Higher DNS-resolution requirements for expanded overlap region and confirmation of a convergence criterion

Direct numerical simulations (DNS) stand out as formidable tools in studying turbulent flows. Despite the fact that the achievable Reynolds number remains lower than those available through experimental methods, DNS offers a distinct advantage: the complete knowledge of the velocity field, facilitating the evaluation of any desired quantity. This capability should extend to compute derivatives. Among the classic functions requiring derivatives is the indicator function, $\Xi(y^+) = y^+\frac{{\rm d}\overline{U}_x^+}{{\rm d}y^+}$. This function encapsulates the wall-normal derivative of the streamwise velocity, with its value possibly influenced by mesh size and its spatial distribution. The indicator function serves as a fundamental element in unraveling the interplay between inner and outer layers in wall-bounded flows, including the overlap region, and forms a critical underpinning in turbulence modeling. Our investigation reveals a sensitivity of the indicator function on the utilized mesh distributions, prompting inquiries into the conventional mesh-sizing paradigms for DNS applications.