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Roe-type Riemann solver for gas-liquid flows using drift flux model with an approximate form of the Jacobian matrix
Abstract
This work presents an approximate Riemann solver to the transient isothermal drift flux model. The set of equations constitutes a non-linear hyperbolic system of conservation laws in one space dimension. The elements of the Jacobian matrix A are expressed through exact analytical expressions. It is also proposed a simplified form of A considering the square of the gas to liquid sound velocity ratio much lower than one. This approximation aims to express the eigenvalues through simpler algebraic expressions. A numerical method based on the Gudunov's fluxes is proposed employing an upwind and a high order scheme. The Roe linearization is applied to the simplified form of A. The proposed solver is validated against three benchmark solutions and two experimental pipe flow data.
... However, the eigenvalues were evaluated numerically. Santim and Rosa [45] also proposed a Roe-type Riemann solver, presenting an approximate analytical form for both the Jacobian matrix and the eigenvalues of the system. It is worth mentioning that all the numerical schemes mentioned above are at best second-order accurate. ...
... For an accurate numerical solution, it would be desirable to have an upwind resolution of all the waves inherent in the two-phase model, for example, building an approximate Riemann solver of Roe [43]. However, this is relatively complex, even for simpler drift-flux two-phase models [22,45]. For the driftflux two-phase models considered in this paper, since u g = u l , there is no tractable exact expression for the Jacobian matrix, as well as for the eigenvalues of the system of equations [50]. ...
... where the expression of the speed of sound of the mixture c m is given in [45]. Numerical experiments led to the choice C ρ g = C ρ l = 0.1. ...
A computational code is developed for the numerical solution of one-dimensional transient gas-liquid flows using drift-flux models, in isothermal and also with phase change situations. For these two-phase models, classical upwind schemes such as Roe- and Godunov-type schemes are generally difficult to derive and expensive to use, since there are no treatable analytic expressions for the Jacobian matrix, eigenvalues and eigenvectors of the system of equations. On the other hand, the high-order compact finite difference scheme becomes an attractive alternative on these occasions, as it does not make use of any wave propagation information from the system of equations. The present paper extends the localized artificial diffusivity method for high-order compact finite difference schemes to solve two-phase flows with discontinuities. The numerical method has simple formulation, straightforward implementation, low computational cost and, most importantly, high-accuracy. The numerical methodology proposed is validated by solving several numerical examples given in the literature. The simulations are sixth-order accurate and it is shown that the proposed numerical method provides accurate approximations of shock waves and contact discontinuities. This is an essential property for simulations of realistic mass transport problems relevant to operations in the petroleum industry.
... The numerical model used to calculate the pressure drop is based on the drift-flux approach proposed by Santim and Rosa, (2016). The model consists of two mass equations representing the liquid and gas phases and one equation presenting the flow mixture momentum conservation. ...
... The entire data set was used to compare the performance of MLP-GA against the homogeneous model, the homogeneous model, (Awad and Muzychka, 2008) Awad and Muzychka, (2008), (Lockhart and Martinelli, 1949) Lockhart and Martinelli, (1949), (Friedel, 1979) Friedel, (1979, (Müller-Steinhagen andHeck, 1986) Müller-Steinhagen andHeck, (1986) and numerical drift flux model proposed by (Santim and Rosa, 2016) Santim and Rosa, (2016). Two statistical parameters of RMSE and AARE% were selected for this comparison, and the results are illustrated in Table 5. ...
... The entire data set was used to compare the performance of MLP-GA against the homogeneous model, the homogeneous model, (Awad and Muzychka, 2008) Awad and Muzychka, (2008), (Lockhart and Martinelli, 1949) Lockhart and Martinelli, (1949), (Friedel, 1979) Friedel, (1979, (Müller-Steinhagen andHeck, 1986) Müller-Steinhagen andHeck, (1986) and numerical drift flux model proposed by (Santim and Rosa, 2016) Santim and Rosa, (2016). Two statistical parameters of RMSE and AARE% were selected for this comparison, and the results are illustrated in Table 5. ...
The two-phase frictional pressure drop has a dominant effect in many industrial applications associated with the multiphase flow. This study investigated the accuracy of several available methods for predicting two-phase frictional pressure drop of different pipe diameters using 4124 experimental data points. It is observed that the performance of the existing methods is poor in a wide range of operating conditions. Then, several Artificial Neural Network models were proposed, including six multilayer perceptron (MLP) and one Radial Basis Function (RBF) using the same data sets. The weights and biases of the ANNs were optimized using Levenberg-Marquardt (LM), Bayesian Regularization (BR), Scaled Conjugate Gradient (SCG), Resilient Backpropagation (RB), Particle Swarm Optimization (PSO) and Genetic Algorithm (GA). Statistical error analysis indicates that neural network incorporated with the Genetic Algorithm (MLP-GA) predicts the entire data set with a Root Mean Square Error of 0.525 and an Average Absolute Relative Error percentage of 6.722. Finally, the sensitivity analysis was carried out, indicating that the mass flux (G) has the highest direct impact on the two-phase frictional pressure drop.
... The numerical model used to calculate the pressure drop is based on the drift-flux approach proposed by Santim and Rosa, (2016). The model consists of two mass equations representing the liquid and gas phases and one equation presenting the flow mixture momentum conservation. ...
... The entire data set was used to compare the performance of MLP-GA against the homogeneous model, the homogeneous model, (Awad and Muzychka, 2008) Awad and Muzychka, (2008), (Lockhart and Martinelli, 1949) Lockhart and Martinelli, (1949), (Friedel, 1979) Friedel, (1979, (Müller-Steinhagen andHeck, 1986) Müller-Steinhagen andHeck, (1986) and numerical drift flux model proposed by (Santim and Rosa, 2016) Santim and Rosa, (2016). Two statistical parameters of RMSE and AARE% were selected for this comparison, and the results are illustrated in Table 5. ...
... The entire data set was used to compare the performance of MLP-GA against the homogeneous model, the homogeneous model, (Awad and Muzychka, 2008) Awad and Muzychka, (2008), (Lockhart and Martinelli, 1949) Lockhart and Martinelli, (1949), (Friedel, 1979) Friedel, (1979, (Müller-Steinhagen andHeck, 1986) Müller-Steinhagen andHeck, (1986) and numerical drift flux model proposed by (Santim and Rosa, 2016) Santim and Rosa, (2016). Two statistical parameters of RMSE and AARE% were selected for this comparison, and the results are illustrated in Table 5. ...
... Notably, although the gas and the liquid phase play a symmetric role in (6), the contribution of the phases are non-100 symmetric in the eigenvalues (9) due to the closure laws (2)- (5). For a detailed analysis of the eigenvalue problem of the conservative DFM, the reader may refer to [31]. The eigenvalue λ 4 shows that there is a stationary wave in the computational domain that becomes visible when the cross-sectional area is discontinuous and ∂A/∂x becomes closer to the impulse function. ...
... Then, according to the chosen ∆x, the temporal discretization ∆t is specified. This section is dedicated to check the steady-state preservation of the numerical approaches proposed in Section 3. Since for the set of algebraic relations (31) in the fourth approach, no slip between the phases is considered, i.e., K = 1 and S = 0, we apply the same condition in this section to perform a fair comparison between different approaches. Figure 4 shows the computational domain for this case study that is a horizontal pipe with one discontinuity in diameter along its length. ...
... Compared to the previous results, pressure and mass flow rate are preserved with significantly higher accuracy. The small deviation from steady-state is due to the error in solving the algebraic relations (31). The simulation results for the set of algebraic constraints in (34) are depicted in Figure 9. ...
This paper presents a modification of a classical Godunov-type scheme for the numerical simulation of a two-phase flow in a pipe with a piecewise constant cross-sectional area. This type of flow can occur in wellbores during drilling for oil and gas as well as after well completion. Contrary to classical finite-volume schemes, the numerical scheme proposed in this paper captures the steady-state solution of the system without generating non-physical discontinuities in the numerical solution close to the locations of discontinuities in the cross-section. Moreover, the proposed scheme can be extended to problems with piecewise continuous cross-sectional area. This extension is achieved by discretization of the area along the spatial domain and converting the piecewise continuous area into a piecewise constant area. The proposed scheme reduces to the classical scheme when the cross-sectional area is constant along the spatial domain. For the purpose of computational efficiency, the modification to the classical scheme is only applied at the locations of area variation and the numerical solver reduces to the classical scheme where the cross-sectional area is constant. It is also shown that the proposed scheme can be effectively used to simulate two-phase flows arising from the perturbation of the steady-state solution. The effectiveness of the proposed scheme is shown through illustrative numerical simulations. Finally, it should be noted that the proposed scheme retains the same order of accuracy as the underlying classical scheme.
... Aiming to extend the analysis content investigated in Masella et al. (1998) and Choi et al. (2013), the present work compares the three methodologies DFM (Santim and Rosa, 2016), TFM (OLGA ® ) and NPW (MARLIM) against the experimental acquisitions presented in Dalla Maria and Rosa (2016), through development of four flow rate transients within the slug flow pattern, making use of two drift correlations (Bendiksen andChoi et al., 1984, 2012) and two mixture viscosity relations Beattie andWhalley et al., 1964, 1982). The main motivation is to assess the better solver accuracy as well as its respective capability to capture the waves´velocities values through analysis of the transients stressed by the pressure and void fraction trends. ...
... (1)-(3), it is used an upwind Godunov´s type approach (see LeVeque (2002)) using an explicit scheme as presented in Santim and Rosa (2016). ...
... Both the dynamic Drift-Flux model (approximate Riemann solver of Santim and Rosa (2016)) and the kinematic (transient MARLIM), calculate the source term related to the friction as follows: ...
The present work compares different methodologies and experimental data in transient isothermal gas-liquid slug flows into a horizontal pipeline. Four cases are addressed: The expansion and compression of a gas-liquid mixture due to the variation of the inlet gas superficial velocity assuming the inlet liquid superficial velocity constant and varying the inlet liquid superficial velocity fixing the gas superficial velocity. The approximate Riemann solver based on the Drift-Flux Model (DFM) proposed in Santim and Rosa (2016), a simplifier Non-Pressure Wave (NPW) used on the MARLIM (Multiphase Flow and Artificial Lift Modelling) and the Two-Fluid Model (TFM) making use of OLGA® (OiL and GAs simulator) are tested against experimental data presented in Dalla Maria and Rosa (2016). Pressure and void fraction trends are used for the comparison. Furthermore the pressure and void fraction wave velocities are also measured. The main motivation is analyzing the transient behavior of the solutions obtained by the models, besides of to verify the solver which presents better accuracy and capability to capture the waves’ velocities magnitude.
... Features of such methods include their strong ability in capturing discontinuities, such as shocks and interfaces, and providing approximation of the intercell fluxes by a family of waves propagating through the cells interface. Among methods of such group for two-phase flows employing the drift-flux model are Roe-type Riemann solvers of [37,38], hybrid flux-splitting scheme of [40,41,42], a combination of the Roe scheme together with the Multi-Stage (MUSTA) scheme is developed in [39] whereas a high-resolution hybrid upwind scheme is also presented in [41] and a semiimplicit relaxation scheme is also developed in [43]. In [44,45], Godunov methods of centred-type are extended to both isothermal and isentropic gasliquid two-phase flow employing the drift-flux model and validated with the developed exact Riemann solvers. ...
... However, a number of numerical methods for the driftflux model are available in literature based on the assumption of mechanical equilibrium between the two phases. See, for example, [37,39,43] and references therein. In addition to that, an exact Riemann solver is also developed in [44,45] with velocity equilibrium and mechanical non-equilibrium between the two phases. ...
This paper focuses on the extension of the Weighted Average Flux (WAF) scheme for the numerical simulation of two-phase gas-liquid flow by imposing velocity equilibrium and without mechanical equilibrium of the transient drift-flux model. The model becomes a hyperbolic system of conservation laws with realistic closure relations where both phases are strongly coupled during their motion. Exploiting this, the drift-flux model discretization, simulation and investigation becomes very fast, simple and robust. The efficiency of the WAF scheme as being a second order in space and time without data reconstruction have been demonstrated in the published literature for compressible single-phase flows. However, the scheme is rarely applied to compressible two-phase flows. Based on a recent and complete exact Riemann solver for the drift-flux model, the model is numerically solved by the WAF scheme. The numerical algorithm accuracy and ability are validated through different published test cases. It is shown that the proposed scheme can be effectively employed to simulate two-phase flows involving discontinuities such as shocks and interfaces. The proposed WAF scheme is also compared with other numerical methods. Simulation results show appropriate agreement of WAF scheme even with the exact solutions. Comparisons of the presented simulations demonstrate that the behaviour of WAF scheme is encouraging, more accurate and fast than other numerical methods.
... 80) where M µ = max (1, 1/µ). Under the assumption|∆F(x, ξ, t; s)| ≤ M m ξ m m! (3.81) |∆G(x, ξ, t; s)| ≤ M m ξ m m! (3.82)where M = (ḡ(ā +b) +d +ē)M µ(3.83) ...
... Other simplifying assumptions are considered inside the computational domain, see[53],[52] and the citations therein.7 see, e.g.[80] for the eigenvalue analysis of the DFM. The standing wave due to the variable geometry corresponding to λ 4 = 0 is analysed in[52]. ...
While drilling an oil well, unwanted influx of fluids from the reservoir may occur. This manuscript studies the dynamics of the resulting fluid flow in the wellbore and the reservoir, during managed pressure drilling (MPD) operations. We study the phenomena using first-principle approach that leads to a modified two phase flow model called the drift-flux model (DFM). The model takes the form of a hyperbolic system of transport equations, whereas the reservoir pressure dynamics is given by a parabolic diffusion equation. We propose appropriate numerical schemes for both. Then, we propose different observer designs to estimate the influx from the reservoir into the wellbore. The observers for the coupled wellbore-reservoir system take different forms, as we combine the distributed and the reduced order models for each system. We propose to use the choke pressure as a measurement that is typically available on a MPD operational site, i.e. topside sensing. However, availability of the bottom hole pressure modifies the observer design, in ways we detail. We show convergence of the observers to the true values of reservoir pore pressure and influx, in each case.
... Moreover, (⋅, ⋅) is the scheme-specific numerical flux function. Various numerical flux functions have been introduced in the literature 11,21,18,29,30,31 . As a case study, the flux function for Rusanov scheme is defined as follows ...
... For this reason, only the case of no-slip is considered in this paper. The reader is referred to 21,29 for a detailed analysis of the speed of sound in the mixture of the gas and liquid. (25) becomes ill-posed when tends to zero or → 1. ...
In this paper, approximate well-balanced finite-volume schemes are developed for the isothermal Euler equations and the drift flux model, widely used for the simulation of single-and two-phase flows. The proposed schemes, which are extensions of classical schemes, effectively enforce the well-balanced property to obtain a higher accuracy compared to classical schemes for both the isothermal Euler equations and the drift flux model in case of non-zero flow in the presences of both laminar friction and gravitation. The approximate well-balanced property also holds for the case of zero flow for the isothermal Euler equations. This is achieved by defining a relevant average of the source terms which exploits the steady-state solution of the system of equations. The new extended schemes reduce to the original classical scheme in the absence of source terms in the system of equations. The superiority of the proposed well-balanced schemes to classical schemes, in terms of accuracy and computational effort, is illustrated by means of numerical test cases with smooth steady-state solutions. Furthermore, the new schemes are shown numerically to be approximately first-order accurate.
... In addition to Constant Total Flow rate (CTF) where the flow rates for both phases are changed to give CTF. The second is to examine the possibility of using the drift-flux model (utilizing the approximate Riemann solver proposed by Santim and Rosa (2016) to predict the pressure drop for two-phase flows by comparing the experimental data with predictions from the model. In addition, the present experimental measurements are also compared with predictions from empirical models in the literature. ...
... A Roe-type Riemann solver based on drift-flux model proposed by Santim and Rosa (2016) was used to calculate the pressure drop numerically. The model is assumed to be isothermal with no mass transfer between the phases. ...
The pressure drop has a significant importance in multiphase flow systems. In this paper, the effect of the volumetric quality and mixture velocity on pressure drop of gas-liquid flow in horizontal pipes of different diameters are investigated experimentally and numerically. The experimental facility was designed and built to measure the pressure drop in three pipes of 12.70, 19.05 and 25.40 mm. The water and air flow rates can be adjusted to control the mixture velocity and void fraction. The measurements are performed under constant water flow rate (CWF) by adding air to the water and constant total flow rate (CTF) in which the flow rates for both phases are changed to give same CTF. The drift-flux model is also used to predict the pressure drop for same cases. The present data is also compared with a number of empirical models from the literature. The results show that: i) the pressure drop increases with higher volumetric qualities for the cases of constant water flow rate but decreases for higher volumetric qualities of constant total flow rate due to the change in flow pattern. ii) The drift-flux model and homogenous model are the most suitable models for pressure drop prediction.
... Two were models embedded in software used by the petroleum industry. The other was an approximate Riemann solver based on the Drift-Flux Model proposed in Santim and Rosa (2016). Maria and Rosa's experimental data showed that even with the transient being introduced by a sudden change in inlet flow rate, the flow parameters varies slightly between the two steady states due to inertia. ...
In the oil and gas industry, the transient flow arises in daily situations, such as the start-up and shutdown of lines, pigging, artificial lifting, and drilling operations. These disturbances affect the temporal behavior of pressure and void fraction, which could cause separator flooding, system vibration, and production line facilities damage. Nevertheless, there are few numerical studies to describe this phenomenon. In this article, the transient slug flow is studied using the slug tracking model, which predicts the transient behaviour and the evolution of slug flow properties. This study demonstrates the slug tracking model's capability to reproduce the transient flow as well as the pressure and void fraction wave's behavior and velocity. Experimental data presented by Maria and Rosa, 2016 were used to assess the model performance. The experimental data show a 6.0 s time delay between the two steady states. This time delay does not influence the void fraction wave, and its numerical maximum deviation is 9.0%. On the other hand, the numerical performance for the pressure wave velocity depends on the time delay. Using the experimental time delay, the maximum deviation for the pressure wave velocity is 6.8%. The slug tracking model is combined with a mass–dashpot–spring analogy to avoid the experimental time delay dependency. Using this methodology, the maximum deviation for pressure wave velocity is 8.2%. Some slug flow properties, such as the bubble nose translational velocity, the bubble and liquid slug lengths, are also compared to experimental data. The comparison is for averaged values and statistical distribution. The bubble nose velocities and pressure deviations are less than 1.5%, while the bubble and liquid slug lengths deviations are 4.1% and 5.3%, respectively. Most slug flow parameters have normal statistical distribution, but slug length follows a close to log-normal distribution. The model reproduces this behavior.
... It is necessary to linearized R at each cell in order to properly evaluate the fluxes at the cell's faces. Many researchers have applied different methods to obtain the linearized form of this Jacobian matrix [47][48][49][50]. So that Eq. (13.b) could be transformed into a similar form of the linear advection equation, which is easier to be solved. ...
The simulation of gas/liquid two-phase flow, considering the heat transfer between the annulus fluid and the surrounding environment, is of significance in predicting temperature and pressure distributions after a gas kick in HTHP deep well drilling. This paper presents the development of a transient non-isothermal two-phase flow model for the dynamic simulation of multiphase flow in the wellbore after a gas kick. The drift-flux model is used to describe gas/liquid two-phase flow, and multiple transient energy conservation equations are used for predicting temperature profiles of fluids in the drillpipe, drillpipe, the fluid in the annulus, casing string, and formation. As for numerical scheme of this strongly coupled model, the advection upstream splitting model (AUSMV) hybrid scheme is adapted for solving flow equations, while the finite difference approach is adapted for simultaneously solving energy conservation equations of the wellbore-formation system. Physical properties of gas and liquid phases are updated at each timestep. Predicted temperature and pressure distributions are validated against the field data. Flow behaviors predicted by the models with and without the heat transfer effect are compared. The effects of some major parameters (reservoir pressure, choke pressure, geothermal gradient, liquid mass flow rate) on temperature, pressure, and gas fraction distributions along the wellbore are investigated.
... A high resolution hybrid upwind scheme for isothermal drift-flux model describing gas-liquid flow in a long pipeline has been presented in [12] . The authors in [13] have also considered the approximate form of the Jacobian matrix for the drift-flux model and introduced a Roe-type Riemann solver for gas-liquid flow. In earlier, the analytical solution of the model equations is based on simplified assumption due to complicated nature of model equations. ...
In the present work, we investigate the Riemann problem and interaction of weak shocks for the widely used isentropic drift-flux equations of two-phase flows. The complete structure of solution is analyzed and with the help of Rankine–Hugoniot jump condition and Lax entropy conditions we establish the existence and uniqueness condition for elementary waves. The explicit form of the shock waves, contact discontinuities and rarefaction waves are derived analytically. Within this respect, we develop an exact Riemann solver to present the complete solution structure. A necessary and sufficient condition for the existence of solution to the Riemann problem is derived and presented in terms of initial data. Furthermore, we present a necessary and sufficient condition on initial data which provides the information about the existence of a rarefaction wave or a shock wave for one or three family of waves. To validate the performance and the efficiency of the developed exact Riemann solver, a series of test problems selected from the open literature are presented and compared with independent numerical methods. Simulation results demonstrate that the present exact solver is capable of reproducing the complete wave propagation using the current drift-flux equations as the numerical resolution. The provided computations indicate that accurate results be accomplished efficiently and in a satisfactory agreement with the exact solution.
... Therefore, the features of void fraction and pressure waves are by themselves a way to typify gas-liquid flows. Furthermore, comparisons of wave velocity prediction offer an excellent means to test the ability of numerical models, such as those based on the mixture model ( Masella et al. 1998;Evje and Fjelde 2002;Malekzadeh et al. 2012 andSantim and, slug tracking models ( Nydal and Banerjee 1996;Taitel and Barnea 1998;Al-Safram et al. 2004;Ujang andHewitt 2006 andRosa et al. 2015 ) and on different forms of the two and multi-fluid models pattern which alternately liquid and gaseous elements are in line, taking the whole pipe cross section as plugs. The. ...
The void fraction and the pressure waves in an air-water mixture flowing in the slug regime are experimentally investigated in a horizontal line. The test section is made of a transparent Plexiglas pipe with 26 mm ID and 26.24 m long, operating at ambient temperature and pressure. The flow induced transients are made by quickly changing the air or the water inlet velocity. The test grid has four operational points. This choice allows one to create expansion and compression waves due to the changes to the gas or to the liquid. Each experimental run is repeated 100 times to extract an ensemble average capable of filtering out the intrinsic flow intermittence and disclosing the void fraction and pressure waves’ features. The slug flow properties such as the bubble nose translational velocity, the lengths of liquid film underneath the bubble and the liquid slug are also measured. The objective of the work is two-fold: access the main characteristics of the void fraction and pressure waves and disclose the mechanics of the transient slug flow as described through the changes of the slug flow properties.
This study investigates the pressure drop in horizontal pipes packed with large particles that result in small pipe-to-particle diameter ratio both experimentally and numerically. Two horizontal pipes of 0.1905 and 0.0254 m ID filled with cylindrical or spherical particles are used to collect the experimental data for single and two-phase flows. The porosity has same value for both pipes when they packed with cylindrical particles which is 0.75, however has different values when packed with spherical particles, 0.7 for the large pipe and 0.57 for the small pipe. The Roe-type Riemann solver proposed by Santim and Rosa Int J Numer Methods Fluids 80 (9), 536–568, [36] which uses the Drift-Flux model is modified aiming to predict the pressure drop in porous media through the implementation of a new source term in the system of equations. Empirical models available in the literature are used to calculate the single and two-phase flows pressure drop. The motivation is to verify the solver capability to reproduce the two-phase flow pressure drop in porous media and to compare some empirical models existing in the literature against the experimental data provided modifying some empirical coefficients when necessary.
The reliable predictions of liquid holdup and pressure drop are essential for pipeline design in oil and gas industry. In this study, the drift-flux approach is utilized to calculate liquid holdups. This approach has been widely used in formulation of the basic equations for multiphase flow in pipelines. Most of the drift-flux models have been developed on an empirical basis from the experimental data. Even though, previous studies showed that these models can be applied to different flow pattern and pipe inclination, when the distribution parameter is flow pattern dependent. They are limited to a set of fluid properties, pipe geometries and operational conditions. The objective of this study is to develop a new drift-flux closure relationship for prediction of liquid holdups in pipes that can be easily applied to a wide range of flow conditions. The developed correlation is compared with nine available correlations from literatures, and validated using the TUFFP (Fluid Flow Projects of University of Tulsa) experimental datasets and OLGA (OiL and GAs simulator supplied by SPTgroup) steady-state synthetic data generated by OLGA Multiphase Toolkit. The developed correlation performs better in predicting liquid holdups than the available correlations for a wide range of flow conditions.
This work analyzes the transient slug flow inside a horizontal pipe both numerically and experimentally. Initially is considered constant superficial velocities of the gas (J g =0.5960m/s) and liquid (J l =0.6040m/s) at the inlet region of the pipe with a pressure of 94100Pa at the outlet boundary. The case proposed assumes that after 30.5s from the steady state is reached, the superficial velocity of the gas drops to J g =0.2986m/s whereas J L remains constant. The main objective of this work is to access the accuracy of the transient drift flux model in transient gas-liquid slug flow comparing the numerical results against the experimental data. An approximate Riemann solver to the transient isothermal drift flux model was developed to solve the problem numerically. The drift flux model consists of one momentum equation of mixture and two mass equations, one for each phase. The set of equations constitutes a non-linear hyperbolic system of the conservation laws with three equations and four unknowns. A kinematic relation proposed by Zuber and Findlay (1965) is used to provide the system closure. A simplified form of the drift flux model was implemented allowing to express the Jacobian matrix analytically, generating simpler eigenvalues expressions and reducing the computational cost to solve this non-linear problem. This problem is linearized employing Roe (1981) method.The upwind scheme and an extension of the Lax-Wendroff scheme using a van Leer limiter were implemented on the solver. Two effective viscosities based on the homogeneous model were analized. The relation proposed by Beattie and Whalley (1982) presented a better agreement with the pressure gradient obtained experimentally at four different axial positions of the pipeline. The relative error between the numerical and experimental mean wave velocity of pressure is about 0.22% whereas for the void fraction this error does not exceed 5%.
The motivation of this work is the development of a Riemann solver to the transient isothermal drift flux model, a set of two mass equations and one momentum equation which describes the transient behavior of a gas-liquid mixture in a pipe. The set of equations constitutes a non-linear hyperbolic system of conservation laws in one space dimension. The hyperbolicity is one of the main features of this system and rules the nature of the numerical methods employed to solve the system. The system is hyperbolic as long as the three eigenvalues of the Jacobian matrix, A, are real and distinct. The present article objective is the development of approximated forms of A to express the eigenvalues by means of analytical expressions in order to reduce the computational cost of the Riemann solver. The simplification hypothesis considers the squared of sound velocities ratio between the gas to the liquid phases much smaller than one. The approximated form of A and the hiperbolicity analysis is performed for a range of gas and liquid superficial velocities spanning from 0.1 to 28 m/s and for operational pressures of 1, 10 and 100 bar. Furthermore the accuracy of the approximated eigenvalues expressions are compared against the exact value resulting in an accuracy better than 3% for applications where the void fraction spans from 0.15 to 0.98.
For two-phase flow models, upwind schemes are most often difficult do derive, and expensive to use. Centred schemes, on the other hand, are simple, but more dissipative. The recently proposed multi-stage (MUSTA) method is aimed at coming close to the accuracy of upwind schemes while retaining the simplicity of centred schemes. So far, the MUSTA approach has been shown to work well for the Euler equations of inviscid, compressible single-phase flow. In this work, we explore the MUSTA scheme for a more complex system of equations: the drift-flux model, which describes one-dimensional two-phase flow where the motions of the phases are strongly coupled. As the number of stages is increased, the results of the MUSTA scheme approach those of the Roe method. The good results of the MUSTA scheme are dependent on the use of a large-enough local grid. Hence, the main benefit of the MUSTA scheme is its simplicity, rather than CPU-time savings. Copyright © 2006 John Wiley & Sons, Ltd.
This paper is devoted to the numerical approximation of the solutions of a system of conservation laws arising in the modeling of two-phase flows in pipelines. The PDEs are closed by two highly nonlinear algebraic relations, namely, a pressure law and a hydrodynamic one. The severe nonlinearities encoded in these laws make the classical approximate Riemann solvers virtually intractable at a reasonable cost of evaluation. We propose a strategy for relaxing solely these two nonlinearities. The relaxation system we introduce is of course hyperbolic but all associated eigenfields are linearly degenerate. Such a property not only makes it trivial to solve the Riemann problem but also enables us to enforce some further stability requirements, in addition to those coming from a Chapman-Enskog analysis. The new method turns out to be fairly simple and robust while achieving desirable positivity properties on the density and the mass fractions. Extensive numerical evidences are provided.
Fromm's second-order scheme for integrating the linear convection equation is made monotonic through the inclusion of nonlinear feedback terms. Care is taken to keep the scheme in conservation form. When applied to a quadratic conservation law, the scheme notably yields a monotonic shock profile, with a width of only 112 mesh.
Several numerical schemes for the solution of hyperbolic conservation laws are based on exploiting the information obtained by considering a sequence of Riemann problems. It is argued that in existing schemes much of this information is degraded, and that only certain features of the exact solution are worth striving for. It is shown that these features can be obtained by constructing a matrix with a certain “Property U.” Matrices having this property are exhibited for the equations of steady and unsteady gasdynamics. In order to construct thems it is found helpful to introduce “parameter vectors” which notably simplify the structure of the conservation laws.
High resolution upwind and centred methods are today a mature generation of computational techniques applicable to a wide range of engineering and scientific disciplines, Computational Fluid Dynamics (CFD) being the most prominent up to now. This textbook gives a comprehensive, coherent and practical presentation of this class of techniques. The book is designed to provide readers with an understanding of the basic concepts, some of the underlying theory, the ability to critically use the current research papers on the subject, and, above all, with the required information for the practical implementation of the methods. Direct applicability of the methods include: compressible, steady, unsteady, reactive, viscous, non-viscous and free surface flows. For this third edition the book was thoroughly revised and contains substantially more, and new material both in its fundamental as well as in its applied parts. © Springer-Verlag Berlin Heidelberg 2009. All rights are reserved.
We construct a Roe-type numerical scheme for approximating the solutions of a drift-flux two-phase flow model. The model incorporates a set of highly complex closure laws, and the fluxes are generally not algebraic functions of the conserved variables. Hence, the classical approach of constructing a Roe solver by means of parameter vectors is unfeasible. Alternative approaches for analytically constructing the Roe solver are discussed, and a formulation of the Roe solver valid for general closure laws is derived. In particular, a fully analytical Roe matrix is obtained for the special case of the Zuber–Findlay law describing bubbly flows. First and second-order accurate versions of the scheme are demonstrated by numerical examples.
In this chapter we apply the mathematical tools presented in Chap. 2 to analyse some of the basic properties of the time–dependent Euler equations. As seen in Chap. 1, the Euler equations result from neglecting the effects of viscosity, heat conduction and body forces on a compressible medium. Here we show that these equations are a system of hyperbolic conservations laws and study some of their mathematical properties. In particular, we study those properties that are essential for finding the solution of the Riemann problem in Chap. 4. We analyse the eigenstructure of the equations, that is, we find eigenvalues and eigenvectors; we study properties of the characteristic fields and establish basic relations across rarefactions, contacts and shock waves. It is worth remarking that the process of finding eigenvalues and eigenvectors usually involves a fair amount of algebra as well as some familiarity with basic physical quantities and their relations. For very complex systems of equations finding eigenvalues and eigenvectors may require the use of symbolic manipulators. Useful background reading for this chapter is found in Chaps. 1 and 2.
The objective of this study is to develop a simplified transient model and a simulator for gas–liquid two-phase flow in pipelines. The reliable predictions of liquid holdup and pressure drop are essential for pipeline design in oil and gas industry. In this study, the drift-flux approach is utilized to calculate liquid holdups. A modification of the power law correlation presented by Al-sarkhi and Sarica (2009) is suggested for pressure drop calculation. The proposed approach and correlation are continuous and flow pattern independent.Additionally, the developed model is simple and presents an easy tuning capability with either experimental data or synthetic data coming from steady state simulators. The simplicity of the model allows quick implementation yielding in a faster simulator as compared to available commercial software.The developed simulator is tested with Vigneron et al. (1995) experimental data, which include two transient conditions; liquid flow rate changes and gas flow rate changes. The results of simulation are compared with OLGA (OiL and GAs simulator supplied by SPT group) simulations and show fair agreement in terms of liquid holdups and pressures.
We present a Roe-type weak formulation Riemann solver where the average coefficient matrix is computed numerically. The novelty of this approach is that it is general enough that can be applied to any hyperbolic system while retaining the accuracy of the original Roe solver. We show applications to the compressible Euler equations with general equation of state. An alternative version of the method uses directly the eigenvectors in the averaging process, simplifying the algorithm. These new solvers are applied in conservative and path-conservative schemes with high-order accuracy and on unstructured meshes. Copyright © 2014 John Wiley & Sons, Ltd.
The main objective of this study is to present new equations for a flow pattern independent drift flux model based void fraction correlation applicable to gas–liquid two phase flow covering a wide range of fluid combinations and pipe diameters. Two separate sets of equations are proposed for drift flux model parameters namely, the distribution parameter (Co)(Co) and the drift velocity (Ugm)(Ugm). These equations for CoCo and UgmUgm are defined as a function of several two phase flow variables and are shown to be in agreement with the two phase flow physics. The underlying data base used for the performance verification of the proposed correlation consists of experimentally measured 8255 data points collected from more than 60 sources that consists of air–water, argon–water, natural gas–water, air–kerosene, air–glycerin, argon–acetone, argon–ethanol, argon–alcohol, refrigerants (R11, R12, R22, R134a, R114, R410A, R290 and R1234yf), steam–water and air–oil fluid combinations. It is shown that the proposed correlation successfully predicts the void fraction with desired accuracy for hydraulic pipe diameters in a range of 0.5–305 mm (circular, annular and rectangular pipe geometries), pipe orientations in a range of -90°⩽θ⩽90°-90°⩽θ⩽90°, liquid viscosity in a range of 0.0001–0.6 Pa-s, system pressure in a range of 0.1–18.1 MPa and two phase Reynolds number in a range of 10 to 5 × 106. Moreover, the accuracy of the proposed correlation is also compared with some of the existing top performing correlations based on drift flux and separated flow models. Based on this comparison, it is found that the proposed correlation consistently gives better performance over the entire range of the void fraction (0 < α < 1) and is recommended to predict void fraction without any reference to flow regime maps.
Transient simulation of two-phase gas-liquid flow in pipes requires considerable computational efforts. Until recently, most available commercial codes are based on the two-fluid model which includes one momentum conservation equation for each phase. However, in normal pipe flow operation, especially in oil and gas transport, the transient response of the system proves to be relatively slow. Thus it is reasonable to think that simpler forms of the transport equations might suffice to represent transient phenomena. Furthermore, these types of models may be solved using less time-consuming numerical algorithms.
Despite the importance of pressure drop in two-phase flow processes, and the consequent extensive research into the topic, there is still no satisfactory method for calculating two-phase pressure drop. In this article a theoretically based flow pattern dependent calculation method is adapted to yield a simple predictive method in which flow pattern influences are partially allowed for in an implicit manner and therefore need not be taken into account when using the method. (A)
From the contents: Introduction: Definitions and Examples.- Nonlinear hyperbolic systems in one space dimension.- Gas dynamics and reaction flows.- Finite Difference Schemes for one-dimensional systems.- The case of multidimensional systems.- An Introduction to Boundary conditions.
Several numerical schemes for the solution of hyperbolic conservation
laws are based on exploiting the information obtained by considering a
sequence of Riemann problems. It is argued that in existing schemes much
of this information is degraded and that only certain features of the
exact solution are worth striving for. It is shown that these features
can be obtained by constructing a matrix with a certain "Property U."
Matrices having this property are exhibited for the equations of steady
and unsteady gasdynamics. In order to construct them, it is found
helpful to introduce "parameter vectors" which notably simplify the
structure of the conservation laws.
Models for predicting flow pattern transitions during steady gas-liquid flow in vertical tubes are developed, based on physical mechanisms suggested for each transition. These models incorporate the effect of fluid properties and pipe size and thus are largely free of the limitations of empirically based transition maps or correlations.
An approximate linearized Riemann solver is presented for the numerical simulation of two-phase flows. This new solver is based on a linearization of nonconservative products and uses an extension of Roe's approximate Riemann solver. The scheme is applied to shock tube problem and to a standard test for two-fluid codes.
This paper deals with the resolution of the hyperbolic system of conservation laws which arises when modeling two-phase flow in an oil and gas pipeline. The model considered here is of the drift–flux type where the slip velocity between the two phases is given by a hydrodynamic model that accounts for the different flow regimes. The complexity of the closure relations prevents us from using classical numerical schemes such as Godunov or Roe's scheme. We propose another finite volume scheme, more rough for which the numerical flux is written as a centered scheme stabilized by a viscous term. The system considered has also the characteristic of having eigenvalues which are of very different orders of magnitude. We present a time discretization which is explicit for the “slow waves” and implicit for the “fast waves”. It allows the use of a time step which is governed only by the “slow waves” and keeps a good precision for these waves.
In this paper an approximate Riemann solver is constructed for solving one class of two-phase models. These models describe the gas–liquid flow in a long tube where the flow behaviour perpendicular to the tube axis is averaged, so that the model is essentially one-dimensional in the direction of the axis. The model consists of equations for the conservation of mass for each of the phases and the conservation of momentum of the mixture. In addition, an equation is supplied which relates the velocities of the two phases at any point, the slip relation, which may have a different shape for each of the flow regimes of interest and change in time. Generally, a slip relation will not be known entirely in algebraic form, but given partly in numerical form. Under certain restrictions the resulting system of conservation laws is hyperbolic and allows discontinuous solutions. The general idea is that the numerical algorithm for solving this model must be able to handle any valid slip relation. As the slip relation affects the Jacobian of the flux function to a large extent, this means that flux vector or flux difference splittings cannot be based on algebraic manipulation of the Jacobian. We propose to use a first-order upwind scheme of Roe-type, where the construction of the approximate Riemann solver is fully numerical. This basic scheme, however, has the same limitation as the original Roe method, namely, that for systems the positivity of the solution is not guaranteed. The modification of the basic scheme to ensure positivity of the solution, is based on the HLL Riemann solver.
A constitutive equation for a vapor drift velocity which specifies the relative motion between phases in the drift flux model is developed for two-phase annular flows. The constitutive equation is derived by taking into account the interfacial geometry, the body force field, and the interfacial momentum transfer, since these macroscopic effects govern the two-phase diffusions. A comparison of the model with three sets of experimental data obtained over a wide range of flow parameters shows a satisfactory agreement.
In view of the practical importance of the drift-flux model for two-phase flow analysis in general and in the analysis of nuclear-reactor transients and accidents in particular, the distribution parameter and the drift velocity have been studied for bubbly flow regime. The constitutive equation that specifies the distribution parameter in the bubbly flow has been derived by taking into account the effect of the bubble size on the phase distribution, since the bubble size would govern the distribution of the void fraction. A comparison of the newly developed model with various fully developed bubbly flow data over a wide range of flow parameters shows a satisfactory agreement. The constitutive equation for the drift velocity developed by Ishii has been reevaluated by the drift velocity calculated by local flow parameters such as void fraction, gas velocity and liquid velocity measured under steady fully developed bubbly flow conditions. It has been confirmed that the newly developed model of the distribution parameter and the drift velocity correlation developed by Ishii can also be applicable to developing bubbly flows.
We are interested in exploring Advection Upstream Splitting Method (AUSM) schemes for hyperbolic systems of conservation laws which do not allow any analytical calculation of the Jacobian. For this purpose, we consider a two-phase model which has been used for modeling of unsteady compressible flow of oil and gas in pipes. The model consists of two mass conservation equations, one for each phase, and a common momentum equation. Since no analytical Jacobian can be obtained it is more difficult to use classical schemes such as Roe- and Godunov-type schemes. We propose an AUSM scheme for the current two-phase model obtained through natural generalizations of ideas described in M.-S. Liou [J. Comput Phys 129(2) (1996) 364]. A main feature of AUSM is simplicity and efficiency since it does not require the Jacobian. In particular, we prove that the proposed AUSM type scheme preserves the positivity of scalar quantities such as pressure, fluid densities and volume fractions. This guarantees that the scheme can handle the important and delicate case of transition from two-phase to single-phase flow without introducing negative masses. Many numerical results are included to confirm the accuracy and robustness of the proposed AUSM scheme. In particular, it is demonstrated that the AUSM scheme gives low numerical dissipation at volume fraction contact discontinuities and is able to produce stable and non-oscillatory solutions, also when more complex slip relations are used, that is, when the relative motion of one phase with respect to the other is more or less complicated. This makes the scheme suitable for simulations of many important two-phase flow processes.
The relative motion of single long air bubbles suspended in a constant liquid flow in inclined tubes has been studied experimentally. Specially designed instrumentation, based on the difference in refractive properties of air and liquid with respect to infrared light, has been constructed and applied to measure bubble propagation rates.A series of experiments were performed to determine the effect of tube inclination on bubble motion with liquid Reynolds and Froude numbers, and tube diameter as the most important parameters.Particular aspects of the flow are described theoretically, and model predictions were found to compare well with observations. A correlation of bubble and average liquid velocities, based on a least squares fit to the data is suggested. Comparisons with other relevant models and data are also presented.
In this paper we deal with the construction of hybrid flux-vector-splitting (FVS) schemes and flux-difference-splitting (FDS) schemes for a two-phase model for one-dimensional flow. The model consists of two mass conservation equations (one for each phase) and a common momentum equation. The complexity of this model, as far as numerical computation is concerned, is related to the fact that the flux cannot be expressed in terms of its conservative variables. This is the motivation for studying numerical schemes which are not based on (approximate) Riemann solvers and/or calculations of Jacobian matrix. This work concerns the extension of an FVS type scheme, a Van Leer type scheme, and an advection upstream splitting method (AUSM) type scheme to the current two-phase model. Our schemes are obtained through natural extensions of corresponding schemes studied by Y. Wada and M.-S. Liou (1997, SIAM J. Sci. Comput.18, 633–657) for Euler equations. We explore the various schemes for flow cases which involve both fast and slow transients. In particular, we demonstrate that the FVS scheme is able to capture fast-propagating acoustic waves in a monotone way, while it introduces an excessive numerical dissipation at volume fraction contact (steady and moving) discontinuities. On the other hand, the AUSM scheme gives accurate resolution of contact discontinuities but produces oscillatory approximations of acoustic waves. This motivates us to propose other hybrid FVS/FDS schemes obtained by removing numerical dissipation at contact discontinuities in the FVS and Van Leer schemes.
New data is presented for horizontal air/water two-phase flow having various flow regimes. It is shown that drift-flux models are able to correlate these data and that the drift velocity, , is normally finite.
A general expression which can be used either for predicting the average volumetric concentration or for analyzing and interpreting experimental data is derived. The analysis takes into account both the effect of nonuniform flow and concentration profiles as well as the effect of the local relative velocity between the phases. The first effect is taken into account by a distribution parameter, whereas the latter is accounted for by the weighted average drift velocity. Both effects are analyzed and evaluated. The results predicted by the analysis are compared with experimental data obtained for various two-phase flow regimes, with various liquid-gas mixtures in adiabatic, vertical flow over a wide pressure range. Good agreement with experimental data is shown.
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Copyright © 2015 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Fluids (2015)
DOI: 10.1002/fld
- Roe-Type Riemann Solver
- Gas
- Liquid
- Using Drift-Flux
- Copyright
ROE-TYPE RIEMANN SOLVER FOR GAS–LIQUID FLOWS USING DRIFT-FLUX MODEL
Copyright © 2015 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Fluids (2015)
DOI: 10.1002/fld
Hyperbolicity of an approximate form of the drift flux model applied to the vertical ascendant gas-liquid flow
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ES Hyperbolicity of an approximate form of the drift flux model applied to the vertical ascendant gas
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Hyperbolicity of an approximate form of the drift flux model applied to the vertical ascendant gas-liquid flow
- Rosaes Santimcgs Limalem