The document compares different equations that model the velocity profile of flows experiencing sudden expansion. It presents equations from Abramovich, Matsumoto et al., and a normal standard curve to model normalized velocity (u/U) against normalized radial distance (r/Rc). The equations from Abramovich, Matsumoto et al., and the normal standard curve all agreed fairly well in modeling the self-similar velocity profiles observed in experimental measurements of such flows. This confirms that self-similar velocity distributions are common in turbulent jet flows experiencing changes in cross-section.

1. Fig.ssvp1. Normal Curve Similarity in Flow-velocity diffusion in Sudden Expansion.
Comparison: A common feature for similarity distribution formats was non-dimensional variants.
Velocity values (u) were divided by the maximum value (U) at the central axis line at particular cross-
section along downstream distances. Radial dimensions (r) were divided by a factor Rc, called ‘Critical
Radius’, which is the radius at each axial station where local velocity is half the maximum at central
line (U/2). Author’s (3) experimental measurement data also were converted into this format of u/U
against r/Rc, over the separated and reattaching part of eddy region immediately downstream of
sudden expansion.
Abramovich (1) gave empirical formulae based on his experiments on free jets. Set of equations
could be approximated to:
u/U = 1/(1+0.402529*(r /Rc) ^2) ^2
Matsumoto et al (2) gave a trigonometric equation for self-similarity distribution:
u/U = (1+ Cos (Pi/2)*(r/Rc))/2
The ‘Normal Standard Curve’ also was modified to meet the requirement of u/U = 1, when r = 0 at
the axis, (by dividing with the maximum). Zero mean with SD =1 simplifies the equation to: u/U = e^-
((r/Rc)^2)/2.
Result: All the four curves agreed fairly well. This confirms that the self-preservation
characteristic of jet flows is also a universal natural distribution, found abundant in nature.
References:
1. Abramovich, G.N.: ‘Theory of turbulent jets’, MIT Press, 1963.
2. Matsumoto, Ryuichi; Kimoto, Kyoji; and Tsuchimoto, Nobutaka: ‘A study on double concentric jets, 1st
report, Experimental results of air to air flow’, Bulletin of J.S.M.E, No 93, March 1973, pp 529-540.
3. Vachali, R.N: ‘An Experimental Study of Sudden Expansion in Pipe Flow’, M. Sc. Engineering thesis.
Indian Institute of Science, Bangalore -12, May1982.
Normal distrn Exist in self similarity curves
-1
-0.5
0
0.5
1
1.5
0 1 2 3 4
r/rc
u/Uc
Abramovich CoFlow Jet e-(r/rc^2)/2
e^-r/rc Cos(r/rc)
All curves except e^-r/rc coincide