We study space-time integrals, which appear in the Caffarelli-Kohn-Nirenberg (CKN) theory for the Navier-Stokes equations analytically and numerically. The key quantity is written in standard notations delta(r) = /1(nu r) f(Qr)vertical bar del u vertical bar(2) dx dt, which can be regarded as a local Reynolds number over a parabolic cylinder Q. First, by re-examining the CKN integral, we identify a cross-over scale r(*) proportional to L (<(parallel to del u parallel to(2)(L2))over bar>/parallel to del u parallel to(2)(L infinity) at which the CKN Reynolds number delta(r) changes its scaling behavior. This reproduces a result on the minimum scale r(min) in turbulence: r(min)(2) parallel to del u parallel to(infinity) proportional to nu, consistent with a result of Henshaw et al. ["On the smallest scale for the incompressible Navier-Stokes equations," Theor. Comput. Fluid Dyn. 1, 65 (1989)]. For the energy spectrum E(k) proportional to k(-q) (1 < q < 3), we show that r* proportional to nu(a) with a = 4/3(3-q) - 1. Parametric repo( v. resentations are then obtained as parallel to del u parallel to(infinity) proportional to nu(-(1+3a)/2) and r(min) proportional to nu(3(a+1)/4) By the assumptions of the regularity and finite energy dissipation rate in the inviscid limit, we derive lim(p ->infinity) zeta(p)/p = 1 - zeta 2 for any phenomenological models on intermittency, where zeta(p) is the exponent of pth order (longitudinal) velocity structure function. It follows that zeta(p) < (1 - zeta(2))(p - 3) + 1 for any p >= 3 without invoking fractal energy cascade. Second, we determine the scaling behavior of delta(r) in direct numerical simulations of the Navier-Stokes equations. In isotropic turbulence around R-lambda approximate to 100 starting from random initial conditions, we have found that delta(r) proportional to r(4)throughout the inertial range. This can be explained by the smallness of a approximate to 0.26,with a result that r. is in the energy-containing range. If the beta-model is perfectly correct, the intermittency parameter a must be related to the dissipation correlation exponent p, 4a as mu = 4a/1+a approximate to 0.8, which is larger than the observed mu approximate to 0.20. Furthermore, corresponding integrals are studied using the Burgers vortex and the Burgers equation. In those single-scale phenomena, the cross-over scale lies in the dissipative range. The scale r(*) offers a practical method of quantifying intermittency. This paper also sorts out a number of existing mathematical bounds and phenomenological models on the basis of the CKN Reynolds number. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4767728]