where c_s^2 is the constant 1/3, which should be
used without units when you proceed to a
dimensional analysis of the above equation.
According to Timm, the conversion factor between P[sub]p[/sub] and P[sub]lb[/sub] is rhop * dxp2 / dtp2, then
P[sub]p[/sub]=rhop * dxp2 / dtp2*P[sub]lb[/sub], so P[sub]lb[/sub]=P[sub]p[/sub]dtp2/(rhopdxp2). Given P=cs[sup]2[/sup]rho,
so rho[sub]p[/sub]=1/cs^2[P[sub]p[/sub]dtp2/(rhopdxp2)].
I wonder where is the first term 1 of the RHS function Latt mentioned comes from.
Hi.
I’m simulating thermal flow in a 2-d micro-couette using LBM. I have a serious problem with thermal unit conversion in LB.
I want to reproduce the results of the DSMC method in (XIAO-JUN GU et. al. “A high-order moment approach for capturing non-equilibrium phenomena in the transition regime”, J. Fluid Mech. (2009), vol. 636, pp. 177–216. c ,Cambridge University Press 2009, doi:10.1017/S002211200900768X) Part 6. Results and discussion, for thermal-micro-couette flow in the transition regime (namely, Knudsen number Kn>0.1). But I don’t know how to convert the values of the wall temperature (Tw) and the gas constant ® to lattice units.
as we know the sound speed in the LBM is Cs=C/SQRT(3) and C=SQRT(3RT). Then what should I do with the value of the R and T in the calculation of the C.
Please help me. I’m working on my thesis and I’m in a bad hurry.
I appreciate any help in advance.
I have a question regarding multicomponent LBM. What is the significance of the density values we put into our simulations? Are they just indicators (showing which part of a domain is occupied by a certain fluid)?
No, of course not. In reality, two different components or phases may have a quite different density (e.g., water and air much more than water and oil). This is of significant physical relevance since the density ratio can influence the flow dynamics. I believe that, in general, not all properties of a multiphase system can be correctly recovered when the density ratio is wrong.
I am simulating Co-current immiscible two-phase flow in a 2D channel, the wetting phase flows along the upper
and lower plate while non-wetting phase flows in the center region using SC model.
My objective is to verify my model with the results of:
" Shan-and-Chen-type multiphase lattice Boltzmann study of viscous coupling effects for two-phase flow in porous media"
by Haibo Huang?, Zhitao Li, Shuaishuai Liu and Xi-yun Lu
I am using the same LB kinematic viscosity \nu_{lb} for both fluids (the value is 1/6) and the density of wetting and nonwetting fluids are 1 and 0.1, respectively. After the end of my simulation, I see a single parabolic velocity profile for velocity in the direction of flow {x-direction} which does not account for phase change (density change). In other words, it is just like plotting velocity profile for flow of SINGLE Phase through a channel!!!
I think that your result is quite logical. The Navier-Stokes equations for Poiseuille flow simplify to
\rho \nu \Delta u = \rho g
where \rho is density (as function of position), \nu is kinematic viscosity (constant throughout your channel), \Delta is the Laplacian operator, u is velocity, and g is gravitational acceleration or any other homogeneous acceleration (constant throughout your channel). You see that the density cancels and that the result MUST be the same as for a single phase fluid. Only if the kinematic viscosity differs, you would expect a modified velocity profile. You should choose different kinematic viscosities.
I am using a constant driving force for both phases (F) and I think applying the Chapman-Enskog expansion procedure to the lattice-Boltzmann equation for multiphase (SC Model), one obtains the following mass and momentum equations
for the fluid mixture treated as a single fluid
(\sum_a F_a )/(\rho) = - \nu \Delta u a=1, ... ,9 for 2D
where
\sum_a F_a = F_ext + F_{f-f} + F_{f-s}
where F_{ext}, F_{f-f} , and F_{f-s} are momenta contributed by external force, interaction between phases, and interaction of ¯fluid with solid, respectively. [Z.L. Yang et al. 2001]
In this case, density is not canceled out and fluids with different density should have different velocity profiles!
Furthermore, I see some peaks in velocity perpendicular to the direction of flow at the interface between fluid phases ( from Navier-Stokes, we know that it should be zero everywhere in the doamin ). I think this accounts for the existence of F_{f-f} at the interface. But it changes the velocity magnitude at points around that!
Do you know any solution for this problem?
I still think that the density cancels because F is momentum, and momentum is proportional to density. Please carefully recheck your equations and make sure that the difference between force and force density is clear.
Again: For two fluids with the same kinematic viscosity the velocity profile must always be a parabola in the presence of a homogeneous acceleration field. But acceleration is not force and also not force density!
I find it a bit strange that you have a velocity perpendicular to the main flow. Due to the symmetry, this would eventually lead to a compressibility of the fluid. Is your interface flat? It is a 2D simulation, right? Is your system symmetric with respect to the main velocity axis?
Hello, could you please tell me where did you get this formula?
nu_{Phys}=dt_{Phys}/(dx_{Phys}*dx_{Phys})*nu_{LB}
Even the dimensions do not match and that’s weird to me !
We have “length^2” vs. “second” (for dt) to be canceled and nu(Phys) dimension becomes equals to that of nu(LB).
Hello, i am preparing an algoritm for natural convection using the LBM. I have a square cavity of 1m * 1m. I have viscosity V_p and Prandtl number(Pr). where suffix ‘p’ indicates in physical units and ‘l’ indicates in lattice units. Now for LBM, i have set dx_l=dt_l=1 lattice unit, so C_l=1 lattice unit. Now i have 100 divisions in X and Y directions, so L_l=100 and also dx_p=dy_p=0.01m.
Now i have somewhere read that the non-dimensional number should be same for lattice units and physical units. In my case the Rayleigh number is 10^5. Now i have assemed the viscosity V_l=0.03 in lattice units. so, the value of alpha_l=Pr / V_l. Now i have found the value of g*beta_l=Ra * V_l * alpha_l / (delta_T * L_l^3).
Now, i want to see the temperature distribution and velocity distribution at a particular time, say 180 s. So, how many iteration i need to be performed?
Hello,
in order to get dt_p you need a quantity that contains time units. Here I guess it’s you viscosity which has units [Length^2/Time]. You can deduce dt_p from the relation
V_p=V_l * dx_p^2/dt_p
Then the number of iteration is iter = 180 / dt_p.