What is your Rayleigh number? Is it approximately 3x104?  If that is the case, then it will simply be a case that the streamfunction is defined differently. One will be scaled with Ra and the other not.  The sign difference is no consequence; some use u=psi_y and v=-psi_x and others u=-psi_y and v=psi_x.

Thanks a lot Dr. Andrew for the reply. Ra is 100000 and the fluid is air with Pr=0.71 I think the author use normal stream function (epsi) with no further changes? if you want to have a look at the paper I will send you it? 

From the picture, I see you are using dimensional value in Fluent while I suspect that the paper shows non-dimensional values. The different order of magnitude seems to confirm. A difference in the sign would be a consequence of a different definition of the vorticity.

Dear Khalid, and Filippo,

My estimate of 30000 was based on a very rough ratio between the maximum values of the streamfunction between the two figures. So this shows that both Filippo and I are correct. So it seems that your computations are ok! Good news.

Regards,

Andrew

No Filippo you wrong I do not use dimensionless analysis for stream function see carefully its units illustrated near the contour title.

Khalid be careful, as I wrote above, FLuent shows the contours in Kg/s. You have to check carefully the formulation used in the paper to assess if it uses the governing equation in non-dimensional form.

Dear Fillippo I do not think its a dimensionless stream function ? however I will be grateful if you check it to me? I was upload the paper with this post. The figure I want to compare with is Fig. 10 on the paper pp. 772. Thanks

The paper shows clearly that the numerical method adopts the non-dimensional formulation. Use the characteristic flow parameters to makes non-dimensional your solution.

How the relation of stream function will be in term of velocity components that the author depend then? may you mentioned it to me? thanks

The stream function is dimensionally homogeneous to a lenght for a velocity. In Fluent you have kg/s therefore you have to make non dimensional this quantity.

I aware this point but I need this relation to use it in Fluent you understand me? Send me the relation of stream function if possible?

u = ∂ ψ/ ∂ y , v = − ∂ ψ/ ∂ x

where u is the flow velocity parallel to and in the direction of the x-axis, and v is the flow velocity parallel to and in the direction of the y-axis.

q = ∂ ψ/ ∂ n 

Now consider two-dimensional plane flow with reference to a coordinate system. Suppose an observer looks along an arbitrary axis in the direction of increase and sees flow crossing the axis from left to right. A sign convention is adopted such that the flow velocity is positive.

pls check the dimensions of flowrate to match and compare the results

Thanks Sreekala for the illustration. I want to understand why stream function unit in fluent is in kg/s? This usually flowrate unit as I think and the correct unit must be m2/s ? So how Fluent evaluate stream function in this case?

kg/s = (kg/m^3)*(m/s)*m^2

therefore you have density*velocity*area.

Thanks Filippo for the answer. But this definition to stream function seems differ from the original definition that given by  Sreekala? However how now I transform the stream function that I was obtained from Fluent to be coincide with  with that given by the paper of comparison (i.e. make it as non-dimensional)?

Fluent defines a compressible form of the stream function, specifically

density*u = ∂ ψ/ ∂ y

, so the units shown are kg/s.

Even these seem not to be correct because this is a 2-D problem; the actual units are kg/(ms), i.e., kg/s per unit depth into the screen. However, Fluent treats 2-D problems as 3-D problems with unit depth in the z direction; the units of kg/s are valid with this three-dimensionality in mind

I think your answer not accurate because:

kg/m3*m/s=kg/(m2*s)

so its not (kg/ms) 

density*u = ∂ ψ/ ∂ y

density*u*∂ y = ∂ ψ

integrating this equation u will get ψ  in kg/ms)

density*u*y =  ψ