I have a bit of a confusion between the speed of sound and the mean thermal speed in liquid water. In LBM when one wants to discretize the LB equation on a grid one uses
c = dx/dt = c[sub]s[/sub]/sqrt(3)
Now is the value of the c[sub]s[/sub] here the mean thermal speed of water molecules at a temperature given by
c[sub]m[/sub] = sqrt(3kT/m) [~550 m/s at 298 K]
or is it the speed of sound given by
c[sub]s[/sub] = sqrt((c[sub]p[/sub]/c[sub]v[/sub])*RT) [~1500 m/s at 298 K].
where c[sub]p[/sub] and c[sub]v[/sub] are the specific heats at constant pressure and volume respectively.
This directly affects the limit on the Mach number of the fluid flow that one can simulate and therefore it is important to know about which one to use in simulations. Would really appreciate any insights related to this that anyone might have…thanks.
Yes, c[sub]s[/sub] is the sound speed in the fluid, which means that an acoustic wave in your simulation would propagate at a speed c[sub]s[/sub]. I think you may have a small error in your first equation there; dx is the distance between nodes and dt is the time step, right? Then
c[sub]s[/sub] = c / sqrt(3) .
A sound wave therefore moves more slowly than the individual particles in the model.
This corresponds with the case of an ideal gas, for which the basic Boltzmann equation is valid. The thermal speed in an ideal gas is a bit higher than the speed of sound, just like in LBM simulations.
For a liquid, your numbers show that this is not the case; in water, the mean thermal speed is lower than the speed of sound. Still, even though the LBM is a discretisation of an equation for gases, you can use the LBM to simulate liquids as the method still gives you the correct Navier-Stokes behaviour.
Thanks for your response and yes the first equation I wrote was erroneous.
So, what confused me was this:
When one expands the equilibrium distribution in a Taylor series about the macroscopic velocity u then in the denominator one ends up with terms involving RT and i[sup]2[/sup][/i] which for ideal gases is passed off in terms of the speed of sound with the relation c[sub]s[/sub] = sqrt(RT). This is ok for ideal gases because these terms can be used interchangeably. But there are instances in articles where instead of the terms involving RT in the denominator c[sub]s[/sub] is directly used and for liquids or non-ideal gases that would involve an extra factor of gamma as defined in my previous post.
"For a liquid, your numbers show that this is not the case; in water, the mean thermal speed is lower than the speed of sound. Still, even though the LBM is a discretisation of an equation for gases, you can use the LBM to simulate liquids as the method still gives you the correct Navier-Stokes behaviour. "
To rephrase then:
For simulating liquids the expansion for the equilibrium distribution function would then involve the speed of sound c[sub]s[/sub] as defined by
c[sub]s[/sub] = sqrt(gamma*(RT/M))
in the denominator terms.
I hope, I am understanding this correctly? Thanks.
is unfortunately only valid for ideal gases. It gives you accurate results for e.g. air, but punching in numbers for water (assuming gamma is 1.33) gives me around 430 m/s. This is about a third of the real value, which is around 1500 m/s.
In fact, as far as I know, there is no equation of state which is as accurate for liquids as the ideal gas law is for dilute gases. (When the marine acousticians in my research group estimate the speed of sound in water, they use an instrument to measure things like temperature and salinity and put these into big ugly empirical formulas that predict c[sub]s[/sub] quite well.)
Thus, when the LBM is applied to liquids, some of the theoretical development (i.e. the connection to ideal gases with R*T) is not really valid anymore. The method still works, but that is because its resulting behaviour corresponds with the Navier-Stokes equation, which describes the behaviour of both gases and liquids.
So, to give a direct answer to your questions: First, you’re correct in saying that the expansion of the equilibrium distribution would contain c[sub]s[/sub]. Many LB articles write the expansion using c[sub]s[/sub], no matter the fluid which is being interested. Second, you’re correct in saying that the particle propagation speed c is connected to the speed of sound as c = sqrt(3)*c[sub]s[/sub], and not to the thermal speed.