One-dimensional Flow Problems Research Articles

Numerical analysis of deformation and failure mechanism of metal truss structure of spacecraft under re-entry aerothermodynamic environment

A coupled aerodynamic-thermoelastic analysis framework is presented to simulate the deformation and failure mechanism of metal truss structure for spacecraft under re-entry aerothermodynamic environment. The three-dimensional dynamic thermo-mechanical coupling finite element algorithm is developed by computable modeling of thermoelasticity and heat conduction equations to solve the thermo-mechanical coupling response of spacecraft structure. The aerodynamic boundary conditions such as temperature, heat flux and pressure distribution are computed and provided by the gas-kinetic unified algorithm (GKUA) based on the Boltzmann model equation. The linear interpolation method is designed and employed to couple the boundary conditions between the external aerodynamic flow field and the solid structure interface. The present thermo-mechanical coupling algorithm is verified on one-dimensional and two-dimensional transient heat flow problems with analytical solutions, and good consistency is achieved. The reliability of the GKUA for solving the boundary conditions of the external flow field and the dynamic thermo-mechanical coupling algorithm for the thermal response behavior of the internal structure material is verified by the consistency calculation of the internal and external temperature distribution. Then, the aerodynamic-thermoelastic analysis framework is further applied to study the deformation behavior of structural thermo-mechanical responses and failure mechanism of a hollow sphere under re-entry aerothermodynamic environment with emphasis placed on the influence of flight speeds, rarefied effects, algorithm schemes and damping effects. The numerical results obtained by the presented algorithm including the internal temperature, thermal stress and deformation distribution law are compared and analyzed. Finally, the deformation and failure mechanism of spacecraft structure during re-entry process are solved and analyzed by the present algorithm framework. The displacement, temperature and stress distribution contours of the structure are presented to make a preliminary understanding and evaluation on the process of spacecraft deformation and failure disintegration. It is indicated that the present coupled aerodynamic-thermoelastic analysis framework can provide a reliable, applicable and efficient platform for the thermo-mechanical coupling responses and failure mechanism of re-entry spacecraft in the end of life.

On the Viability of Quantum Annealers to Solve Fluid Flows

This paper explores the suitability of upcoming novel computing technologies, particularly adiabatic annealing based quantum computers, to solve fluid dynamics problems that form a critical component of several science and engineering applications. For our experiments, we start with a well-studied one-dimensional simple flow problem, and provide a framework to convert such problems in continuum to a form amenable for deployment on such quantum annealers. Since the DWave annealer returns multiple states sampling the energy landscape of the problem, we explore multiple solution selection strategies to approximate the solution of the problem. We analyze the continuum solutions obtained both qualitatively and quantitatively as well as their sensitivities to the particular solution selection scheme.

Analytical solution for supercritical flow in rectangular channel contraction

Abstract Analysis of the supercritical flow pattern in nonprismatic open channels is an important issue in engineering practice and often a challenging question principally due to the oblique standing wave structure. In this paper, an original analytical solution for supercritical free-surface flow in rectangular channel contraction is proposed. The two-dimensional steady depth-averaged shallow water equations are simplified to an asymptotic one-dimensional (1D) unsteady flow problem in the transverse direction. A first-order approximation is then obtained using small perturbation theory, and the 1D asymptotic model is solved analytically using the Laplace integral transformation. The two-dimensional flow field solution is obtained following a compilation of sequence planes in the time domain. Free-surface profiles along the side wall and the channel axis were compared against experimental data. For this, laboratory tests were performed in addition to a comparison with classical experimental results available in the literature. Within the limits underlying the model, good agreement in water surface profiles is obtained by comparing experimental and theoretical predictions. Principal flow features, such as wave amplitude and positions, are well captured, which makes the solution model of interest for the design and analysis of contracted open channels.

Riemann problem for constant flow with single-point heating source

This work focuses on the Riemann problem of Euler equations with global constant initial conditions and a single-point heating source, which comes from the physical problem of heating one-dimensional inviscid compressible constant flow. In order to deal with the source of Dirac delta-function, we propose an analytical frame of double classical Riemann problems (CRPs) coupling, which treats the fluids on both sides of the heating point as two separate classical Riemann problems and then couples them. Under the double CRPs frame, the Riemann solution is self-similar, and three types of structures are found. The theoretical analysis is also supported by the numerical simulation. Furthermore, the uniqueness of the Riemann solution is established with some restrictions on the Mach number of the initial condition.

Solution of One-Dimensional Contaminant Flow Problem Incorporating the Zero Order Source Parameter by Method of Eigen-Functions Expansion

A semi – analytical study of a time dependent one – dimensional advection – dispersion equation (ADE) with Neumann homogenous boundary conditions for studying contaminants flow in a homogenous porous media is presented. The governing equation which is a partial differential equation incorporates the advection, hydrodynamic dispersion, first order decay and a zero order source effects in the model formulation. The velocity of the flow was considered exponential in nature. The solution was obtained using Eigen function expansion technique after a suitable transformation. The results which investigate the effect change in the parameters on the concentration was discussed and represented graphically. The study revealed that as the zero order source coefficient increases, the contaminant concentration decreases with time.

Pulsating Flow of a Viscous Fluid over a Cavity Containing a Compressible Gas Bubble

A two-dimensional pulsating flow of a viscous fluid in a plane channel whose wall has rectangular microcavities partially or completely filled with a compressible gas is investigated. This problem formulation can clarify the friction reduction mechanism in a laminar sublayer of a turbulent viscous boundary layer flow over a textured stripped superhydrophobic surface containing periodically arranged rectangular micro-cavities filled with gas. It is assumed that the dimensions of the cavities are much smaller than the channel thickness. On the macroscale, the problem of one-dimensional unsteady viscous flow in a plane channel with no-slip conditions on the walls and a harmonic variation of the pressure difference is solved. The solution obtained in this way is used for formulating non-stationary in time and periodic in space boundary conditions for the flow on the scale of a chosen cavity (microscale), with the instantaneous volume of the gas bubble in the cavity depending on the instantaneous pressure over the cavity. The flow on the microscale near a cavity with a gas bubble occurs in the Stokes regime. The numerical solution is obtained using an original version of the boundary element method. A parametric numerical study of the flow field in a pulsating shear flow over a cavity with a compressible gas bubble is performed. The averaged parameters characterizing the effective ‘velocity slip’ of viscous fluid and the friction reduction in a pulsating flow over a stripped superhydrophobic surface are calculated.

A suitability analysis of transient one-dimensional two-fluid numerical models for simulating two-phase gas-liquid flows based on benchmark problems

Gas-liquid two-phase flows in pipes are present in a variety of engineering applications. These applications require transient one-dimensional mathematical models coupled with robust numerical methods to simulate the flow behavior in time and space. As there exist several possible combinations of models and numerical methods, questions arise as to the suitability of these numerical models to best accommodate the problem features. Aiming to partially answer this question, we present in this work numerical simulations of two well-known gas-liquid benchmarks, the water faucet, and the shock tube problems, carried out with two two-fluid models – 4E1P (four equations and one pressure) and 5E2P (five equations and two pressures) – in association with three numerical methods – FCT (Flux-Corrected Transport method), FORCE (First-Order Centered Scheme) and ModFORCE. The numerical methods used herein are written in a general form and do not require the solution of the associated Riemann problem. The obtained results are sought to serve as a reference to choose the best approaches, among the ones studied, to tackle some transient one-dimensional two-phase flow problems, highlighting the advantages and disadvantages of each model and method combination regarding hyperbolicity, diffusive and dispersive effects. Overall, the combination of the 5E2P model with the ModFORCE numerical method provides the most accurate results compared to the analytical solutions available.

Automatic differentiation approach for solving one-dimensional flow and heat transfer problems

Traditional reactor system analysis tools are confronted with challenging difficulties in model development, low accuracy, and poor convergence. To solve these problems, the automatic differentiation (AD) method that allows the automatic numerical calculation of derivatives of functions was adopted to develop the reactor system code in this paper. For the simulation of single-phase models, the steady and transient responses were presented to investigate the effects of the spatial and temporal discretization schemes on modeling accuracy and efficiency. Meanwhile, the comparison of convergence performance between the automatic differentiation using operator overloading (ADOO) and the traditional hand-coded method was completed. Further, in the case of two-phase flow problems, the high-order discrete schemes were applied in this code. It was demonstrated that the reactor system code with single-phase and five-equation two-phase flow models, which adopted the high-order discretization and the ADOO method, performed very well for one-dimensional flow and heat transfer problems.

Modeling shallow geological flows on steep terrains using a specific differential transformation

Shallow geological flows such as landslides, debris flows, and floods are frequently modeled mathematically in the depth-averaged framework. As the bed topography becomes steep and irregular, the classical shallow-water model loses accuracy. How to handle bed topography and gravity effects in depth-averaged models becomes important, and the validity of the model needs verifying. In this study, a differential transformation called the G2L transformation linking the global height to the local depth of the surface material is proposed. This transformation helps to reconstruct the landslide/avalanche surface for one-dimensional and two-dimensional terrains that are obviously curved as well as real landslide examples such as the Huangtian and Rongsong landslides with shallowness ratios as high as 1/4. With this G2L transformation and closures of the lateral and bed stress, two versions of a model governing the geological flow dynamics are developed to avoid difficulties in terrain treatment when using differential geometry. Version 1 of the model is a coupled hyperbolic–differential system; version 2 is a modified hyperbolic system. The two proposed models are verified by a simple one-dimensional granular flow problem on a steep slope. Both mitigate problems associated with computational differential geometry of geological flows and the singularity problem of shallow-water models. In particular, the reduced gravity of several current models in landslide dynamics is a special instance of version 2. The two proposed models are well able to mathematically model the gravity geological hazards and may provide the tools required for prediction and mitigation of natural geological hazards.

Transient pressure behavior for unconventional gas wells with finite-conductivity fractures

Horizontal wells and hydraulic fracturing technologies are the only way that makes commercial production from the low and ultra-low permeability unconventional plays possible. The development of analytical and semi-analytical tools to analyze transient behavior of multi-fractured horizontal gas wells have been largely biased towards solving the simplified problem of one-dimensional (1D) Cartesian gas flow in matrix domain, which are only applicable to the production from infinite-conductivity fractures. This study presents a novel semi-analytical method for transient behavior analysis of horizontal gas wells producing from finite-conductivity fractures. Unlike the available transient analysis tools for finite-conductivity fractures which are developed based on empirical and approximate solution, the proposed solution is rigorously derived by solving the governing gas diffusivity equations written for gas flow in two contiguous regions of matrix and fracture simultaneously. Nonlinear, pressure-dependent gas properties remaining in pseudopressure-based diffusivity equations—namely viscosity and compressibility—are rigorously and straightforwardly captured in both fracture and matrix systems. The validity of proposed solution is verified against commercial numerical simulator (COMSOL Multiphysics®) for a series of synthetic cases considering constant and variable -rate production constraints. The case studies presented in this paper also showcase the applicability of proposed solution to the analysis of transient behavior of multi-fractured horizontal gas wells in shale plays under the following two state-of-the-art interpretations: 1) production from finite-conductivity hydraulic fractures in a single-porosity matrix; and 2) production from infinite-conductivity hydraulic fractures in a naturally-fractured matrix.

Application of the Chebyshev spectral method to the simulation of groundwater flow and rainfall-induced landslides

In this paper, the potential of the Chebyshev spectral method (CSM) is examined for the numerical solution of time-dependent variably saturated Darcian flow problems described by the Richards equation (RE). Generally, the computational efficiency of the traditional finite difference method (FDM) is not high because it requires a high mesh density to improve accuracy. Therefore, the Chebyshev spectral method (CSM) was extended to model groundwater transient flow in unsaturated soils. Furthermore, the Gardner model (soil-water characteristic curve) was adopted to linearize the RE for an inclined slope. The test results demonstrated that the accuracy, efficiency, and robustness of the CSM are better than those of the traditional FDM. A comparison of the analytical solutions of each method showed that the computational accuracy (L2(h*)) of the CSM was less affected by the grid size than that of the FDM. Simultaneously, the CSM was not sensitive to the initial conditions. The results demonstrated that the numerical accuracy of the CSM was stable in the order of 10−6 to 10−7, whereas that of the FDM was in the order of 10−3 to 10−6; it achieved higher accuracy with fewer grid points. The CSM was successfully applied to solve the one-dimensional transient flow problem on unsaturated soil slopes. The numerical results indicated that the proposed method solved the transient flow problem related to rainfall-induced landslides with high accuracy. This proposed approach can be developed to analyze rainfall-induced landslides.

Approximate analytical solution for supercritical flow in rectangular curved channels

In this study, we present an asymptotical mathematical model and an analytical solution for a supercritical flow in curved rectangular open channels. An original approach is proposed for solving the free-surface configuration and features of the flow in the presence of cross shock waves. The two-dimensional steady depth-averaged shallow water equations are transformed into an equivalent one-dimensional (1D) unsteady flow problem and a first order approximation is then obtained using small perturbation theory. Furthermore, the 1D asymptotic model is solved analytically by Laplace integral transformation and the two-dimensional flow field solution is reconstructed according to the translating planes. The free-surface profile along the outer chute wall and downstream channel was compared with the available experimental data, and the results indicated the satisfactory agreement of the maximum flow depth, peak positions, and wavelength. The proposed approach provides accurate predictions of the flow features and it facilitates the safe design of curved channel transitions.

Exact analytical solutions of non-Darcy seepage flow problems of one-dimensional Bingham fluid flow in finite long porous media with threshold pressure gradient

In the paper, the study on the exact analytical solution of the moving boundary problem of one-dimensional Bingham fluid seepage flow is extended from the infinite long porous media (Liu et al., 2012) to the finite long porous media. Two exact analytical solutions are presented by appropriately relying on some methods of mathematical physics and mathematical techniques. One is for the model with finite closed outer boundary condition; the other is for the model with finite constant pressure outer boundary condition. The existence and the uniqueness of the exact analytical solutions are also strictly proved theoretically. In addition, the numerical solutions of the two models by the finite difference method are also provided. Through the comparison, it is found that these exact analytical solutions have very excellent agreement with the numerical solutions although few terms of the infinite function series existent in the exact analytical solutions have to be retained for the calculation. Furthermore, for the two models, the effect of the threshold pressure gradient on the transient pressure and the transient pressure derivative at the inner boundary for the whole flow process is analyzed through the analytical solutions. Finally, through the comparison of the relevant model solutions, it is concluded that it is very necessary to incorporate the process of moving boundary for the modeling of non-Darcy Bingham fluid flow in finite long porous media with threshold pressure gradient; otherwise, large errors can be introduced in predicting the transient pressure and the transient pressure derivative in the porous media. The presented work can support solid theoretical foundations for the experiment design of measuring the threshold pressure gradient and the pressure transient analysis in the field of inverse problems in the petroleum engineering, which have been widely involved in the development of low-permeable oil reservoirs and heavy oil reservoirs.

Coupled reactive flow and dissolution with changing reactive surface and porosity

Mineral dissolution flows in porous media occur in numerous industrial and natural processes. We investigate the effects of varying rock-liquid interface on mineral dissolution transport in porous media. The one-dimensional mineral-dissolution flow problem that accounts for varying reacting interface and porosity is essentially non-linear. However, a novel exact solution is derived. The exact solution reveals a four-zone structure of the flow pattern with typical mineral concentration curves in all zones. The exact solution allows for a simplified inverse solver, facilitating determination of the surface function from laboratory reactive flow tests. Accounting for surface area evolution in the governing system of equations allows for significant improvement of matching the experimental data if compared with the constant-surface model. Moreover, the comparison between the analytical model and laboratory data reveals high agreement. The values of equilibrium mineral concentration as obtained from the matching and by thermodynamic calculations exhibit close agreement.

Optimising the heat balance integral method in spherical and cylindrical Stefan problems

The Heat Balance Integral Method (HBIM) is generally applied to one-dimensional Cartesian heat flow and Stefan problems. The main reason for this being that solutions in spherical and cylindrical coordinates are less accurate than in Cartesian. Consequently, in this paper we examine the application of the HBIM to Stefan problems in spherical and cylindrical coordinates, with the aim of improving accuracy. The standard version as well as one designed to minimise errors will be applied on the original and transformed system. Results are compared against numerical and perturbation solutions. It is shown that for the spherical case it is possible to obtain highly accurate approximate solutions (more accurate than the first order perturbation for realistic values of the Stefan number). For the cylindrical problem the results are significantly less accurate.

Non-monotonic permeability variation during colloidal transport: Governing equations and analytical model

Permeability decline associated with the migration of natural reservoir fines impairs the well index of injection and production wells in aquifers and oilfields. In this study, we perform laboratory corefloods using aqueous solutions with different salinities in engineered rocks with different kaolinite content, yielding fines migration and permeability alteration. Unusual permeability growth has been observed at high salinities in rocks with low kaolinite concentrations. This has been attributed to permeability increase during particle detachment and re-attachment of already mobilised fines by electrostatic attraction to the rock in stagnant zones of the porous space. We refine the traditional model for fines migration by adding mathematical expressions for the particle re-attachment rate, particle detachment with delay relative to salinity decrease, and the attached-concentration-dependency of permeability. A one-dimensional flow problem that accounts for those three effects allows for an exact analytical solution. The modified model captures the observed effect of permeability increase at high water salinities in rocks with low kaolinite concentrations. The developed model matches the coreflooding data with high accuracy, and the obtained model coefficients vary within their usual intervals.

Semi-analytical study of a one-dimensional contaminant flow in a finite medium

The Bubnov-Galerkin weighted residual method was used to solve a one-dimensional contaminant flow problem in this paper. The governing equation of the contaminant flow, which is characterized by advection, dispersion and adsorption was discretized and solved to obtain the semi-analytical solution. The adsorption isotherm was assumed to be of Freudlich type. The results obtained were expressed in graphical form to show the effect of change in the parameters on the concentration of the contaminants. From the analysis of the results, it was discovered that the contaminant concentration decreases with increase in the distance from the origin as the dispersion and velocity coefficient decrease.Keywords: Adsorption, advection, Bubnov-Galerkin weighted residuals, contaminant, dispersion

A space-time adaptive method for reservoir flows: formulation and one-dimensional application

This paper presents a space-time adaptive framework for solving porous media flow problems, with specific application to reservoir simulation. A fully unstructured mesh discretization of space and time is used instead of a conventional time-marching approach. A space-time discontinuous Galerkin finite element method is employed to achieve a high-order discretization on the anisotropic, unstructured meshes. Anisotropic mesh adaptation is performed to reduce the error of a specified output of interest, by using a posteriori error estimates from the dual-weighted residual method to drive a metric-based mesh optimization algorithm. The space-time adaptive method is tested on a one-dimensional two-phase flow problem, and is found to be more efficient in terms of computational cost (degrees-of-freedom and total runtime) required to achieve a specified output error level, when compared to a conventional first-order time-marching finite volume method and the space-time discontinuous Galerkin method on structured meshes.

Two-dimensional numerical modeling of dam-break flow using a new TVD finite-element scheme

A new numerical scheme based on the finite-element method with a total-variation-diminishing property is developed with the aim of studying the shock-capturing capability of the combination. The proposed model is formulated within the framework of the one-step Taylor–Galerkin scheme in conjunction with the Van Leer’s limiter function. The approach is applied to the two-dimensional shallow water equations by different test cases, i.e., the partial, circular, and one-dimensional dam-break flow problems. For the one-dimensional case, the sub- and supercritical flow regimes are considered. The results are compared with the analytical, finite-difference, and smoothed particle hydrodynamics solutions in the literature. The findings show that the proposed model can effectively mask the sources of errors in the abrupt changes of the flow conditions and is able to resolve the shock and rarefaction waves where other numerical models produce spurious oscillations.

Nested sparse grid collocation method with delay and transformation for subsurface flow and transport problems

In numerical modeling of subsurface flow and transport problems, formation properties may not be deterministically characterized, which leads to uncertainty in simulation results. In this study, we propose a sparse grid collocation method, which adopts nested quadrature rules with delay and transformation to quantify the uncertainty of model solutions. We show that the nested Kronrod–Patterson–Hermite quadrature is more efficient than the unnested Gauss–Hermite quadrature. We compare the convergence rates of various quadrature rules including the domain truncation and domain mapping approaches. To further improve accuracy and efficiency, we present a delayed process in selecting quadrature nodes and a transformed process for approximating unsmooth or discontinuous solutions. The proposed method is tested by an analytical function and in one-dimensional single-phase and two-phase flow problems with different spatial variances and correlation lengths. An additional example is given to demonstrate its applicability to three-dimensional black-oil models. It is found from these examples that the proposed method provides a promising approach for obtaining satisfactory estimation of the solution statistics and is much more efficient than the Monte-Carlo simulations.