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Application of the Lattice Boltzmann method to open channel flow with liquid-solid phase changes 長岡技術科学大学 大学院工学研究科 Ayurzana Badarch 長岡技術科学大学 環境社会基盤工学 細山田 得三 1. Introduction In the northern sphere of earth, a common state of natural water is an ice in winter time. Ice and water mixture is one of the major subjects of the cold region engineering and motivation of this study. Ice acts like a solid in water flow and situation will become even more difficult when thermodynamic is involved in system. The liquid-solid phase changes are often referred as Stefan problem, which has complexity having moving boundary showing liquid-solid interface in time. Solving moving body boundary brings many difficulties for conventional methods. State-of-Art methods for phase change in closed region are fixed grid based approach and front tracking approach, however they require to solve additional sets of equation and boundary condition on the moving boundary interface. Recently, particle based method appear to solve phase change in free surface flow and performance of those approach is limited for laboratory scale. In this paper, we present numerical procedure for liquid-solid phase change of water with Lattice Boltzmann method (LBM). The numerical procedure is composed as follows. Two distribution functions approach will be used to account flow field and temperature field on fixed grids with 9 directional lattices. The enthalpy-based non iterative method is used for phase change of fluid depending on local enthalpy updates 1) . For open channel fluid flow, free surface formulation of single phase LBM is employed 2) . 2. Numerical procedure For fluid flow with free surface representation, discretized Boltzmann equation with the Bhatnagar-Gross-Krook collision operator and modification of the immersed moving boundary is to be solved by collision and streaming steps as: f i (x+c i ∆t,t+∆t)-f i (x,t)= - ∆t(1-β) τ tot (f i (x,t)-f i eq (x,t)) +βf i m (x,t)+∆tF i , (1) where c i is the discrete unit velocity in the i direction, τ tot is the dimensionless relaxation time with respect to the lattice viscosity v and is adjusted with the sub-grid scale turbulent model and is the parameter given by β(l f )= (1-l f )(τ-0.5) l f +(τ-0.5) , (2) in which l f (x,t) is the liquid fraction value of the cell, which takes a value between 0 and 1. Liquid fraction values of 0 and 1 represent ice and water, respectively. In Eq. (2), the total relaxation τ tot can be used instead of the relaxation time τ. The immersed boundary modification can be used for not only dynamic separation of solid (ice) and liquid (water) phases, but also for a moving body (moving ice) in a fluid flow. Therefore, an additional collision term f i m is for cells partially or fully covered by a solid, i.e., ice cell, is given as f i m (x,t)=f ̅ (x,t)-f i (x,t)+f i eq (ρ, u s )-f ̅ eq (ρ, u), (3) where u s is the velocity of the moving solid. The macroscopic variables, namely density ρ and velocity u, can be computed based on the order of the moments of the distribution functions f i as ρ= f i 8 i=0 , ρu= c i f i 8 i=0 + F∆t 2 . (4) Free surface of the liquid is tracked by volume fraction value defined by ratio of cell mass to node density. The cell mass is evaluated on interface cell by distribution functions. Solution of free surface in LBM is known as single phase model, so that no physic variables will be defined for gas state. Influence on gas state realized by a free surface boundary condition in terms of distribution function. The free surface boundary condition assumes that the fluid has a much lower kinematic viscosity than the gas state and is expressed in terms of the following distribution function on free surface interface node: f ̅ ' (x,t+∆t)=f i eq (ρ A ,u)+f ̅ eq (ρ A ,u)-f i (x,t), (5) where ρ A is the gas density implicitly acting as air pressure onto the free surface. In the modeling of heat transport with phase change, the temperature field is considered to be an essential variable and can be calculated by the following thermal lattice Boltzmann equation with latent heat of fusion: g i (x+c i ∆t,t+∆t)-g i (x,t)=- (g i (x,t)-g i eq (x,t)) τ h -w i L h c p (l f (x,t-∆t)-l f (x,t)), (6) where g i (x,t) is the distribution function for the temperature field, τ h (=3α+1/2 ) is the dimensionless relaxation time with

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Page 1: Application of the Lattice Boltzmann method to …cds.nagaokaut.ac.jp/niigata_form/symposium2016_pdf/2-207.pdfApplication of the Lattice Boltzmann method to open channel flow with

Application of the Lattice Boltzmann method to open

channel flow with liquid-solid phase changes

長岡技術科学大学 大学院工学研究科 Ayurzana Badarch

長岡技術科学大学 環境社会基盤工学 細山田 得三

1. Introduction

In the northern sphere of earth, a common state of natural

water is an ice in winter time. Ice and water mixture is one of

the major subjects of the cold region engineering and

motivation of this study. Ice acts like a solid in water flow and

situation will become even more difficult when thermodynamic

is involved in system.

The liquid-solid phase changes are often referred as Stefan

problem, which has complexity having moving boundary

showing liquid-solid interface in time. Solving moving body

boundary brings many difficulties for conventional methods.

State-of-Art methods for phase change in closed region are

fixed grid based approach and front tracking approach,

however they require to solve additional sets of equation and

boundary condition on the moving boundary interface.

Recently, particle based method appear to solve phase change

in free surface flow and performance of those approach is

limited for laboratory scale.

In this paper, we present numerical procedure for liquid-solid

phase change of water with Lattice Boltzmann method (LBM).

The numerical procedure is composed as follows. Two

distribution functions approach will be used to account flow

field and temperature field on fixed grids with 9 directional

lattices. The enthalpy-based non iterative method is used for

phase change of fluid depending on local enthalpy updates1).

For open channel fluid flow, free surface formulation of single

phase LBM is employed2).

2. Numerical procedure

For fluid flow with free surface representation, discretized

Boltzmann equation with the Bhatnagar-Gross-Krook collision

operator and modification of the immersed moving boundary is

to be solved by collision and streaming steps as:

fi(x+ci∆t,t+∆t)-fi(x,t)= -∆t(1-β)

τtot

(fi(x,t)-fi eq(x,t))

+βfi m(x,t)+∆tFi, (1)

where ci is the discrete unit velocity in the i direction, τtot is

the dimensionless relaxation time with respect to the lattice

viscosity v and is adjusted with the sub-grid scale turbulent

model and 𝛽 is the parameter given by

β(lf,τ)=(1-lf)(τ-0.5)

lf+(τ-0.5), (2)

in which lf(x,t) is the liquid fraction value of the cell, which

takes a value between 0 and 1. Liquid fraction values of 0 and

1 represent ice and water, respectively. In Eq. (2), the total

relaxation τtot can be used instead of the relaxation time τ.

The immersed boundary modification can be used for not only

dynamic separation of solid (ice) and liquid (water) phases, but

also for a moving body (moving ice) in a fluid flow. Therefore,

an additional collision term fi m

is for cells partially or fully

covered by a solid, i.e., ice cell, is given as

fi m(x,t)=f𝑖̅(x,t)-fi(x,t)+fi

eq(ρ, us)-f𝑖̅ eq(ρ, u), (3)

where us is the velocity of the moving solid.

The macroscopic variables, namely density ρ and velocity u,

can be computed based on the order of the moments of the

distribution functions fi as

ρ=∑ fi

8

i=0

, ρu=∑ cifi

8

i=0

+F∆t

2. (4)

Free surface of the liquid is tracked by volume fraction value

defined by ratio of cell mass to node density. The cell mass is

evaluated on interface cell by distribution functions. Solution of

free surface in LBM is known as single phase model, so that no

physic variables will be defined for gas state. Influence on gas

state realized by a free surface boundary condition in terms of

distribution function. The free surface boundary condition

assumes that the fluid has a much lower kinematic viscosity

than the gas state and is expressed in terms of the following

distribution function on free surface interface node:

f𝑖̅ '(x,t+∆t)=fi

eq(ρ

A,u)+f𝑖̅

eq(ρ

A,u)-fi(x,t), (5)

where ρA

is the gas density implicitly acting as air pressure

onto the free surface.

In the modeling of heat transport with phase change, the

temperature field is considered to be an essential variable and

can be calculated by the following thermal lattice Boltzmann

equation with latent heat of fusion:

gi(x+ci∆t,t+∆t)-g

i(x,t)=-

(gi(x,t)-g

i

eq(x,t))

τh

-wi

Lh

cp

(lf(x,t-∆t)-lf(x,t)), (6)

where gi(x,t) is the distribution function for the temperature

field, τh(=3α+1/2 ) is the dimensionless relaxation time with

Page 2: Application of the Lattice Boltzmann method to …cds.nagaokaut.ac.jp/niigata_form/symposium2016_pdf/2-207.pdfApplication of the Lattice Boltzmann method to open channel flow with

respect to the thermal diffusivity α, Lh is the dimensionless

latent heat of fusion, and cp is the specific heat capacity of

water or ice. The equilibrium distribution function for the

temperature field can be given as

gi

eq=wiθ [1+

ci∙u

cs2] with θ =∑ g

i

8

i=0

, (7)

and the macroscopic temperature T can be converted into

dimensionless temperature θ as follows:

T=Tmax-Tmelt

θmax-θmelt

(θ-θmelt)+Tmelt . (8)

After the dimensionless temperature evolution, the local

enthalpy, obtained by En = cpθ+lf(x,t-∆t)Lh, can be used to

linearly interpolate the liquid fraction,

lf(x)=

{

1 for En>Ens+Lh=Enl

0 for En<Ens=cpθmelt

En-Ens

Enl-Ens

for Ens≤En≤Ens+Lh

, (9)

and the liquid fraction defines the liquid (water) and solid (ice)

phases in the domain. The phase, i.e., ice or water, is assigned

to F cells because ice, which acts like a solid, will become

water after melting. Moreover, if ice interacts with the air (G

cells), boundary cells between the ice and air must be IF cells

because IF cells have a certain water content. At the interface

between ice and water, 𝑙f(𝒙) takes a value of between 0 and 1.

The interaction between the free surface flow module and the

heat transport with phase change module is such that the

temperature difference produces a buoyance force in the flow

field, and the flow field affected by the buoyance force forms a

temperature field in the domain. Although the buoyance force

is negligible in a turbulent flow, it must be included in the

computation. The lattice viscosity is related to the lattice

thermal diffusivity of a fluid as αwater = ν/Pr, where Pr is

the Prandtl number, so that the relation between the

computational modules is maintained.

3. Model validation

We have solved several phase change phenomena in closed

region to validate numerical accuracy and performance. First

validation was ice layer melting in rectangular cavity and

results3) was in good agreement with result in literature4). Here

we show other numerical simulation of ice melting in square

enclosure, 0.2 m in each side. Initially, enclosure is filled with

ice in ready melt case, i.e. temperature of ice is tmelt = 0°C and

latent heat is omitted. In the bottom side, constant heat is

maintained while walls in right/left are assumed to be insulated

by heat. Top wall is maintained with tmelt. Heat transferred by

conduction in the initial stage of simulation. After creating

sufficient space for convection flow by conduction dominated

melting, convection flow has created by density difference in

melted zone. Density influence by the temperature is calculated

by a force term expressed with non- Boussinesq approximation

in the numerical procedure. Depending on the Rayleigh number

(Ra), number of fingering and its height was different. In Fig. 1,

melting interfaces of Ra=107 is shown with velocity vector at

time 11.91 min. Convection flow created several fingering and

they joined to two big circulation flow immediately. Those two

big circulations dominated for further heat transport and melted

ice until top boundary. If Rayleigh number is low, conduction

heat transfer is dominated and took long time to start

convection heat transport. Maximum velocity induced by heat

was around 0.01 m/s, while maximum Nusselt number at

bottom wall was around Nu=40, shown in Fig. 2. After ice

melted until top boundary, temperature field in middle area of

enclosure was almost constantly distributed. Numerical results

were what we expected and same as results of researcher in

same field, which has not yet been published elsewhere.

Figure 1 Ice melting from below in enclosure.

Figure 2 Measured Nusselt number at bottom wall.

Page 3: Application of the Lattice Boltzmann method to …cds.nagaokaut.ac.jp/niigata_form/symposium2016_pdf/2-207.pdfApplication of the Lattice Boltzmann method to open channel flow with

Figure 4 Snapshots of ice bed melting by weir over flow at different times.

4. Application to open channel flow with phase

changes

We had coupled the free surface LBM and heat transport

solution of LBM in together first time. To evaluate

performance, we had solved an ice bed melting problem by

flow over weir in two dimension channel.

In this study, ice with thickness of 0.06 m and temperature of

-30℃ is located in downstream of the 0.2 m height weir and

water at temperature of 30℃ is released from upstream of the

weir to downstream and surrounding air temperature set to be

20℃. Grid number for the weir was selected as 80 grids, which

gives grid space as ∆x=2.5e-3 m and time step as

dt=5.41e-3 sec. Prandtl number was fixed at Pr=11.59.

Total melting time of fixed ice bed was 3.25 minute and Fig. 4

shows fluid phases (water and ice) in left hand, temperature

field in right hand. We have measured local Nusselt number

through the melting interface between ice and water to evaluate

heat transport and it is plotted in Fig. 5. Nappe flow play

important role in melting of ice bed and average Nusselt

number was 60 to 110 during the simulation. Therefore heat

transfer and phase changes are strongly induced by convection

in open channel flow. In our simulation, the Nusselt number in

turbulent flow was observed in the exact referred range of the

Nusselt number. We had tested numerical simulation on the

coarse grid, however there was no significant difference to

remark. Only things to be careful is that choosing right

parameters according to right physical relation. For example,

temperature transfer in water requires big relaxation parameter,

which is almost near to its limit 2.0. Value near 2.0 develops

instability for simulation. To avoid this issue we need to change

relaxation parameter, thus time step of heat transport simulation

will be different from time step of free surface module. Figure 3 Problem conditions and geometry

Page 4: Application of the Lattice Boltzmann method to …cds.nagaokaut.ac.jp/niigata_form/symposium2016_pdf/2-207.pdfApplication of the Lattice Boltzmann method to open channel flow with

5. Conclusion

We have tested our numerical procedure on the ice melting in

enclosure and the ice bed melting in open channel flow.

Numerical procedure successfully validated on liquid solid

phase changes in closed region, which can be found our

previous publications. The numerical procedure does not

require any additional equation or condition to treat moving

boundary of melting/solidification fronts. Identifying liquid

fraction value is useful value for multi-physics simulation,

which involving solid and liquid motion in system. Because

this value is suitable for boundary treatment idea of LBM and

as well partial bounce back or immersed boundary method. We

have adopted the Immersed boundary method in separate liquid

and solid state dynamics in our numerical procedure. Immersed

boundary method gives smooth force value on the solid surface

to enhance numerical stability rather than partial bounce back

approach. The performance of the numerical model was in

good agreement. However, the accuracy of the numerical

procedure needs to be validated against experimental study in

the open channel flow. In future studies, ice motion and ice-ice

interaction need to be considered in order to study ice jam or ice

water mixture in the river. The large simulation involving the

large time and space scale can be solved in parallel

computational architectures, in which LBM is perfectly suitable.

We have been working on the parallel code to perform such

large scale simulation.

References

1) Huber, C, Parmigiani, A, Chopard, B, Manga, M. and Bachmann,

O: Lattice Boltzmann model for melting with natural convection.

International Journal of Heat and fluid flow, Vol 29(5), pp.1469-

1480, 2008.

2) Ayurzana, B, Hosoyamada, T. and.Nasanbayar, N: Hydraulics

application of the Free-surface Lattice Boltzmann method.

Proceedings of IFOST-2016, Vol 1, 2016

3) Ayurzana, B and Hosoyamada, T: Phase change simulations of

water near its density inversion point by Lattice Boltzmann method.

Proceedings of the 23rd IAHR International Symposium on Ice, Vol

1, pp.1-8, 2016

4) Arid, A, Kousksou, T, Jegadheeswaran, S, Jamil, A. and Zeraouli,

Y: Numerical simulation of Ice Melting Near the Density Investion

Point under Periodic Thermal Boundary conditions. Fluid dynamics

and Material processing, Vol 8(3), pp.257-275, 2012

Figure 5 Local and average Nusselt number at interface