A spectral model for the wall-normal distribution of velocity variance in turbulent wall layers

We analyse velocity spectra from direct numerical simulations (DNS) of turbulent pipe flow up to a friction Reynolds number italic Re Subscript tau Baseline almost equals 12 000 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> <mml:mrow> <mml:mi>τ</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≈</mml:mo> <mml:mn>12</mml:mn> <mml:mrow> <mml:mspace width="thinmathspace"/> </mml:mrow> <mml:mn>000</mml:mn> </mml:math> $ \textit{Re}_{\tau } \approx 12{\,}000$ . The spectral maps naturally partition into four regions, corresponding to small- and large-scale motions, and to wall-attached eddies, and are further divided into near-wall and outer contributions. Physically grounded models are formulated for each region, with particular emphasis on the near-wall range beneath the wall-attached eddies. Integration of the spectral models yields the distribution of the streamwise velocity variance in the lower part of the logarithmic layer, which agrees well with the DNS data. The analysis shows that the emergence of an outer-layer peak in the velocity variance at sufficiently high Reynolds number arises from the increasing relative contribution of wall-attached eddies, and can be interpreted without invoking additional physical mechanisms beyond this classical framework. Extrapolation to extremely high Reynolds numbers suggests a saturation of the velocity variance, characterised by two peaks of comparable magnitude.