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Procedia Engineering 205 (2017) 3946–3953
10th International Symposium on Heating, Ventilation and Air Conditioning, ISHVAC2017, 1922 October 2017, Jinan, China
Numerical Simulation of the Fluid Traverses the Porous Media by Single Domain Method Based on UDF Xianbo Niana, Hai Liub ,Yanwei Lia, Yong Liua, Chunsheng Guoa,b,*, Fangyi Qua a a
Dynamics and mechanical & electrical equipment engineering technology research center , Shandong University, Weihai 264209, China b b School of Material Science & Engineering, Shandong University, Jinan 250061, China ccSchool of mechnical, electrical&information engineering, Shandong University, Weihai 264209, China
Abstract The flow pattern and heat and mass transfer in the porous media and fluid mixing area are widely found in the production and engineering, such as heat pipe, building envelope and insulation materials. Especially the slip effect at the at the interface between the fluid and the porous media has a great impact on the flow and heat and mass transfer. Therefore, the processing of the interface in the mixed area is the focus of many studies. In this paper, the mixed area of fluid and porous media is studied by means of theoretical analysis and numerical simulation. The numerical simulation of the mixed region is carried out by using the single domain method. The purpose of controlling the entire mixing area with a set of control equations is achieved by controlling the source terms which are defined by the user-defined function (UDF) in Fluent. And the accuracy and feasibility of the procedure are verified by comparison with numerical simulation of double domain method. Then, by discussing the influence of different physical parameters on the slip characteristics at the interface between fluid and porous media, the mechanism of velocity slip effect is summarized, which lays the foundation for the study of heat transfer and mass transfer. © 2017 The Authors. Published by Elsevier Ltd. © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the 10th International Symposium on Heating, Ventilation and Air Peer-review under responsibility of the scientific committee of the 10th International Symposium on Heating, Ventilation and Conditioning. Air Conditioning. Keywords: porous medium, slip effect, numerical simulation, single domain method, UDF
1. Introduction * *
Corresponding author. Tel.: 13573728778 E-mail address:
[email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility ofthe scientific committee of the 10th International Symposium on Heating, Ventilation and Air Conditioning.
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the 10th International Symposium on Heating, Ventilation and Air Conditioning. 10.1016/j.proeng.2017.09.851
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Porous media and fluid composite systems, and the fluid flow and heat transfer mass transfer phenomenon in systems can be found in many fields, such as building envelope, insulation materials, nuclear energy utilization, microelectronics technology. The key to many problems is the coupling of fluid and porous media. Fluid region and the porous media region have different flow mechanisms, so they have different flow control equations. Therefore, it is necessary to establish a suitable flow coupling condition for the interface between the free fluid and the porous medium. In 1967, Joesph and Beavers[1]were the first to study the boundary conditions of porous media interface. They use the Navier-Stokes equation to describe the free fluid region, and the porous medium to penetrate the Darcy equation to describe the fluid flow in the porous media region. Due to the difference between the two regional control equations, the interface between the two out of the velocity distribution can not be explained. The Darcy's law has some limitations in application. To solve this problem, Beavers proposed a method of introducing a tangential slip velocity at the interface between free flow and porous media, resulting in a velocity slip phenomenon at the interface (Figure 1). Neale, and Nader[2], Vafai and Kim [3] used the Darcy mode which was extended by Brinkman-Forchheimer to explore the fluid motion in porous media, and proposed that the velocity and the shear of the interface should be treated simultaneously.
Fig. 1. Interface velocity slip diagram
Ochoa-Tapia and Whitaker [4] pointed out that when solving the volume integral transfer equation, due to the differences of the integration program for the fluid variable between the two regions, and considering the effect of fluid effective viscosity in the porous medium (Effective Viscosity), it is recommended to using the Shear Stress Jump Condition at the interface. The problem in the application of shear jumps conditions is similar to that of Darcy's law. Estimating the jump degree of the shear force at the interface are the empirical coefficients, and different coefficients will lead to different calculation results. Kuznetsov [5] has studied the effects of shear jump conditions on the velocity of the free-flow region. And it is found that the shear lag and the effective viscous dynamic coefficient can affect the flow velocity distribution. Chandesris and Jamet [6] have found that the shear boundary condition is related to the pressure gradient and presented the corresponding method to deriving the interface boundary condition. Silva and de Lemos [7] applied the mathematics model established by the shear jump condition to analyze the influence of the physical properties of the porous media to the flow velocity distribution in the flow of the mixed region. Many researchers assume that the effective viscous power coefficient is equal to the external fluid viscous power coefficient, and then derived another way to deal with boundary conditions. That is when the dynamic viscosity coefficient is consistent, considering the fluid inside and outside the porous media as two kinds of fluid which have same characters (Single-Domain method) without special handling of interface boundary conditions. When using a single-domain method to solve a mixed-area flow problem, the value automatically satisfies the continuous conditions of velocity, pressure, shear, temperature and flux at the boundary. Vafai and Kim [8] found that the single-region method performed well in verifying the boundary conditions of the interface, and it was possible to predict the change of the gradient of fluid properties at the interface more precisely. Later Chandesris, Goyeau, Ochoa-Tapia [9,10]and some other scientists applied further tests on the single-region method generally, and developed a general-purpose model which includes the single-region method to calculate the
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change of the flow rate profile in the mixed area. Chan [11] studied the turbulent flow of the fluid flowing through the porous medium by the single domain method and predicted the turbulence stress using the k-ε model. As for the turbulence, the general k-ε turbulence mode is inapplicable when treating the turbulent region of the turbulence, such as the region near the solid boundary or the interface of the porous medium which has low porosity. The available literature rarely mentions the boundary conditions of turbulence at the interface, most apply directly the boundary conditions of laminar flow to the turbulence [12]. Silva et al. [13,14] using continuous boundary conditions of turbulent kinetic energy and turbulent dissipation rate at the interface, and introduced the combat flow conditions of Ochoa-Tapia and Whitaker to solve the turbulent flow of the mixed flow problem, At present, the treatment of boundary conditions of fluid turbulence through porous media is still one of the focuses and hot spots in this field. 2. The theoretical model of single domain method In this paper, the single region method is used which treat the fluid in the free fluid region and the porous medium as the same material to solve the fluid flow of mixed area. It can simultaneously solve two different areas of fluid movement with high efficiency. By using the single-region method to solve the problem of the mixing region, there is no need for any iteration of the interface to satisfy the boundary condition of temperature, shear, velocity, pressure and associated flux at the interface. The transmission equation in different regions is different from the source. We use the user-defined function (UDF) in Fluent to define the inlet velocity function and the source function of the transport equation to realize the interaction between the fluid and the porous medium, and then achieve that use a unified equation that single domain method to control the entire model. Theoretically, the Reynold average Navier-Stokes equation can also solve the turbulent flow field, but in the porous media the pore structure is often unknown, in this case, taking the volume average for the Reynold averaged Navier-Stokes equation, and the control equation for the turbulent flow field is derived as follows: Quality equation:
∇ ⋅γ u = 0
Momentum equation: Where
(1)
∇ ⋅ ( ργ uu ) = −∇γ p + μ∇ 2u D +∇ ⋅ ( − ργ u ′u ′ ) + R
is the Darcy velocity,
is the porosity of the porous medium,
(2)
is the average of the time
is the total resistance of the porous average velocity in the characterization of the fluid inside the unit, and medium pores acting on the fluid unit. Modelling with Forchheimer-extended Darcy model mode expressed as:
R=−
μγ K
u D−
CF γρ u D u D K
(3)
The pore features on the pore structure of the porous medium are integrated by volume, and are determined by the permeability K and the porosity . is a non-linear Forchheimer coefficient. Using the general pattern, the formula (2) is similar to the Navier-Stokes equation and is equal to zero when the porous medium is absent. In free . As long as the fluid, (1) and (2) become the Reynold average Navier-Stokes equations when appropriate interface boundary conditions, by adjusting the value of K and ,we can use the single system of equations to solve the problem of fluid migration through porous media and interface. The emergence of the Reynolds shear stress introduces new unknowns, and the k-ε turbulent model is used to close the equation. The equation of turbulence kinetic energy and turbulence dispersion introduced by the k-ε turbulence model. Due to the difficulty of measurement, the turbulence model used in the porous medium has high uncertainty, and the volume average turbulence kinetic energy and turbulence dissipation rate equation of inside the porous medium is obtained by the volume integral of k-ε the turbulent model. The transport equation of the turbulent kinetic energy
k
:
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∇ ⋅ ( ρ uD k
μt
k
) = ∇ ⋅ μ + σ φ ∇γ k − ργ
The transport equation of turbulent energy dissipation rate
u ′u ′ ∇u D + Ck ργ
ε
k uD K
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− ργ ε
(4)
:
2 ε ε uD ε μt ∇ ⋅ ( ρ uD ε ) = ∇ ⋅ μ + ∇ ( γ ε ) − C1ε ( ρ u′u′ ∇uD ) + C2ε ργ Ck − (5) σε k k K Where, σ k , σ ε , C1ε , C2ε is dimensionless experience constant, Ck is dimensionless experience constant. The
third term on the right side of the equation (6) reflects the turbulence kinetic energy generation in the REV due to the emergence of the porous medium, and is associated with turbulent energy and pore structure. and when γ → 1 and K → ∞ it returns to the original transport equation, Similarly, the turbulence energy dissipation rate term for the third term on the right side of equation (7) is the same. For the turbulence model, the numerical simulation using the single domain method is based on the macro of the UDF that comes with the source in the Fluent, and the source kU and of the k and ε transport equation in the porous medium region is standard, − ρε + G + C k ργ K
C1ε G
ε
k
+ C 2ε ρ( C k γ
εU
K
−
ε2 k
).
The above equations are partial differential equations. The control equations and the quoted boundary conditions all contain nonlinear terms, and the exact analytical solution is not simple. It is necessary to change solving the equation problem into solving the algebraic equation problem by numerical method. 3. Numerical simulation results by UDF and discussion The computational domain of numerical simulation, shown in Figure 2, is the problem of the flow of fluid through the mixed region filled with porous media. The fluid enters the channel from the left side by way of steady h f , the flow, the shaded portion of the figure h p represents the porous medium. The height of free fluid area is , and the total depth in channel is H. The following assumptions were made thickness of porous media region is in this study before solving the fluid flow in the mixing region: If the lateral variation is neglected, the motion of the fluid is two-dimensional. The fluid in the channel is incompressible fluid. The fluid is isothermal. The porous that consists of homogeneous material is rigid and non-deformable.
Fig. 2. Fluid swept porous media model
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According to the previously proposed mathematical and physical model of the free fluid region as well as the porous media region, the simulation of the velocity slip effect of the interface between free fluid and porous media was studied by means of ANSYS Fluent. Setting of a virtual 0.2 * 2m^2 rectangle calculation area, the parameters of the porous media region are set: The permeability coefficient is set to 1 × 10 -3 m 2 the porosity is set to γ = 0.6, hp= 0.1 m, hf= 0.1 m, D a= 1 × 10 -3 . The fluid flows from the left side of the fluid region in a way of steady flow. The turbulence model of the source of the fluid crossed movement is simulated under the standard k-ε turbulence model. The turbulence model of FLUENT is compared with the single domain method by UDF, and the correlation and difference of single and double domain are compared. The physical model chooses the standard k-ε model in the turbulent model, where the liquid material is selected as water-liquid. The velocity, X-direction momentum source, turbulence kinetic energy k and the source of the turbulence dissipation ε source are introduced to simulate, and the CFD-Post is used for post-processing. A velocity profile in the above model is created as follows: Rf ≈ 4200 can be obtained by the relevant data from the Reynolds calculation formula: Rf = ρν d / μ .When Rf> 4000, the turbulence mode is determined. The eddy phenomenon occurs in the tail of the runner according to the cloud diagram analysis.
Fig. 3. Turbulence model velocity cloud
Fig.4. Comparison of single and double domain of velocity distribution
CFD-Post establishes the line to select the fluid velocity at the section x = 1m as the object of study. The data table of the single and double domain method is derived by origin. The velocity distribution curve in the same coordinate system is shown in figure 4. According to the analysis of figure 4, under the same initial conditions, the single-domain method controlled by the UDF program and the dual-domain method of the turbulence model in Fluent are similar to the velocity curves simulated in the error range. The correctness of the formula and the
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applicability of the procedure are verified. The single domain method of program control still needs to be optimized to make it closer to the real situation. According to the velocity profile, in the turbulent model, Whether the single domain method or the double domain method, the free fluid region through the fluid velocity is relatively large, and the velocity closed to the wall is very close to zero. While the velocity of fluid flow in porous media is relatively small and uniform, the slip transition at the interface has a phenomenon of velocity drop, and there is a slip phenomenon, that is, at 0-0.1m, the velocity distribution in the free fluid region is set as a parabola, the velocity closed to the wall is very small and close to 0. In the porous medium area of 0→-0.1 m, the abrupt change of velocity decreases suddenly, the initial position of the runner is smaller, and the velocity increases further during the forward flow of the fluid, thus, at the entrance, the fluid through the porous media area is relatively slow, and the velocity increases as the velocity approaches the average velocity.
Fig. 5. Velocity distribution at different Reynolds numbers
The k-ε turbulence model is used for numerical simulation, and the simulation results show that the different Reynolds numbers have no influence on the slip velocity distribution at the interface(figure5). The results are in agreement with the laminar flow calculation results of Choi and Waller (1997), and it shows that the model agrees well with the prediction of bottom bed flow in porous media. Velocity distribution of flow in different porosity showed in figure 6a, we can learn the Darcy velocity increases slowly with the increase of porosity. The Darcy velocity is mainly determined by the Darcy term in the governing equation. So, the effect of porosity is not significant. The main reason for this phenomenon is that the porosity is between 0-1. When it is used for the analysis of series variation of parameters, the influence on the velocity distribution is not obvious. In addition, the velocity in the transition region of porous medium increases with the increase of porosity. The thickness of the transition region increases as well. As the transition region velocity and Darcy velocity increase, the fluid flow in the whole porous medium increases with the increase of porosity. Therefore, in order to maintain the continuity of the overall flux through the interface, the flow velocity in the free flow region increases with the increase of porosity, but the dimensionless slip velocity of the interface decreases with the increase of porosity. We can learn velocity distribution of flow with different Darcy number in figure 6b. The slip velocity at the interface, the velocity in transitional region and Darcy velocity are all decreased significantly because of Darcy number reduced. Free flow velocity increased with Darcy number decreased in order to maintain the continuity of the overall flux through interface. It also can be found that the slip velocity decreases with the decrease of the Darcy number. When the Darcy number is reduced, the resistance effect of the porous medium to the fluid is obvious, and the distribution of the velocity in the excessive area decreases sharply in the short distance.
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Fig. 6. velocity distribution at different (a)porosities and (b)Darcy numbers
4. Conclusion This paper is based on the theoretical analysis of the combination of microscale and macroscale. Firstly, we establish the physical model and mathematical model of free fluid region and porous media region. The fluid region takes the Reynolds-averaged Navier-Stokes equations and low Reynolds numbers k-ε turbulence model as the control equation. While the porous media region uses macroscopic concepts and the Forchheimer-extended Darcy model. We study the flow between the fluid and porous media region with the method of theoretical analysis and numerical simulation, and analyze velocity slip effect on the interface between fluid and porous media by the simulation. The main conclusions are as follows: With same initial conditions, the velocity curves, which are simulated by the single-domain method with UDF and the double-domain method using the turbulence model in Fluent, are almost matched in the error-allowed range, that verifies the correctness of the formula and the applicability of the program. The program-controlled singledomain method still needs to be optimized, making it match the true situation quite well. The study on the influence on flow field of physical quantities such as Reynolds number, Porosity and permeability found that Reynolds number has no effect on slip velocity distribution on the interface, Darcy velocity increases with the increase of porosity, but the variation is small. The Darcy number dropping makes the prominent decrease in slip velocity on the interface, transition velocity and Darcy velocity. Acknowledgements This Study was supported by: 1. National Natural Science Foundation of China (Grant No.61473174); 2. Natural Science Foundation of Shandong Province (Grant No.ZR201702140069); 3. Focus on research and development plan in Shandong Province (Grant No.2017GHY15122); 4. The Open Project Program of State Key Laboratory of Petro-leum Pollution Control (Grant No. PPC2017018), CNPC Research Institute of Safety and Environmental Technology. References [1] [2] [3]
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