Research Article  Open Access
Pan Wu, Fei Chao, Dan Wu, Jianqiang Shan, Junli Gou, "Implementation and Comparison of HighResolution Spatial Discretization Schemes for Solving TwoFluid SevenEquation TwoPressure Model", Science and Technology of Nuclear Installations, vol. 2017, Article ID 4252975, 14 pages, 2017. https://doi.org/10.1155/2017/4252975
Implementation and Comparison of HighResolution Spatial Discretization Schemes for Solving TwoFluid SevenEquation TwoPressure Model
Abstract
As compared to the twofluid singlepressure model, the twofluid sevenequation twopressure model has been proved to be unconditionally wellposed in all situations, thus existing with a wide range of industrial applications. The classical 1storder upwind scheme is widely used in existing nuclear system analysis codes such as RELAP5, CATHARE, and TRACE. However, the 1storder upwind scheme possesses issues of serious numerical diffusion and high truncation error, thus giving rise to the challenge of accurately modeling many nuclear thermalhydraulics problems such as long term transients. In this paper, a semiimplicit algorithm based on the finite volume method with staggered grids is developed to solve such advanced wellposed twopressure model. To overcome the challenge from 1storder upwind scheme, eight highresolution total variation diminishing (TVD) schemes are implemented in such algorithm to improve spatial accuracy. Then the semiimplicit algorithm with highresolution TVD schemes is validated on the water faucet test. The numerical results show that the highresolution semiimplicit algorithm is robust in solving the twopressure twofluid twophase flow model; Superbee scheme and Koren scheme give two highest levels of accuracy while Minmod scheme is the worst one among the eight TVD schemes.
1. Introduction
In many industrial applications especially in nuclear industries, twophase flows exist widely and are the most important phenomenon. Accurate calculation of twofluid flows is a subject of intense interest and of great importance in research topics. There are many important models in literature for describing twophase flows. Herein, the twofluid sixequation model seems to be the most complete approximation for twophase flows. Hence, current main reactor thermalhydraulics analysis codes such as RALAP5, TRACE, TRAC, and CATHARE are all based on such sixequation model.
However, due to an instantaneous equilibrium pressure assumption, such sixequation model has been proved to be illposed, which means that the initial value problem of the sixequation system is nonhyperbolic and could lead to numerical oscillations. From the book of De Bertodano et al. [1], which is of great reference value for investigating the stability of twofluid model, the oscillations are caused by the Kelvin–Helmholtz instability (KH). When the relative velocity exceeds the Kelvin–Helmholtz instability criterion, the oscillations occur. In this book, De Bertodano et al. [1] show how the KH instability behaves in horizontal stratified flow for a wellposed fixedflux model with surface tension compared to an illposed one without surface tension. For the wellposed model, the shortwavelength ripples die out and the large wave grows with time while for the illposed model the shortwavelength has a larger growth rate and dominates the solution after a short time.
For simulating complicated twophase flow phenomenon of many nuclear power plant accidents, the illposedness of the differential equations presents a problem for higher order numerical methods or finer grids. Phasic vapor/liquid velocities are generally different, especially in the (fast) transients of nuclear power plant accidents where twophase flow phenomenon is very complicated and phasic relative velocity may exceed Kelvin–Helmholtz criterion. To overcome the illposed issue of twofluid singlepressure model and obtain wellposed numerical model, there are three important ideas: implementation of the interfacial pressure term used in CATHARE [2], implementation of the virtual mass force term used in RELAP5 [3], and application of the twopressure model. The interfacial pressure differential term and the virtual mass force differential term are adding to phasic momentum equations to restore the hyperbolicity. The present authors investigated the illposed characteristic and analyze illposed regions of the twofluid singlepressure model and the effect of the virtual mass force and the interfacial pressure on improving the illposedness [4]. The results showed that such twofluid sixequation singlepressure model cannot completely avoid the illposedness with the virtual mass force and the interfacial pressure, only the twofluid twopressure model in which each phase is assumed to have its own pressure is a wellposed model in all situations.
Many researchers carried out the study of twopressure model due to the wellposed advantage since 1976. The most important twopressure model is the twofluid sevenequation model to which much attention has been devoted [5–14]. Such sevenequation twopressure model consists of two mass conservation equations, two momentum conservation equations, two energy conservation equations, and a volume fraction transport equation. Using the volume fraction propagation as a basic conservation equation is widely used because it changes the structure of conservation equation system and makes sevenequation model unconditionally hyperbolic (wellposed) in the sense of Hadamard [15]. In recent years, the development of advanced thermalhydraulics system analysis codes has aroused great interest such as RELAP7 [16], NEPTUNE project [17], CASL [18], and CATHARE3 [19]. RELAP7 code developed by the Idaho National Laboratory is ongoing now. It should be noted that RELAP7 code is based on such sevenequation model.
In the last few decades, a volume of work has been conducted on the numerical computation of this sevenequation twopressure model. Liang et al. [20] and Zein et al. [21] presented the operator splitting approach to decompose the sevenequation model into the hyperbolic operator and the relaxation operators. They used Godunov scheme with the HLLC flux to gain the numerical scheme for solving such sevenequation model. Gallouët et al. [22] proposed two finite volume methods based on Rusanov scheme and Godunov scheme to solve such twopressure model. The source terms such as relaxation terms, the phase change, and gravity were calculated by a fractional step approach in his scheme. Ambroso et al. [23] constructed a new approximate Riemann solver for the numerical approximation of the sevenequation model. Berry et al. [6] and Abgrall and Saurel [24] developed the discrete equation method (DEM) with a HLLC scheme to solve the sevenequation model. They analyzed two pressure relaxation cases: infinitely fast and bounded with a realistic specific interfacial area. Moreover, Berry et al. took into account the interphase heat and mass transfer in this twopressure model. The semiimplicit method with the staggered grid mesh is widely used in existing reactor thermalhydraulics analysis codes (RELAP5, TRAC, TRACE, etc.) due to the advantage of high efficiency and stability. However, there is little information about the semiimplicit scheme to solve the sevenequation twopressure model in existing public literature. Consequently, more investigations are essential to studying the semiimplicit algorithm for solving such twopressure model.
In addition to great interest in the development and simulation of advanced twofluid model, highorder accuracy schemes have also attracted great increasing attention. Many current reactor thermalhydraulics codes like RELAP5 and CATHARE were developed based on classical firstorder upwind scheme. Using such loworder scheme to make the convection terms of conservation equations discrete produces excessive numerical diffusion. This disadvantage has been realized in many nuclear thermalhydraulics applications such as hydraulic load analysis of loss of coolant accident [25], long term transient natural circulation flow [26, 27], flow instability [28] or stability analysis [29], and boron solute transport [30]. In the past, there are many classical linear schemes like centraldifference scheme (CD), QUICK, thirdorder upwind scheme (TOU), Fromm scheme [31], and secondorder upwind scheme (SOU) to reduce high numerical diffusion [32]. These linear schemes are at least of 2ndorder accuracy and they are unbounded. However, Godunov [33] proved that linear unbounded highorder schemes are not mathematically monotonic as compared to 1storder upwind scheme, allowing unphysical oscillations under some circumstances. Only bounded nonlinear highorder schemes can be monotone and stable. Nonlinear flux limiter schemes which fulfill total variation diminishing (TVD) criteria are one of the most effective methods to achieve highorder accuracy and stability and many researchers have investigated these nonlinear flux limiters. Tiselj and Petelin [34] developed the PDE2 code based on the sixequation model with flux limiter Minmod in achieving secondorder accuracy. Wang et al. [29] have implemented highresolution flux limiters into TRACE code to improve spatial accuracy of convection terms in mass/energy equations and the performance of six nonlinear flux limiters was assessed. Wang et al. [35, 36] also used LaxWendroff (LW) scheme with flux limiters to achieve the 2ndorder accuracy for both spatial discretization and time integration and added the ENO limiter scheme into TRACE for BWR stability analysis. Abu Saleem et al. [37] developed a new flux limiter based on a combination of 1storder upwind scheme and 3rdorder QUICK scheme to achieve highresolution of the solver for the twophase singlepressure model. Zou et al. [30] adapted an existing highresolution spatial discretization scheme to reduce numerical errors. In his work, the highresolution scheme was applied for the mass and momentum equations only, and energy equations were neglected. In all the work mentioned above, the highorder schemes are performed for twofluid sixequation singlepressure model only; few studies have been done on implementing highresolution scheme in solving twopressure model.
In the present work, a semiimplicit algorithm using highresolution TVD schemes is derived to calculate the twopressure model. The accuracy and robustness of the proposed numerical scheme are validated with the water faucet benchmark test. Eight highresolution flux limiter schemes, Minmod [38], Superbee [39], Harmonic [40], OSPRE [41], Van Albada [42], SMART [43], Koren [44], and MUSCL [45], are implemented to evaluate the performance of their accuracy. Consequently, this research will potentially lay the foundation for simulating twophase flows based on the twopressure model with higher fidelity algorithm. This paper is organized as follows. Section 2 introduces twofluid twopressure mathematical model. The numerical algorithm using highresolution TVD scheme for solving this model is described in Section 3. Next, the validation test for the proposed scheme is presented in Section 4. At last, Section 5 is devoted to the conclusion.
2. TwoFluid TwoPressure Mathematical Model
In nuclear industries, the onedimensional form of twofluid model is more economical and used widely. The twofluid sevenequation twopressure model used in this paper is summarized as follows [16, 22, 23, 46], which allows for a change of the pipeline cross section [34].The gas and liquid volume fractions and are related by the saturation constraint ; the pressure relaxation coefficient stands for the relaxation rate at which the phase pressures equilibrate. As derived in [8], the interfacial pressure and velocity are, respectively, determined byin which The pressure relaxation coefficient can be expressed in the following form (see [22, 23, 46]):where is the pressure relaxation time (s). Here we chose an empirical constant s and the numerical results in this paper show reasonable agreement with the exact/analytical solutions. The net interfacial mass transfer per unit volume can be calculated by [3, 37, 47]
3. Numerical Scheme
In this paper, the finite volume method based on the staggered mesh is used to discretize seven conservation equations of the twopressure model and the semiimplicit solution algorithm using highresolution TVD schemes is developed to solve such twopressure model. Figure 1 shows the staggered mesh schematic. Based on the staggered mesh, the scalar variables such as phasic pressures , specific internal energies , phasic temperatures , and phasic volume fractions are described at the volume centers and the vector quantities (velocities ) are defined at the volume edges.
3.1. HighResolution TVD Scheme
Using the convection flux of phasic mass conservation equations and the convection flux of phasic energy conservation equations as examples, these convection fluxes can be, respectively, discretized asIt should be noted that the terms and in energy equations are treated similarly to the convection flux. Order of accuracy for spatial discretization depends significantly on the scheme to calculate the donor quantities with an overdot in numerically evaluated fluxes. There are many numerical methods calculating these quantities; one of the most effective methods is to construct flux limiters based on total variation diminishing (TVD). In this paper, the traditional 1storder upwind scheme and standard highorder schemes are also discussed for the comparisons with highresolution flux limiter schemes.
For a donor quantity (this can be , or ), the numerically evaluated general form can be written as [32, 47]in which is termed the flux limiter function and the gradient ratio is given byLet ; the traditional 1storder upwind scheme (FOU) can be obtained from (12).Though FOU scheme is a monotone scheme avoiding nonphysical numerical oscillations, the scheme is of 1storder accuracy and has significant numerical diffusion resulting in relatively large spatial discretization error. This will be shown in Section 4.
Let the limiter function be a linear unbounded function; standard highorder linear schemes can be derived from (12), as summarized in Table 1.

These linear schemes are of at least 2ndorder accuracy, reducing numerical diffusion effectively, but often lead to unphysical spatial oscillations for the case of sharp gradients which will be shown in the next section.
Let the limiter function be a bounded function; many researchers [32] have proposed monotonic highresolution schemes which satisfy TVD [48]; here eight flux limiters are selected, as Table 2 shows.

These bounded flux limiter functions are illustrated in Figure 2. All the limiter schemes in Table 2 fulfill TVD and preserve monotonicity to avoid unphysical numerical oscillations under the circumstance of steep gradients. Hereinto, Minmod is the lower bound of secondorder TVD region and Superbee is upper bound of secondorder TVD region. SMART, Koren, and MUSCL are bounded standard highorder schemes such as QUICK, thirdorder upwind scheme, and Fromm’s scheme. Harmonic, OSPRE, and Van Albada are symmetric polynomialratio schemes.
3.2. SemiImplicit Scheme
RELAP5 code uses some numerical convenient set to overcome serious numerical instabilities associated with gas/liquid phase appearance and disappearance when solving the sixequation singlepressure model [3]. These numerical challenges arise from the degeneration of the model to the singlephase case or opposite resulting in a singular equation system. Similar to RELAP5 [3], for dealing with numerical challenges in phase appearance and disappearance in solving the advanced twopressure model, mass and momentum conservation equations (1)–(4) are rearranged into sum and difference equation forms and the time derivative terms in conservation equations (1)–(6) are expanded in this numerical scheme.
Such semiimplicit scheme is achieved by treating implicitly phasic velocities in mass and energy conservation equations, pressure gradient in momentum conservation equations, and all interfacial exchange terms in conservation equations and all the rest terms are treated effectively at the new time [49, 50]. The time advancement for all conservation equations (1)–(7) is achieved by the firstorder approximation. Superscripts “” and “” in the following discretization equations denote the old and new time step, respectively, and subscripts containing “” determine the spatial position of scalar variables and subscripts containing “” determine the spatial position of vector variables; the variables with a dot are donor quantities calculated from (12); the variables with an overtilde are the intermediate new time variables that will be corrected in the final.
Expanding the time derivative in phasic mass conservation equations (1) and (2) and adding these two expanded mass conservation equations yield the sum mass conservation equation. The corresponding finite discretization equation in the control volume can be modeled asIn the same way, expanding the time derivative in the mass conservation equations (1) and (2), subtracting these two expanded mass conservation equations, and substituting (10) for yield the difference mass conservation equation. The finite discretization form of the difference mass conservation equation can be given byBy substituting (10) for , the finite difference equation for the volume fraction transport equation (7) readsThe expanded forms of the gas energy equation are defined by expanding the time derivative in gas energy equation (5) and substituting (10). The finite discretization form for the expanded gas energy equation is Expanding the time derivative in liquid energy equation (6) and substituting (10) yield the expanded liquid energy equation. The finite discretization form for the expanded liquid energy equation is expressed asMomentum equations (3)(4) are firstly rearranged into the expanded forms by substituting mass conservation equations (1)(2).Adding (20) and (21) yields the sum momentum equation. The sum momentum equation can be converted into the following discretization form.The difference momentum equation is obtained by dividing gas momentum equation (20) by and dividing liquid momentum equation (21) by , respectively, and subtracting them. The finite difference equation for the difference momentum equation can be written asA variable with an overbar is calculated as an averaged quantity in the following.In order to close the system, equations of state for water properties are supplemented. The properties are based on IAPWS IF97. In this paper, phasic densities and temperatures are provided as functions of phasic pressures or/and phasic specific internal energies and the linearized equations of state are given as
On substituting directly (25) into these seven discretization equations (15)–(19) and (22)(23), seven discretization equations correspond to seven unknown solution variables (, and . For a system containing volumes, these seven discretization equations form a linear algebraic equation set with unknown solution variables which can be solved by Gauss elimination solver [51]. The intermediate new time variables , and are solved by expanded time derivative forms of conservation equations such that there are linearization errors in these solutions. To alleviate these errors, the final , and are solved again by unexpanded forms of mass and energy conservation equations where , and are used for evaluation of the source terms of conservation equations, similar to RELAP5 [49].
4. Numerical Test
In this section, the water faucet problem is simulated to assess the proposed solution scheme based on twopressure model. Here, highresolution TVD schemes are implemented in such scheme to demonstrate the highorder of spatial accuracy. For comparisons, results from traditional 1storder upwind scheme and classical highorder linear schemes are shown.
4.1. Water Faucet Problem
The water faucet problem proposed by Ransom [52] is one of the most important benchmark tests for validating the ability of numerical scheme to calculate twophase flows. The geometric configuration is a vertical tube of 12 m in length with the diameter of 1 m. Initially, the tube is composed of a uniform annulus of gas with initial velocity and a surrounded uniform column of water with initial velocity The initial gas volume fraction is 0.2 and all the initial phasic pressures are set to 0.1 MPa. The wall and interfacial drags are ignored and no phase transition is assumed; then the flow goes down driven by the gravity. A schematic diagram of the time evolution for this water faucet problem is shown in Figure 3. Based on the above initial conditions and assumptions, analytical solutions for the gas volume fraction and liquid velocity distribution with time along the pipe length direction have been obtained and are given in the following [53, 54].The gravity accelerates the rate of the liquid; then the liquid column becomes thinner with time; there is a moving discontinuity in the profile (see Figure 3), where it is a very important region for testing the accuracy of the numerical scheme and its stability near discontinuities.
All the following numerical results are calculated using a fixed initial liquid Courant number of CFL_{f} = 0.2. Numerical simulations using the classical FOU scheme are performed first. Figure 4 shows results of gas volume fraction distribution with 96 cells and 192 cells at times 0.2, 0.5, and 0.75 s. This figure illustrates the numerical stability to handle discontinuities and the numerical results are excellently consistent with the corresponding analytical solutions except the near discontinuities (. The difference near discontinuities between numerical and analytical solutions is caused by the numerical diffusion from the FOU scheme. As shown in this figure, the numerical results with 192 cells are more accurate than those with 96 cells which means that the finer the grid is, the smaller the numerical diffusion is. However finer meshes require more computation time and decrease the efficiency; highorder schemes are needed to reduce numerical errors more effectively. The numerical results with 96 cells of liquid velocity distribution are presented in Figure 5. It can be observed that the calculated liquid velocities are in very good agreement with the corresponding analytical solutions such that the calculated results with 192 cells are not given in this figure.
Furthermore, finer grid FOU resolutions ranging from 384 to 1152 cells are studied with singlepressure model and twopressure model, as shown in Figures 6 and 7, and the comparisons of them are shown in Figure 8. For the case of 1152 cells, numerical solutions of singlepressure model show a big undershoot at the discontinuity which is due to the nature of illposedness. As compared to singlepressure model, numerical solutions of twopressure model are more stable under the same conditions.
Secondly, numerical simulations using standard highorder linear schemes (Table 1) are performed with 96 cells. Figure 9 shows the gas volume fraction distribution using standard highorder linear schemes at 0.75 s. For the comparisons, the result from FOU scheme is also shown in the figure (dash line). It is observed that, as compared to FOU scheme, standard highorder linear schemes reduce the numerical dissipation effectively and TOU scheme seems to be more accurate than other standard highorder linear schemes due to its 3rdorder precision. But unphysical oscillation occurs near discontinuities in some schemes (TOU, Fromm, CD, and SOU scheme). This oscillation can be explained by Godunov’s order barrier theorem [33] which states that linear unbounded highorder schemes used to solve the advection equation are not monotonic, allowing unphysical oscillations under some circumstances such as this case of discontinuities. To achieve monotonic highorder schemes, constructing bounded flux limiters based on TVD is one of the most effective methods. Some flux limiters are piecewiselinear functions constructed from bounded standard highorder linear schemes without compromising their highorder precision as shown in Figure 9.
Finally, eight highresolution TVD schemes in Table 2 are analyzed with 96 cells. Figure 10 shows the gas volume fraction distribution using highresolution TVD schemes at 0.75 s and results for various highresolution TVD schemes are presented in Figure 11, respectively. All of the flux limiter schemes have effectively reduced numerical diffusion and improved the prediction of gas volume fraction. As expected, when compared to unstable results from unbounded SOU, TOU, CD, and Fromm scheme, the corresponding bounded TVD schemes Minmod, Koren, and MUSCL produce stable and more accurate results. Among all the flux limiter schemes, Minmod performs worst to reduce numerical diffusion at the discontinuity point Results from Superbee seem to be more accurate than those from the other seven schemes at the discontinuity point.
In order to analyze the order of accuracy of considered highresolution schemes, L_{1} norm of errors between the numerical results and exact solutions on gas volume fraction is assessed and given aswhere is the number of control volumes in the domain; here ; is the numerical solution at and is the exact solution from (26) at . Table 3 shows the order of decreasing accuracy for the TVD schemes. Obviously, Superbee is the most accurate scheme while Minmod is more smeared than other schemes. Koren performs the second best due to a bounded thirdorder upwind scheme based on TVD.

5. Conclusions
In this paper, a semiimplicit numerical algorithm based on the finite volume method associated with staggered grids has been developed to solve the advanced wellposed twofluid sevenequation twopressure model. To overcome the challenge of excessive numerical diffusion from FOU scheme, eight highresolution TVD schemes are implemented in such numerical algorithm to improve spatial accuracy. Then such new numerical algorithm is validated on the water faucet test, demonstrating highorder spatial accuracy of TVD schemes and robustness near discontinuities. Numerical comparisons reveal that Superbee and Koren give two highest levels of accuracy while Minmod seems to be the worst scheme as compared to other schemes. This research will lay the foundation for more accurately simulating twophase flows based on the twopressure model with higher fidelity algorithm.
Nomenclature
Symbol:  Gas density (kg·m^{−3}) 
:  Liquid density (kg·m^{−3}) 
:  Gas velocity (m·s^{−1}) 
:  Liquid velocity (m·s^{−1}) 
:  The crosssectional area in the pipeline (m^{2}) 
:  The net gasliquid mass transfer per unit volume (kg·m^{−3}·s^{−1}) 
:  Gas pressure (Pa) 
:  Liquid pressure (Pa) 
:  Interfacial pressure (Pa) 
:  The gravity acceleration (m·s^{−2}) 
:  The interfacial velocity (m·s^{−1}) 
:  The wall friction coefficient for phase 
:  The hydraulic diameter (m) 
:  The interfacial area between two phases per unit volume (m^{−1}) 
:  The continuous phase density (kg·m^{−3}) 
:  The interfacial drag coefficient between two phases 
:  Vapor/gas specific internal energy (J·kg^{−1}) 
:  Liquid specific internal energy (J·kg^{−1}) 
:  Interfacetogas convective heat transfer coefficient per unit volume (W·m^{−3}·K^{−1}) 
:  Interfacetoliquid convective heat transfer coefficient per unit volume (W·m^{−3}·K^{−1}) 
:  Temperature for phase (K) 
:  The saturation temperature under the interfacial pressure (K) 
:  The gas specific enthalpy evaluated at the interfacial mass transfer condition (J·kg^{−1}) 
:  The liquid specific enthalpy evaluated at the interfacial mass transfer condition (J·kg^{−1}) 
:  The pressure relaxation coefficient function (Pa^{−1}·s^{−1}) 
:  The interfacial density corresponding to the liquid saturated density with (kg·m^{−3}) 
:  The phase acoustic impedance (kg·m^{−2}·s^{−1}) 
:  The phase sound velocity (m·s^{−1}) 
:  The length of the control volume (m) 
:  Volume of the control cell (m^{3}) 
:  Number of control volumes in a system. 
:  Gas phase 
:  Liquid phase 
:  Phasic interface 
:  Gas phase () or liquid phase () 
Initial:  Initial condition. 
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The research is supported by Natural Science Foundation of China (Approval no. 11675125) and Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China (Approval no. LRSDT2017401).
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