## Fluids

#### Possible unit inconsistency in k – epsilon equations applied to each phase of multiphase flow

• KeFon
Subscriber

Good Morning

I was looking at the transport equations for multiphase flow (Equations 14-387 and 14-388 in Section 14.5.18.1.3.1, 2021 R2 Fluent theory guide) and I cannot understand why, in the 4th and 5th terms of the right end side, the turbulent viscosity is divided by what I believe is the surface tension.

"σ" appears but the text does not say what it is, however it was used to refer to the surface tension or turbulent Prandtl number in this chapter.

The problem is that, dimensionwise, the equations only make sense if "σ" has a dimension of a density [M]/[V]^3.

I do not know if this sigma refers to some parameters that I have never heard of before, but has a dimension of a density, or if this "σ" should be a "ρ", the density of one of the phase.

Regards,

KF

• DrAmine
Ansys Employee
"Sigma" is the turbulent Prandtl Number. Please have a look into the single Phase Transport Equations for TKE and Dissipation.
• KeFon
Subscriber
Dear DrAmine,
Thank you for your answer. I do know that "sigma" in the first right hand term is the turbulent Prandtl number. If you mean that the "sigma" in the terms I mentioned (4th and 5th right hand terms of the Equation 14-387), is also the turbulent Prandlt number (Note that the subscript is not k nor epsilon but a refers to a phase) then the dimensions do not make sense:
K (upper case, not the TKE in m^2/s^2) was defined as the interphase exchange coefficient previously in the chapter with the following unit: kg/(m^3.s)
U was defined as the phase weighted velocity in m/s
"mu" was defined as the turbulent viscosity in Pa.s = kg/(m.s)
"alpha" being the volume fraction has no dimension
"sigma" as the turbulent Prandlt number has no dimensions
This gives a dimension kg^2/(m^4.s^2) whereas all the other terms have a dimension of kg/(m.s^3).
For the dimensions to be the same, either "sigma" is in kg/m^3, or, maybe, K is no longer the interphase exchange coefficient and is in 1/s, but this is not stated in the theory guide.
Regards KF

• DrAmine
Ansys Employee
I thought you are referring to the other equations and I looked into new documentation let me check what you are mentioning.
• DrAmine
Ansys Employee
I got your point and I will look deeper into it. Thanks for reporting!
• DrAmine
Ansys Employee
I can confirm now that the documentation will require a modification. The term should be divided by the density. In other word we require kinematic viscosity in the term.