naofluid scale

hello every body
iam working on nanofluid which is in the single phase and is in cavity (natural convection) and i want to convert them into boltzmann scale the properties are as follow
rhos=3880; %solid density
rhof=997.1; %fluid density
phi=0.0; %volume fraction
cps=765; cpf=4179; %specific heat coefficient
betas=0.8510^-5; betaf=2110^-5; %thermal expansion coefficient
kf=0.613; ks=45; %conductivity
viscosityf=0.001004; %dynamic viscosity of water
pr=5.6; %prantle of water
Ra=10.^4; %Rayleigh Number
the rayleigh and prandtle number are fixed
any body knows how can i use these inputs and convert them into boltzmann scale
appreciated in advance

Hi there,

The densities are the easiest to convert, as these can be defined arbitrarily for the system you wish to model and have very little influence on . Usually you would use a fluid density of 1 in lattice Boltzmann simulations to maximise the precision in calculations, which would mean the solid density would be (3880/997.1) = 3.891.

The relaxation time for collisions (tau) can be determined using the kinematic viscosity of the fluid (nu):

nu = 1/3 * (tau - 0.5) * (dx)^2 / (dt)

where dx and dt are the lattice spacing and time step respectively. Ideally, you would want a relaxation time that is greater than 0.5 for good numerical stability of calculations, so this will help dictate what the lattice spacing and time step would be. Another property to help determine dx and dt would be the speed of sound ©, as:

c = 1/sqrt(3) * dx/dt

which is a property of the lattice scheme you are using. Using this in combination with the kinematic viscosity, which is the dynamic viscosity divided by fluid density (i.e. 0.001004/997.1 = 0.000001007 m^2 s^(-1)), you can determine the physical sizes of the lattice spacing and time step.

You have already hinted at the easiest way to obtain lattice-based properties, which is to use dimensionless numbers such as the Prandtl and Rayleigh numbers. So long as these numbers are constant and obtained by using numbers with consistent units, the lattice system will represent the real system you wish to model.

For instance, as the Prandtl number is defined as the ratio of kinematic viscosity (momentum diffusivity) and thermal diffusivity (alpha), i.e.

Pr = nu/alpha

and this value is fixed for your fluid at 5.6, you can work out what alpha should be from nu. Incidentally, if you are using a thermal distribution function, the value of alpha will probably be defined as:

alpha = 1/3 * (tau_t - 0.5) * (dx)^2 / (dt)

which is similar to the expression for nu - this can be used to determine the thermal relaxation time (tau_t) you will need in your simulation (as dx and dt will already be fixed). It should also be noted that temperature is similar to density in lattice Boltzmann simulations, i.e. its values can be chosen arbitrarily, so your choice will have an effect only on the lattice value of thermal expansion as used in the Rayleigh number. (This will be needed if you are using the Boussinesq approximation to apply a buoyancy force for convective flow.)

I hope the above is helpful.