# natural convection 2D square cavity:about boundary conditions for temperature

Hello Dear LBM Friends;
i would like to ask you how to implement D2Q5 for adiabatic walls? my case is : left wall=>> hot / right wall =>> cold / top wall and bottom wall are both adiabatic;
Thank you so much

Bounce-back is sufficient to implement adiabatic boundaries with the D2Q5 lattice. This is because it sets the normal derivative to be be zero at the wall (a Neumann boundary condition). This can be seen from an asymptotic study of the algorithm, which tells us the derivatives of a function are embedded in the non-equilibrium parts of the particle distribution function. A recent article by Wang et al (CAMWA, 2012) gives a detailed study of how good LBE is for simulating natural convection.

Hello Pleb01, Thank you so much
wat do u mean? the boundary conditions for temperature(adiabatic) wall is:

g(i,jmax) = (4*g(i,jmax-1)-g(i,jmax-2))/3 ? g is the function distribution for temperature

and wat about the boundary conditions for fluid?
by the way ,i looked for that document (wang and al. 2012) on the internet but i didnt find it,could u give me the link please?

have a nice day
Thank you very much My mistake - it is a 2013 publication http://www.lions.odu.edu/~lluo/Reprints-luo/2013/WangJ_CAMWAv65n2-2013.pdf

I think this article will answer all your questions. It looks like you are using a finite difference approximation for dT/dy at the wall, where T is temperature. But we can say that g=g^0+g^1, where 0 means equilibrium and g^1 is the first correction. The temperature flux is sum(g_ic_i) and contains equilibrium and non equilibrium parts. The first order (in tau) non-equilibrium part contains the gradients of T (as shown by the Chapman-Enskog expansion) and the equilibrium part is uT. Since u=0 on the wall, if you have no-slip, the equilibrium part to the flux is zero. If you want the normal gradient to be zero at the wall then you just have to make sure that sum(g_ic_{iN})=0, where N means the normal direction. Taking the bottom wall as an example, you’d need flux_Y=sum(g_ic_{iY})=0, which means g_2=g_4.

Good luck!

thank u so much Pleb01
c:wat does it mean : lattice speed?

Hello LBM Friends ’

i have to simulate natural convection ““laminar”” using a Matlab code based on LBM (take a look on rayleigh-benard convection matlab’s code on this website) in square cavity filled with air (Pr = 0.71) where : left wall assumed HOT and the right one is COLD : so for these 2 walls i applied macroscopic temeperature as equal to the sum of Ti (D2Q5)

for the upper and the bottom walls (adiabatics) i applied Neuman boundary condition : Tk(i,jmax)=(4Tk(i,jmax-1)-Tk(i,jmax-2))/3

for the fluid i applied the model D2Q9 ,and the domain walls are : 1:lx ; 1:ly
the problem is the average Nusselt value is good ,however the isotherms lines are not similar to those obtained by simulation using Fluent Ansys 6.3