Effect of quadratic pressure gradient term on a one-dimensional moving boundary problem based on modified Darcy’s law

A relatively high formation pressure gradient can exist in seepage flow in low-permeable porous media with a threshold pressure gradient, and a significant error may then be caused in the model computation by neglecting the quadratic pressure gradient term in the governing equations. Based on these concerns, in consideration of the quadratic pressure gradient term, a basic moving boundary model is constructed for a one-dimensional seepage flow problem with a threshold pressure gradient. Owing to a strong nonlinearity and the existing moving boundary in the mathematical model, a corresponding numerical solution method is presented. First, a spatial coordinate transformation method is adopted in order to transform the system of partial differential equations with moving boundary conditions into a closed system with fixed boundary conditions; then the solution can be stably numerically obtained by a fully implicit finite-difference method. The validity of the numerical method is verified by a published exact analytical solution. Furthermore, to compare with Darcy’s flow problem, the exact analytical solution for the case of Darcy’s flow considering the quadratic pressure gradient term is also derived by an inverse Laplace transform. A comparison of these model solutions leads to the conclusion that such moving boundary problems must incorporate the quadratic pressure gradient term in their governing equations; the sensitive effects of the quadratic pressure gradient term tend to diminish, with the dimensionless threshold pressure gradient increasing for the one-dimensional problem.     

低渗透多孔介质渗流动边界模型的解析与数值解

考虑启动压力梯度的低渗透多孔介质非达西渗流模型属于强非线性动边界问题, 分别利用相似变量变换方法和基于空间坐标变换的有限差分方法, 对内边界变压力情况下、考虑启动压力梯度的一维低渗透多孔介质非达西渗流动边界模型进行了精确解析与数值求解研究. 研究结果表明:该动边界模型存在唯一的精确解析解, 且所求得的精确解析解可严格验证数值解的正确性;且当启动压力梯度值趋于零时, 非达西渗流动边界模型的精确解析解将退化为达西渗流情况下的精确解析解. 由求解结果作出的非零无因次启动压力梯度下的地层压力分布曲线表现出紧支性特点, 其与达西渗流模型的有显著不同. 因此, 研究低渗透多孔介质中非稳态渗流问题时, 应该考虑动边界的影响. 研究内容完善了低渗透多孔介质的非达西渗流力学理论, 为低渗透油气藏开发的试井解释与油藏数值模拟技术提供了理论基础.      

Numerical study on effects of check valve closure flow conditions on pressure surges in pumping station with air entrainment

This paper proposes a numerical procedure to better compute the characteristics of pressure surges when check valves close under different flow conditions in a pumping station. Studies have shown that the effects of check valve closure on the pressure transients are predominantly dependent on the magnitude and gradient of the flow velocities immediately downstream of the check valve at the instant of valve closure. Through the present study, it was noted that the transient flow velocities near the check valve of a fluid system are also dependent on the characteristics of the air entrained into the fluid system. An improved numerical computational procedure for the fluid system with air entrainment under different transient conditions downstream of the check valve is also proposed in this paper. With a fluid system operating within the critical range of air entrainment values, the present analysis showed that there is a possibility of ‘high pressure surges’ when the check valves were closed at flow rates other than the positive flow conditions. This phenomenon was confirmed through field observations. This study thus concludes that a detailed numerical transient analysis of the fluid system, with various assumed amounts of entrained air, is necessary whenever there is the possibility of air entrainment into the fluid system, and that the flow conditions at the instant of check valve closure need to be modelled. Copyright © 2001 John Wiley & Sons, Ltd.     

Degassing of water-solved gases in thin pipings during fast pressure drops

 A one-dimensional model is presented, which describes the transient two-phase flow in thin pipes during fast pressure drops and degassing by use of Eulerian and Lagrangian systems. The reduction in dimension is obtained by introduction of a geometry model for bubbly and slug flow regimes. The complete model includes the transient two-phase flow, bubble formation and bubble growth. The flow model predicts rising velocities of bubbles and plugs in arbitrary inclined highly accurate pipes. The mass transfer (diffusion) of the dissolved phase is calculated by the bubble growth model. The quality of the model was examined by simulation of experimental series, whereby water was depressurised from the saturation pressure of the dissolved gas mixture (air), by variation of saturation pressure, pressure gradient and pipe geometry. The results of numerical simulation fit the experimental data well. Received on 17 January 2000     

CE/SE方法在内弹道瞬态两相流中的应用

根据CE/SE方法思想,构造在变换坐标系下内弹道两相流动数值计算的 格式. 采用此格式研究了膛内瞬态二维两相流动点火过程. 数值计算结果反映实际 规律并和试验结果符合. 结果表明,对中心点火方式,在点传火过程初期,膛内压力在 径向和轴向上存在较大的压力梯度,当火药全部着火后,药室内压力在径向上逐渐趋于 一致. 这表明CE/SE方法能够完全模拟瞬态二维两相流动的情况.     

Numerical study of adverse pressure gradient generation over a flat plate using a rotating cylinder

Generating an adverse pressure gradient (APG), using a rotating cylinder in the proximity of a plane wall under a laminar freestream flow, is studied numerically in this work. The magnitude of the generated APG is a function of the gap, G, between the cylinder and the wall, and the rotational speed of the cylinder, Ω. The flow in such a configuration is characterized by periodic transient vortex shedding at high Reynolds number. A numerical model for the computation of the transient flow for this configuration is developed using the ANSYS CFD simulation tool. The model is validated against published experimental and numerical data for similar flow configurations and excellent agreement is observed. A parametric study is carried out for different combinations of G and Ω for two different Reynolds numbers of 200 and 1000 to examine the development of the resulting separation bubble due to the generated APG. The mechanism of the boundary layer separation over the plane wall and the corresponding wake dynamics is investigated. Results are presented in terms of the distribution of the pressure coefficient as well as skin friction coefficient along the wall and flow patterns around and downstream of the cylinder in the proximity of the wall. The results of these computations confirm that using a rotating cylinder over a plane wall in a freestream flow is an effective technique to generate a controlled range of adverse pressure gradients.     

Transient electro-osmotic and pressure driven flows of two-layer fluids through a slit microchannel

By method of the Laplace transform, this article presents semi-analytical solutions for transient electroosmotic and pressure-driven flows (EOF/PDF) of two-layer fluids between microparallel plates. The linearized Poisson-Boltzmann equation and the Cauchy momentum equation have been solved in this article. At the interface, the Maxwell stress is included as the boundary condition. By numerical computations of the inverse Laplace transform, the effects of dielectric constant ratio ε , density ratio ρ , pressure ratio p, viscosity ratio μ of layer II to layer I, interface zeta potential difference △ψ, interface charge density jump Q, the ratios of maximum electro-osmotic velocity to pressure velocity α , and the normalized pressure gradient B on transient velocity amplitude are presented.We find the velocity amplitude becomes large with the interface zeta potential difference and becomes small with the increase of the viscosity. The velocity will be large with the increases of dielectric constant ratio; the density ratio almost does not influence the EOF velocity. Larger interface charge density jump leads to a strong jump of velocity at the interface. Additionally, the effects of the thickness of fluid layers (h1 and h2 ) and pressure gradient on the velocity are also investigated.     

Fourier analysis of an equal‐order incompressible flow solver stabilized by pressure gradient projection

Fourier analysis techniques are applied to the stabilized finite element method (FEM) recently proposed by Codina and Blasco for the approximation of the incompressible Navier–Stokes equations, here denoted by pressure gradient projection (SPGP) method. The analysis is motivated by spurious waves that pollute the computed pressure in start‐up flow simulation. An example of this spurious phenomenon is reported. It is shown that Fourier techniques can predict the numerical behaviour of stabilized methods with remarkable accuracy, even though the original Navier–Stokes setting must be significantly simplified to apply them. In the steady state case, good estimates for the stabilization parameters are obtained. In the transient case, spurious long waves are shown to be persistent when the element Reynolds number is large and the Courant number is small. This can be avoided by treating the pressure gradient projection implicitly, though this implies additional computing effort. Standard extrapolation variants are unfortunately unstable. Comparisons with Galerkin–least‐squares (GLS) method and Chorin's projection method are also addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

A novel method to calculate the pressure interaction between dust and fluid using SPH

Unstable behavior of smoothed particle hydrodynamics (SPH) dust particles, such as clumping or fingering under certain conditions, has been reported by several researchers who have conducted studies on dusty fluid SPH. The simulation results in this study show that this instability is numerical, and the instability is mainly attributable to the ill‐interpolated pressure gradient in the interaction term between 2 phases. In this paper, we introduce a new method to calculate the pressure force interaction term between dust and fluid particles. The key idea is to first interpolate the pressure gradient at SPH fluid particles and then use the values to calculate the pressure gradient at SPH dust particles, in a consecutive manner. To compare the new method with the existing method, we first conducted an interpolation of pressure gradient at hydrostatic equilibrium under gravity to estimate any error. The results show that the new method is more accurate. We then conducted additional numerical tests, namely, dust‐liquid counterflow, sedimentation in a confined tank, and sedimentation in the presence of turbulence. The unphysical unstable behavior of SPH dust particles such as clumping or fingering was significantly reduced in the new method. The results also show that the instability becomes more significant when using the existing method especially for the case when simulating a flow with relatively high concentration of dust or for the case in which inertia dominates the dynamics of dust particles. Especially, in those cases, the existing method should be avoided, and the newly proposed method is highly recommended.     

A numerical method for the computation of the effects of an air vessel on the pressure surges in pumping systems with air entrainment

This paper describes an improved numerical method and computational procedure for the implementation of typical air vessel responses and their influence on the pressure transient for unsteady flow in a pipeline system with air entrainment. The proposed numerical method and computational procedure is without the necessity of an excessive iterative procedure as required previously by the conventional approach. The effects of air in the transient fluid system with the air vessel were then studied through the improved numerical computational method. Free and dissolved gases in the transported fluid, and cavitation at vapour pressure, are included. © 1998 John Wiley & Sons, Ltd.     

Numerical computation of fluid pressure transients in pumping installations with air entrainment

In pumping installations such as sewage pumping stations, where gas content and air entrainment exist, the computation of fluid pressure transients in the pipelines becomes grossly inaccurate when constant wave speed and constant friction are assumed. A numerical model and computational procedure have been developed here to better compute the fluid pressure transient in a pipeline by including the effects of air entrainment and gas evolution characteristics of the transported fluid. Free and dissolved gases in the fluid and cavitation at the fluid vapour pressure are modelled. Numerical experiments show that entrained, entrapped or released gases amplify the pressure peak, increase surge damping and produce asymmetric pressure surges. The transient pressure shows a longer period for down-surge and a shorter period for up-surge. The up-surge is considerably amplified and the down-surge marginally reduced when compared with the gas-free case. These observations are consistent with the experimental observations of other investigators. Numerical experiments also show that the use of a variable loss factor in the pressure transient analysis produces marginally higher maximum and lower minimum pressure transients when compared with the constant-loss-factor model for pipelines where the pressures are above the fluid vapour pressure.     

On stagnation pressure increases in calorically perfect,ideal gases

When stagnation pressure rises in a natural or numerically simulated flow it is frequently a cause for concern, as one usually expects viscosity and turbulence to cause stagnation pressure to decrease. In fact, if stagnation pressure increases, one may suspect measurement or numerical errors. However, this need not be the case, as the laws of nature do not require that stagnation pressure continually decreases. In order to help clarify matters, the objective of this work is to understand the conditions under which stagnation pressure will rise in the unsteady/steady flows of compressible, viscous, calorically perfect, ideal gases. Furthermore, at a more practical level, the goal is to understand the conditions under which stagnation pressure will increase in flows simulated with the Reynolds averaged Navier–Stokes equations and eddy-viscosity turbulence models. In order to provide an improved understanding of increases in stagnation pressure for both these scenarios, transport equations are derived that govern its behavior in the unaveraged and Reynolds averaged settings. These equations are utilized to precisely determine the relationship between changes in stagnation pressure and zeroth, first, and second derivatives of fundamental flow quantities. Furthermore, these equations are utilized to demonstrate the relationship between changes in stagnation pressure and fundamental non-dimensional quantities that govern the conductivity, viscosity, and compressibility of the flow. In addition, based on an analysis of the Reynolds averaged equation (for eddy-viscosity turbulence models), it is shown that stagnation pressure is particularly likely to experience a spurious rise at the outer edges of shear layers that are undergoing convex curvature. Thereafter, numerical experiments are performed which confirm the primary aspects of the theoretical analysis.     

A New Mathematical Model for Force Gravity Drainage in Fractured Porous Media

In force gas/oil gravity drainage process in fractured porous media, gas is flowing in both matrix and fractures leading to produce a finite gas pressure gradient. Consequently, viscous force plays an important role for displacing matrix oil toward fractures in addition to gravity force that is required to be modeled appropriately. A new analytical model for estimation of steady state oil saturation distribution with assumption of fixed gas pressure gradient throughout the matrix is presented. Moreover, based on some results of this analytical model a different numerical formulation is developed to predict the performance of oil production process. Comparison of the results obtained from this numerical model with the results of a conventional simulator demonstrates that the newly developed model can be applied with satisfactory accuracy. Numerical simulations show that the viscous displacement in fractured porous media can reduce the capillary threshold height, and thus it suggests the force gravity drainage as a favorable production mechanism when the matrix length is close to the threshold height.     

An SPH pressure correction algorithm for multiphase flows with large density ratio

In this paper, we propose an interfacial pressure correction algorithm for smoothed particle hydrodynamics (SPH) simulation of multiphase flows with large density ratios. This correction term is based on the assumption of small deformation of the interface, and derived from perturbation expansion analysis. It is also proven to be applicable in cases with complex interfaces. This correction algorithm helps to overcome the discontinuities of the pressure gradient over the interfaces, which may cause unphysical gap between different phases. This proposed correction algorithm is implemented on a recent multiphase SPH model, which is based on the assumption of pressure continuity over the interfaces. The coupled dynamic solid boundary treatment is used to simulate solid walls; and a cut‐off pressure is applied to avoid negative particle pressure, which may cause computational instabilities in SPH. Three numerical examples of air–water flows, including sloshing, dam breaking, and water entry, are presented and compared with experimental data, indicating the robustness of our pressure correction algorithm in multiphase simulations with large density ratios. Copyright © 2015 John Wiley & Sons, Ltd.     

Theoretical analysis on hydraulic transient resulted by sudden increase of inlet pressure for laminar pipeline flow

Hydraulic transient, which is resulted from sudden increase of inlet pressure for laminar pipeline flow, is studied. The partial differential equation, initial and boundary conditions for transient pressure were constructed, and the theoretical solution was obtained by variable-separation method. The partial differential equation, initial and boundary conditions for flow rate were obtained in accordance with the constraint correlation between flow rate and pressure while the transient flow rate distribution was also solved by variable-separation method. The theoretical solution conforms to numerical solution obtained by method of characteristics (MOC) very well.     

Near-wall scaling for turbulent boundary layers with adverse pressure gradient

A new extended inner scaling is proposed for the wall layer of wall-bounded flows under the influence of both wall shear stress and streamwise pressure gradient. This scaling avoids problems of the classical wall coordinates close to flow separation and reattachment. Based on the proposed extended velocity and length scales a universal nondimensional family of velocity profiles is derived for the viscous region in the vicinity of a wall that depend on wall distance and a parameter α quantifying the importance of the streamwise pressure gradient with respect to the wall shear stress in the momentum balance. The performance of the proposed extended scaling is investigated in two different flow fields, a separating and reattaching turbulent boundary layer and a turbulent flow over a periodic arrangement of smoothly contoured hills. Both flows are results of highly resolved direct numerical simulation (DNS). The results show that the viscous assumptions are valid up to about two extended wall units. If the profiles are scaled by the extended inner coordinates, they seem to behave in a universal way. This gives rise to the hope that a universal behavior of velocity profiles can be found in the proposed extended inner coordinates even beyond the validity of the extended viscous law of the wall.      

地下岩体的垂向渗透率测试及分析

本文介绍了在真实油井中使用阵列压力计测量油藏薄层垂向渗透率的实验方法。通过注水-停注并多次重复此过程使地层产生压差,并在井中不同深度放置压力计接收压力信号。为分析压力的变化,本文从渗流力学基本方程出发,对其无量纲化并采用Laplace变换及数值反变换技术,得到考虑垂向渗透率的地层压力分布表达式,利用这一表达式,对测试的压力资料进行解释。最后根据非线性回归理论对其中的4只压力计的压力资料进行分析,得到了各层垂向渗透率。     

Macroscopic Flow Potentials in Swelling Porous Media

In swelling porous media, the potential for flow is much more than pressure, and derivations for flow equations have yielded a variety of equations. In this article, we show that the macroscopic flow potentials are the electro-chemical potentials of the components of the fluid and that other forms of flow equations, such as those derived through mixture theory or homogenization, are a result of particular forms of the chemical potentials of the species. It is also shown that depending upon whether one is considering the pressure of a liquid in a reservoir in electro-chemical equilibrium with the swelling porous media, or the pressure of the vicinal liquid within the swelling porous media, a critical pressure gradient threshold exists or does not.     

A comparison of second-order and fourth-order pressure gradient algorithms in a σ-co-ordinate ocean model

In stratified three-dimensional models the use of a boundary-fitted vertical co-ordinate is known to produce errors in the horizontal pressure gradient calculation near steep topography. The error is due to the splitting of the horizontal pressure gradient term in each of the momentum equations into two parts and the subsequent incomplete cancellation of the truncation errors of those parts. In order to minimize these pressure gradient errors, a fourth-order-accurate pressure gradient calculation has been implemented and installed in SPEM, a three-dimensional primitive equation ocean model. The stability and accuracy of the new scheme are compared with those of the original second-order-accurate model in a series of calculations of unforced flow in the vicinity of an isolated seamount. The new scheme is shown to have much smaller pressure gradient errors over a wide range of parameter space as well as a greater parametric domain of numerical stability.     

PRESSURE AND PRESSURE GRADIENT IN AN AXISYMMETRIC RIGID VESSEL WITH STENOSIS

Based on an improvement of the Karman-Pohlhausen's method, using nonlinear polynomial fitting and numerical integral, the axial distributions of pressure and its gradient in an axisymmetric rigid vessel with stenosis were obtained, and the distributions related to Reynolds number and the geometry of stenotic vessel were discussed. It shows that with the increasing of stenotic degree or Reynolds number, the fluctuation of pressure and its gradient in stenotic area is intense rapidly, and negative pressure occurs subsequently in the diverging part of stenotic area. Especially when the axial range of stenosis extends, the flow of blood in the diverging part will be more obviously changed. In higher Reynolds number or heavy stenosis, theoretical calculation is mainly in accordance with nast experiments.