# Correlating lattice time to physical time in Lattice boltzmann Method

In LBM literature, more often the time-dependent behavior is plotted with respect to the number of time-step(lattice time $\Delta t$), I also read this note from Jonas Latt, however it stills seems unclear to me how to correlate $\Delta t$ to physical time t.

For instance, I want to simulate some fluid(Renolds=20), on a 2D grid(1 meter by 1 meter, $N_x$=100 in each dimension), then according to Jonas's note, we are advised to choose a time step ~${N_x}^2$, i.e. $N_t$ ~ $10000$, however, could someone demonstrate with these information available, what is the exact physical time corresponding to a single lattice time-step $\Delta t$ ?

Another curiosity: It's suggested to choose time step ~${N_x}^2$, otherwise the numerical scheme may be unstable, so if we choose less time steps, then the intermediate results could be unphysical or numerically incorrect? Does this mean LBM is more suitable for steady-state simulation rather than transient-behavior studying?

• @nluigi appreciate you can help ! – Lorniper Oct 10 '16 at 14:20
• Jonas' notes are useful but another resource i found more useful was a lecture on lbm units conversion from lbmworkshop which i attended back in 2011. This was particularly useful for other conversion such as the scaling of an external force for which no analytical solution is available. I may convert this to an answer later on but i don´t have time at the moment. – nluigi Oct 10 '16 at 19:49
• @nluigi Thanks in advance! I also read the material from lbmworkshop, it's also advised to keep Mach number low through keeping the velocity $u/C_s^2 << 1$, but how to limit the velosity u in LBM? – Lorniper Oct 11 '16 at 20:21
• Woops, i wrongly linked the previous lbm unit conversion pdf. – nluigi Oct 11 '16 at 20:50
• To answer your question: It depends on how the flow is driven; if using zhou-he bcs to specify an inlet boundary condition then $u$ is specified in the bc, if using e.g. body force to drive the system then the acceleration may be analytically related to the maximum velocity in the channel. The message of the lecture i linked to is to try a certain set of parameters and see if they are within the desired (and stable) range. If not then tweak them using the conversion to get parameters which are in range. – nluigi Oct 11 '16 at 20:50

In those notes, it is stated that time $t_p$ is divided by some reference time-scale $t_{0,p}$. And the time step in the discretised system is $\delta_t=t_{0,p}/N_\mathrm{iter}$. So if you perform $N_t$ time-steps then this corresponds to a time interval of $\Delta t_p=N_tt_{0,p}/N_\mathrm{iter}$.
• can you deomonstrate the number here with simple math? Please look at eq.(8) and eq.(9) in the note, "recall that both reference variables are unty in the dimensionless system", doesn't it mean here we use $l_0=1$ and $t_0=1$? – Lorniper Oct 10 '16 at 14:36
• @Lorniper If $l_0=1$ and $t_0=1$ then the physical system and the dimensionless system are equivalent, yes. – lemon Oct 10 '16 at 14:45