Inlet Thermal Boundary Condition

Hello Everyone, Can someone tell how to implement the Inlet Thermal Boundary condition?

Suppose you are using D2Q9 LBM-BGK model and the incoming distributions are 1, 5 and 8 then inlet thermal boundary condition can be implemented as :

f(i,j,1) = inlet_temperature * (w(1) + w(3)) - f(i,j,3)
f(i,j,5) = inlet_temperature * (w(5) + w(7)) - f(i,j,7)
f(i,j,8) = inlet_temperature * (w(8) + w(6)) - f(i,j,6)

where, inlet_temperature is the non-dimensionalized temperature of fluid at the inlet ((T_inlet - T_max)/ (T_min - T_max)), w is the weighting factor and 3, 7, 6 are the particles opposite to 1, 5, 8 respectively.

Obviously the above boundary condition works if you are using two distribution functions, momentum and thermal distribution function in LBM. Since, the incoming of fluid particles are already handled by momentum distribution function boundary condition and there is coupling relationship between momentum and thermal distribution function as well, you only need to specify the temperature of incoming fluid at the inlet thermal boundary condition.

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thanks for a detailed response. Just one thing more that for non-dimensionalization of temperature, how to define the maximum and minimum values of temperature. I mean if I have a heater installed in my domain then I can set the maximum temperature as the temperature of the heater but for minimum temperature, how would I define the Tmin??

Hi, it indeed is a good question. See, if you consider basic thermal LBM-BGK model that employs two distribution functions, the non-dimensionalization of temperature can be a problematic one as this model is not very good in modelling such kind of flows, one of which is a example mentioned by you.

However, you may try one thing. When you try to initialize the temperature of every node in the domain before beginning the simulation, try to set it at T_min. For example, for a heater installed in the closed room, you may assume that initially the room is at 300 K, so you initialize the domain at T = 0 (T_min). The boundary nodes that are occupied by heater nodes can be set up by implying inlet thermal boundary condition as mentioned above by setting inlet_temperature = 1 (T_max). Accordingly, the other boundary conditions can be implied depending upon the conditions, like bounce back for adiabatic walls, t_wall * (w(?) + w(?)) - f(i,j,?) for defined temperature of the wall (? is the required distribution functions which you can figure out).

Hope I am clear to you.

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Thanks a lot. it is really helpful for me.