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Seamus O'Shaughnessy November 27, 2007 08:58

Non dimensionalisation Problems - Please Advise
I am trying to solve a problem involving Marangoni convection in which an air bubble is surrounded by a liquid oil layer. The governing equations are Navier-Stokes and The energy Eqn. I am trying to benchmark against data in the literature but my data points are all out by a constant factor, so I am assuming I have made some error in the non-dimensionalisation of the problem, or rather, in the implementation of the non-dimensional problem when using a commercial CFD code.

The original eqns were non-dimensionalised using the same reference scales as the author to which I am comparing results, and I get the same non-dimensional form of the eqns.

Original NS: (ρ_α)*(∂u/∂t + (u.V)u) = V[-pI + (μ_α)*(Vu + (Vu)^t)] - (ρ_op)*(β_α)*g*(T-T_op)

Non_Dimensional NS: (ρ_α/ρ_oil)*(∂u'/∂t' + (u'.V')u') = V'[-p'I +(μ_α/μ_oil)* (V'u' + (V'u')^T)] - (ρ_α/ρ_oil)*(β_α/β_oil)*(Ra/Pr)*T'z

Original Energy: (ρ_α)*(Cp_α)*(∂T/∂t + u.VT) = (k_α)*(V^2)T

Non_Dimensional Energy: (ρ_α/ρ_oil)*(Cp_α/Cp_oil)*Pr*(∂T'/∂t' + u'.V'T') = (k_α/k_oil)*(V'^2)T'

The important boundary condition (BC) is at the liquid-vapor interface, where the shear stress is related to surface tension effects.

Dimensional BC: -μ(∂u_tang/∂n) = (∂σ/∂T)*(∂T/∂s)

Non-Dimensional BC: (∂u_tang'/∂n') = (Ma/Pr)*(∂T'/∂s')

Apologies for the appearance of the eqns. The prim symbol denotes dimensionless quantities. ρ,μ,Cp,k,β refer to density, dynamic viscosity, specific heat, thermal conductivity and volume expansion coefficient respectively. u is the velocity, and V is the differential operator. g is gravitational acceleration, T is temperature, and Ra, Pr, Ma are the Rayleigh, Prandtl and Marangoni numbers. ∂σ/∂T is the tempeture derivative of surface tension. The subscript alpha refers to the reference fluid, so if the reference fluid is chosen to be the oil (as properties of liquid generally much greater in magnitude to properties of vapor/air), should all those ratios in the non-dimensional form should go to unity?

I have tried to construct this problem using both FLUENT and COMSOL codes, but with no success so far. In both cases, I have non-dimensionlised the original geometry (0.5mm radius bubble, 10mm liquid layer depth) using the reference length scale, the radius. Hence I have a dimensionless geometry with bubble radius of 1 and liquid layer depth 20. I must set the following parameters: density, dynamic viscosity, specific heat, thermal conductivity, volume expansion coefficient, gravitational acceleration, and temperature derivative of surface tension.

My problem is what values do I give these properties??? I am also confused by the appearance of the Pr term in my non-dimensional energy eqn. Any help will be greatly appreciated. Thanks you.

pankaj saha November 30, 2007 16:30

Re: Non dimensionalisation Problems - Please Advis
First, be sure whethere the commercial software solver is based on dimension or non-dimensional....i hope Fluent is a dimensional cannot take values non-dimensonally....just check....i am also not sure ...

Seamus O'Shaughnessy November 30, 2007 19:00

Re: Non dimensionalisation Problems - Please Advis
I think FLUENT is dimensional, but I thought it shouldn't matter if the ratios of the fluid properties and the geometry are such that the relevant dimensionless properties like Re, Pr, Ra, Ma are the same for both the dimensional and dimensionless cases?

lyes43 February 28, 2011 13:23

fluent trat only dimensional equations but we can use dimensionless equations and change something so that Fluent will understand them like dimensional equation when u write the dimensional and non diemnsonal u see that if we set ro=1 and viscosity + 1/Re they apear the same and like that everything is ok but results of Fluent are dmensionless

lyes43 February 28, 2011 13:29

hello i'm working on simulate a turbulent flow in a rectangularr channel were heat blocks are amounted i want have a Gr of 2.5 E +7 but i dont know how to set the thermal expansion coefficient and delta T in Fluent ?

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