Similarity in 2-D spatially developing and long shear layers

2D shear layers are studied using a high accuracy vortex-in-cell (VIC) method. The case investigated is $U_2/U_1=0.38$, as in the Brown and Rhosko experiment. The inflow corresponds to a regularized vortex sheet with momentum thickness $\theta_0=\pi/4$ and $Re_0=54$. It then growths and smoothly undergoes transition, through TS waves and then K-H instabilities, to a ``turbulent shear layer'' developed at $x\approx 500$. Two computational domains are used: $L_1=2500$ and $L_2=3500$. Various outflow conditions are also used with $L_1$. We focus on self-similarity: profiles of $U/U_1$, $-\overline{u'v'}$, etc. as a function of $\eta=y/(x-x_0)$ (with $x_0$ virtual origin), and slopes $d\theta/dx$, etc. The results of the $L_1$ simulations agree well with each other; the region $x\in[1800, 2500]$ being affected by the outflow and thus dismissed. They also agree well with the results of the $L_2$ simulation, thus confirming the $L_1$ simulations validity. The region $x\in[2800, 3500]$ is dismissed in the $L_2$ simulation. A remarkable result is that we do not obtain one long region of self-similarity but, instead, multiple such regions: the region $x\in[900, 1200]$ with $d\theta/dx=0.0180$ and $-\overline{u'v'}_{\rm max}/(\Delta U)^2=0.0135$, then the region $x\in[1400, 1900]$ with $0.0146$ and $0.0115$, then the region $x\in[2100, 2600]$ with $0.0177$ and $0.0140$ (thus almost identical to the first region, potentially hinting at a recurring pattern).