a representative benchmark test case, we consider a two-dimensional quasigeostrophic ocean circulation model which, in general, displays an enormous range of fluctuating spatial and temporal scales. We first demonstrate that the conventional Galerkin projection-based reduced order modeling of such systems requires a high number of POD modes to obtain a stable flow physics. In addition, ROM-Galerkin projection (ROM-GP) does not seem to capture the intermittent bursts appearing in the dynamics of the first few most energetic modes. However, the proposed nonintrusive ROM framework based on LSTM (ROM-LSTM) yields a stable solution even for a small number of POD modes. We also observe that the ROM-LSTM model is able to capture quasiperiodic intermittent bursts accurately, and yields a stable and accurate mean flow dynamics using the time history of a few previous time states, denoted as the lookback time window in this paper. We show several features of ROM-LSTM framework such as significantly higher accuracy than ROM-GP, and faster performance using larger time step size. Throughout the paper, we demonstrate our findings in terms of time series evolution of the field values and mean flow patterns, which suggest that the proposed fully nonintrusive ROM framework is robust and capable of predicting chaotic nonlinear fluid flows in an extremely efficient way compared to the conventional projection-based ROM framework.

Predictor–Corrector Nodal Integral Method for simulation of high Reynolds number fluid flow using larger time steps in Burgers ’ equation

Nodal Integral Methods (NIM), although frequently used for solving neutron transport equations, have not found wider acceptance for solving fluid flow problems. One of the pivotal reasons behind this is the lack of efficient non-linear solvers for the system of coupled algebraic equations obtained by these methods, thus making such methods prohibitively expensive for highly nonlinear (higher Reynolds number) fluid flow problems. Successive improvements in earlier attempts resulted in an improved version called Modified NIM (MNIM) scheme; following which, a further modified version appeared (called as M2NIM), that used the concept of the delayed coefficients. Upon comparing the solutions obtained using MNIM and M2NIM schemes, respectively, it was observed that although the modified scheme with delayed coefficients (M2NIM) has significantly faster convergence, it is less accurate. The convergence and accuracy are very likely to suffer to a significant extent in the high Reynolds number fluid flows, especially when a large time-step is to be considered. In order to resolve these difficulties, a new type of physics-based predictor–corrector algorithm is proposed in the present study. In the proposed computational algorithm, the linearized M2NIM scheme is used to predict the solution, whereas the MNIM scheme is utilized to improve the predictive guess. The novelty of such a hybrid numerical algorithm comes from the fact that it combines the advantage of the faster convergence of M2NIM with the accuracy of the MNIM scheme. The algorithm also benefits from the unique inclusion of Jacobian-free Newton-Krylov (JFNK) method, which helps in getting rid of the formation of large Jacobian matrices, thereby reducing unnecessary computational overhead. The proposed methodology has been applied to solve a non-linear convection–diffusion problem, represented by the Burgers’ equation. The computational results for bothone-dimensional and two-dimensional Burgers’ equation are presented to demonstrate the effectiveness of the developed novel algorithm, as well as, the advantage it offers over the existing numerical methods.

Sparse feature map-based Markov models for nonlinear fluid flows

Data-driven Markov linear models of nonlinear fluid flows using maps of the state into a sparse feature space are explored in this article. The underlying principle of low-order models for fluid systems is identifying maps to a feature space where the system evolution (a) is simpler and efficient to model accurately and (b) the state can be recovered accurately from the features through inverse mapping. Such methods are useful when real-time models are needed for online decision making from sensor data. The Markov linear approximation is popular as it allows us to leverage the well established linear systems machinery. Examples include the Koopman operator approximation techniques and evolutionary kernel methods in machine learning. The success of these models in approximating nonlinear dynamical systems is tied to the effectiveness of the feature map in accomplishing both (a) and (b) above as long as the system provides a feasible prediction horizon using data. We assess this by performing an in-depth study of two different classes of sparse linear feature transformations of the state: (i) a pure data-driven POD-based projection that uses left singular vectors of the data snapshots – a staple of common Koopman approximation methods such as Dynamic Mode Decomposition (DMD) and its variants such as extended DMD; and (ii) a partially data-driven sparse Gaussian kernel (sGK) regression (a mean sparse Gaussian Process (sGP) predictor). The sGK/sGP regression equivalently represents a projection onto an infinite-dimensional basis characterized by a kernel in the inner product reproducing kernel Hilbert space (RKHS). We are particularly interested in the effectiveness of these linear feature maps for long-term prediction using limited data for three classes of fluid flows with escalating complexity (and decreasing prediction horizons) starting from a limit-cycle attractor in a cylinder wake followed by a transient wake evolution with a shift in the base flow and finally, a continuously evolving buoyant Boussinesq mixing flow with no well-defined base state. The results indicate that a purely data-driven POD map is good for full state reconstruction as long as the basis remains relevant to the predictions whereas the more generic sparse Gaussian Kernel (sGK) basis is less sensitive to the evolution of the dynamics but prone to reconstruction errors from lack of parsimony. Contrastingly, the sGK-maps outperform POD-based maps in learning the transient nonlinear evolution of the state for the same feature dimension in systems that contain a well-defined attractor(s). Consequently, both POD and sGK-maps require additional layer(s) to help mitigate these shortcomings. For example, POD-maps require nonlinear functional extensions for improved feature space predictions whereas sGK-maps require dimensionality reduction to balance the large feature dimension needed for accurate full state reconstruction. However, both classes of multilayer feature maps fail to predict the highly evolving buoyant mixing flow for very different reasons.

Corrigendum to “A Review of Critical Conditions for the Onset of Nonlinear Fluid Flow in Rock Fractures”

Corrigendum to “A Review of Critical Conditions for the Onset of Nonlinear Fluid Flow in Rock Fractures”

Resonance in laminar pipe flow of non-linear viscoelastic fluids

Resonance phenomenon is investigated in pulsating flows in straight circular tubes for the class of constitutively non-linear affine viscoelastic fluids represented by the Johnson–Segalman constitutive model. The relaxation response of non-linear viscoelastic fluids associated with the natural frequencies of oscillation that arise as a consequence of the structure of the constitutive equation is explored. An analytical solution, which circumvents the high-Weissenberg number problem, which plagues numerical solutions of highly viscoelastic fluids, based on an asymptotic expansion in terms of a small material parameter is developed. At the lowest order of the asymptotic expansion the velocity field of the Maxwell fluid in round tubes is recovered, therefore providing validation of the computations. The analytical solution allows insights into the high Weissenberg number behavior of highly elastic fluids and thus avoids the difficulties the numerical solutions of high Weissenberg number problems are fraught with such as instability and lack of convergence. The analysis reveals that the forcing frequency associated with the pressure gradient generates a sequence of resonances of rapidly decaying intensity with increasing forcing frequency for a fixed value of the Weissenberg number. The intensity of the resonance is a rapidly increasing function of the increasing Weissenberg numbers, in particular at the first natural frequency. The latter is a decreasing function of the increasing elasticity of the fluid that is of the increasing Weissenberg numbers. The intensity of the resonance at the first natural frequency is a rapidly increasing function of the increasing elasticity. We show the existence of a limit point for the enhancement in energy savings with increasing elasticity. The effect of resonance on the flow rate, maximum amplitude of the average velocity, the mean flow rate and the oscillating velocity field is explored.

Transversal field and heat transfer in the flow of non-linear viscoelastic fluids in tear-drop shaped tubes including viscous dissipation

Steady, laminar flow of a class of non-linear viscoelastic fluids in tear-drop shaped ducts is investigated for constant heat flux at the wall including viscous dissipation. Field equations are solved analytically through expanding the field variables in double asymptotic series in terms of the Weissenberg number and a mapping parameter coupled with the shape factor method, a continuous and one-to-one mapping, to map the circular contour onto the tear-drop shaped tube boundary. Analytical solution for the transversal and axial velocity fields is derived for the first time, and the influence of the viscoelastic constitutive parameters in shaping the transversal vortex structure is discussed in detail. The temperature field and the dependence of the Nusselt Nu = f (Br, Pe, Re, We) number on the Brinkman Br, Péclet Pe, Reynolds Re and Weissenberg We numbers as well as combinations of the material parameters are investigated. At the lowest order in the Weissenberg number the velocity and temperature fields of Newtonian fluids in tear-drop shaped straight tubes as well as in straight round tubes with viscous dissipation are recovered. The approach the analysis is based on can be easily adapted to yield the velocity and temperature fields of Newtonian and constitutively non-linear viscoelastic fluids in non-circular tubes other than tear-drop shaped including viscous dissipation.