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The temperature distribution and pollutant distribution in large reservoirs have always been a hotspot in the field of hydraulics and environmentology, and the three-dimensional numerical modeling that can effectively simulate the interactions between the temperature fields, concentration fields, and flow fields needs to be proposed. The double-diffusive convection lattice Boltzmann method is coupled with a single-phase volume of fluid model for simulating heat and contaminant transfer in large-scale free surface flows. The coupling model is used to simulate the double-diffusive natural convection in a cubic cavity and the temperature distribution of a model reservoir. The mechanism of convection-diffusion, gravity sinking flow, and the complexity of the temperature and the pollutant redistribution process are analyzed. Good agreements between the simulated results and the reference data validate the accuracy and effectiveness of the proposed coupling model in studying free surface flows with heat and contaminant transfer. At last, the temporal and spatial variations of flow state, water temperature stratification, and pollutant transport in the up-reservoir of a pumped-storage power station are simulated and analyzed by the proposed model. The obtained variations of the flow field agree well with the observations in the physical model test and in practical engineering. In addition, the simulated temperature field and concentration field are also consistent with the general rules, which demonstrates the feasibility of the coupling model in simulating temperature and pollutant distribution problems in realistic reservoirs and shows its good prospects in engineering application.

A great number of large reservoirs have been built in China. By 2017, there are 732 reservoirs with a water capacity of over 100 million cubic meters [

At present, the general methods for studying heat and contaminant transfer in LS-FS flows are mainly two-dimensional (2D) numerical models, especially shallow water equations coupling convection-diffusion equations for simulating planar flow problems [

Single-phase free surface lattice Boltzmann (SPFS-LB) model was originally proposed by Thürey in 2003 [

To study the 3D temperature and pollutant distribution in large reservoirs, the double-diffusive convection LB method is coupled with the SPFS-LB model. The rest of the paper is organized as follows. Firstly, the basic ideas and algorithms of methods are briefly described, and the treatments of the coupling scheme for improving stability and precision are addressed. Then, the accuracy and effectiveness of the proposed method are verified by two benchmark cases: the double-diffusive natural convection in a cubic cavity and the temperature distribution of a model reservoir. At last, as an attempt of simulating engineering practice, the process of the flow state, water temperature, and pollutant variations in a practical reservoir is simulated and analyzed.

Unlike traditional two-phase flow methods that set up different distribution functions (DFs) for water and air, the SPFS-LB model only simulates water flows (the denser and more viscous phase) by the single-phase LB equations and tracks the free surface by a constructed boundary treatment. The following assumptions should be made before the simulation by the SPFS-LB model [

Cell types of SPFS-LB model.

The movement of the interface in the SPFS-LB model is realized by the transformation of cell types. And the cell type is judged by the volume fraction _{i}, where Δ_{i} denotes the mass variation on _{i} and _{inv(i)} (see equations (_{inv(i)} = −_{i} [

Therefore, the mass

In the SPFS-LB model, the propagation and collision of LB equations can be normally performed in filled cells for that they are not adjacent to empty cells. However, the interface cells are always surrounded by empty cells, where physical quantity and DFs are not defined (shown in Figure _{i}Δ

It should be pointed out that not only the missing DFs but also all the DFs whose discrete velocity _{i} satisfies _{i}

Interface cells need special treatments after the updating of their mass ^{ex} and

From a large-scale point of view, the water flows in reservoirs are normally considered incompressible, whose viscosity, density, thermal diffusivity, and contaminant diffusivity can also be assumed to be constant except for the buoyant. Therefore, the governing equations for simulating heat and contaminant transfer in reservoirs can be expressed with the Boussinesq assumption [

_{i}, _{i} refer to the DFs for the velocity field, temperature field, and concentration field. Variable

The equilibrium distribution functions

To derivate governing equation (_{i} (

_{i} is the weight coefficient, _{0} = 1/3, _{1∼6} = 1/18, and _{7∼18} = 1/36 in the D3Q19 model. The macroscopic quantities

Simulating 3D temperature fields and concentration fields in large-scale flows by the D3Q6 model can reduce calculation amount and save computer memory usage. However, when precisely analyzing the small-scale flow problems with complex boundary conditions, using the D3Q19 model instead of the D3Q6 model can retain better accuracy and stability.

The number of computational grids for simulating the model reservoir (in Section

The SPFS-LB model is firstly coupled to the DDC-LB method in the present work, and several key treatments need to be adopted to make the coupling scheme compatible and to obtain reasonable simulated results.

As described in Section

Some empty cells would convert to interface cells after the update of cell types, and the temperature _{i} should be initialized. This coupling model neglects the exchange (absorption or release) of heat and contaminant between water and air. Therefore, the macroscopic _{i}, DFs _{i} in new converted interface cells can be initialized by equations (

The flow velocity dominates the transfer of heat and contaminant, while the buoyancy induced by the nonuniform distribution of temperature and concentration has an impact on the water flows. To simulate the buoyancy flows and realize the two-way coupling between the three fields, an additional body forcing term Fi needs to be attached to equation (_{T} (_{0}) + _{C} (_{0})).

Turbulence exists in practical large-scale water flows with heat and contaminant transfer. The subgrid-scale model based on large eddy simulation is introduced to the coupling scheme. The physical variables are separated into large-scale variables and small-scale variables by a certain filter function

The following equation should be obtained through filtering equation (

The relaxation time _{e} is not a constant. It varies with space and time, which can be calculated by _{e} can be understood as the total viscosity _{e} = _{1} + _{t}, in which _{1} and _{t} denote the molecular viscosity and eddy viscosity (or turbulence viscosity), respectively. _{1} is determined by the physical property of water, and _{t} can be computed by

This section makes a summary of the calculation procedure of the coupling scheme proposed in the paper. Before the iteration, the geometry needs to be initialized, and the reasonable boundary condition should be adopted in accordance with the simulated flow problems. Then given all the variables at time step

Perform LB propagation with the boundary conditions

Compute the mass flux, and update the mass of cells as described in Section

Calculate the volume fraction of nonempty cells

Reconstruct the DFs of the flow field, temperature field, and concentration field by equations (

Compute the macroscopic velocity, pressure, temperature, and concentration through equation (

Obtain relaxation time _{e} (Section

Allocate the excessive or negative mass through equation (

Initialize the macroscopic quantity and distribution functions of new emerging interface cells as described in Sections

Perform the analysis of convergence

To validate the feasibility and accuracy of the proposed coupling scheme, the double-diffusive natural convection in a cubic cavity and the temperature distribution of a model reservoir are simulated, and the detailed comparison and analysis are made between the simulated results and the reference data. If not otherwise stated, the acceleration of gravity, water density, and kinematic viscosity are set as ^{2}, ^{3} kg/m^{3}, and ^{−6} m^{2}/s, respectively.

In the heat and contaminant transfer flows, the double-diffusive natural convection in a cubic cavity reflects the essential problem of two-way coupling between temperature field/concentration field and flow field. At present, there have been many classic physical experiments or numerical research results [_{0}/_{0} and _{1}/_{1}, respectively (_{0} < _{1}; _{0} < _{1}); the top and bottom are adiabatic for heat transfer and impermeable for contaminant transfer; that is, the derivatives of temperature fields and concentration fields are always 0 along the normal direction of the boundary. Initially, the cubic cavity is filled with homogeneous fluid with velocity _{0} + _{1})/2, concentration _{0} + _{1})/2, and Pr = 0.71. The flow characteristics of double-diffusive natural convection can be estimated by the following three dimensionless numbers: temperature Rayleigh number _{0}/_{1} and concentration _{0}/_{1} can be assigned to arbitrary numbers that satisfy the formulas above, such as _{0}/_{0} = −1 and _{1}/_{1} = 1 in the present simulation.

Schematic diagram of double-diffusive natural convection in a cubic cavity.

In order to quantify the simulated results, several dimensionless numbers along the right hot wall are calculated below, which are local Nusselt number, average Nusselt number, local Sherwood number, and average Sherwood number, respectively, as shown in the following equations:

Firstly, the Rayleigh number and Lewis number are set as Ra_{T} = 10^{4}, Ra_{C} = 10^{4}, and Le = 1, respectively. The fluid in the cavity is driven by both temperature field and concentration field. The added driving flows are generated if the driving directions of temperature and concentration are the same. Otherwise, the opposed driving flows are generated. Both flows are simulated in this section. The computational grid is set as 128 × 128 × 128. It should be noted that the simulated distribution of temperature field and concentration field is exactly the same, for the same convection-diffusion equation is applied to simulate heat transfer and contaminant transfer, and the same calculation parameters are set in the present simulation in order to compare with the research results in [

Figures

Isothermal surfaces. (a) Added driving flows. (b) Opposed driving flows.

Streamline and velocity gradients of added driving flows.

For the opposed driving flows, the buoyancy caused by temperature difference and concentration difference is exactly equal in magnitude and opposite in direction, and their driving effect on the flow field also cancels each other out. The fluid in the cavity does not flow at all (see Figure

The local Nusselt number of the added driving flows along the right wall (hot wall) is calculated and plotted in Figure

Local Nu/Sh number of the added driving flows along the right wall.

Additionally, the added driving double-diffusive convection with the temperature Rayleigh Ra_{T} = 10^{7} and concentration Rayleigh number Ra_{C} = 10^{6}, 5 × 10^{6}, 1.5 × 10^{7}, and 5 × 10^{7} is simulated. The average Nusselt number and average Sherwood number are calculated along the hot wall. According to Table

Comparison of average Nusselt number and average Sherwood number along the hot wall.

Rayleigh number | Dimensionless number | Present | Reference [ |
---|---|---|---|

Ra_{T} = 10^{7} | 16.036 | 16.0 | |

Ra_{c} = 10^{6} | −16.036 | −16.0 | |

Ra_{T} = 10^{7} | 13.684 | 13.6 | |

Ra_{c} = 5 × 10^{6} | −13.684 | −13.6 | |

Ra_{T} = 10^{7} | 13.647 | 13.6 | |

Ra_{c} = 1.5 × 10^{7} | −13.647 | −13.6 | |

Ra_{T} = 10^{7} | 23.603 | 23.7 | |

Ra_{c} = 5 × 10^{7} | −23.603 | −23.7 |

In this section, the variation process of temperature field and flow field in a model reservoir is simulated by the proposed coupling model, so as to demonstrate its effectiveness in simulating practical physical problems. The simulation is based on the data of gravity undercurrent experiments, tested by the US Army Corps of Engineers, Waterways Experiment Station [^{3}/s, while the water flows out from the outlet at the same flow rate.

General layout of reservoir model.

Due to the lateral symmetry of the simulation, only a half is selected for calculation and results show (see Figure

Figures

Jet penetrating stage: once the cold water flows into the reservoir, it sinks and forms an undercurrent that moves along the bottom of the reservoir due to its higher density. And a large temperature gradient zone appears near the inlet, as shown in Figures

Tongue-shape isothermal surface stage: when the cold current forward reaches the vertical expansion section of the reservoir along the bottom slop, it accelerates and keeps a high speed due to the gravity, and the tongue-shaped isothermal surfaces form (see Figures

Horizontal thermal stratification stage: when the tongue-shaped cold current comes to the right wall of the reservoir, the thermal stratification formally formed. The backflow area at the top of the reservoir stretches across the entire reservoir and reaches the longest, as shown in Figure

Isothermal surfaces distribution of the model reservoir at different times: (a)

Water surface and velocity of the model reservoir at different times: (a)

Based on the above analyses, the flow field and the temperature field are strongly coupled. The accurate simulation of buoyant flows is the key to study the thermal stratification of the reservoir water. Caused by the temperature difference, the inflow water sinks and the transfer of vertical heat is restricted during the whole moving process of the cold current, which strongly affects the formation, development, and stabilization of flow state and temperature distribution. It also explains the mechanism of horizontal thermal stratification in reservoirs.

Figure

Velocity

Histories of the discharge water and

An upper reservoir of a pumped-storage power station is about 700 m in length and width, and the maximum depth is about 25 m (shown in Figure _{0} = 1) in the center of the reservoir bottom (

The changing processes of the flow field, temperature field, and concentration field in the upper reservoir under the pumping condition are simulated. The simulation conditions are as follows: ① at the time of _{0} = 0°C flows into the reservoir at a flow rate of 640 m^{3}/s; ② at _{1} = −1°C, and remain the flow rate unchanged; and ③ at

Figure

The upper reservoir of a pumped-storage power station: (a) topographic map; (b) geometry.

Under the pumping operation modes, the water bypasses Mountain A and Mountain B (Figure

Water area and velocity distribution at different times: (a)

Figure

Isoconcentration surface distribution of an upper reservoir: (a)

The point pollution source is located between the two mountains. According to the analysis of flow fields in Section

At the time of

Isothermal surface distribution of an upper reservoir: (a)

When the cold water just enters the reservoir (

This paper summarizes the general conclusion that explains the spatial-temporal variation process of water temperature in the case of the injection of cold water into a warm reservoir. On the whole, the coupling model proposed in this paper is effective and feasible to simulate the large-scale free surface flows with the heat and contaminant transfer in reservoirs.

The SPFS-LB model is firstly coupled to the DDC-LB method in this paper to simulate large-scale flows with heat and contaminant transfer. To make the coupling scheme compatible and to obtain reasonable results, several key treatments described in Section

Compared with the traditional 2D numerical models, 3D temperature, pollutant, and velocity distribution can be obtained in this work. In addition, the multifield coupling mechanism of this model makes the simulation more accurate. These are the main advantages of the proposed coupling model. It should be noted that the proposed model ignores the exchange (absorption or release) of heat and contaminant between water and air. The nonpoint source term will be introduced in equation (

Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.

The authors declare no conflicts of interest.

Wei Diao contributed to methodology, investigation, writing of the original draft; Hao Yuan contributed to resources and validation; Cunze Zhang contributed to funding acquisition; Liang Chen contributed to software; Xujin Zhang contributed to supervision and writing, review, and editing.

The authors are grateful for the financial support from the Science and Technology Research Program of Chongqing Municipal Education Commission, Grant no. KJQN201800712.