Sketch of the ratio of the mean viscous force (VF) to mean effect of turbulent inertia (TI). This ratio reveals the four layer force balance structure of turbulent wall-bounded flows.8 Note that this sketch is for a fixed Reynolds number, as the layer boundaries depend on δ+. Note also that layer I in the zero pressure gradient turbulent boundary layer (dotted line in I) differs from that of channel or pipe flow. In the boundary layer all of the terms in the mean momentum equation approach zero as y → 0.
Ratio of the mean viscous force to the mean effect turbulent inertia in pipe flow for 90 ≲ δ+ ≲ 1000.
Inner-normalized mean viscous force and time-averaged turbulent inertia profiles in pipe flow for 90 ≲ δ+ ≲ 1000.
Positions of the inner and outer maxima of the dT +/dy + profiles of Fig. 3. Note that the inner peak approaches a fixed y + value (), while the outer peak approaches a fixed y/δ value (y/δ ≃ 0.5). The vertical line (δ+ ≃ 180) indicates the present estimate for the onset of the four layer regime.
Profiles of T + and WdT +/dy + at δ+ = 171.
Actual and predicted values of versus δ+: actual values, •; values predicted from Eq. (10), ■.
Layer width distribution of the L β hierarchy for transitional and four layer regime pipe flow; (a) linear axes and (b) logarithmic axes. Note that the Reynolds number of each profile is given by the end point position y + = δ+ in (b). The curve-fit of the W(y +) distribution at δ+ = 1002 is over the range and given by 7.52 + 0.644y +. The dashed profile is that of Wu and Moin18 at δ+ = 1142.
Comparison of the linear portion of the W(y +) profiles from channel and pipe flow at δ+ ≃ 1000. Data are from the present simulation, δ+ = 1002, the pipe flow simulation of Wu and Moin,18 δ+ = 1142, and the channel flow simulation of Kawamura et al.,21 δ+ = 1016.
Scaling behaviors of the layer thicknesses and velocity increments associated with the mean momentum equation in turbulent wall-flows in the four-layer regime, see Fig. 1. Note that the layer IV properties are asymptotically attained as δ+ → ∞, see Ref. 5.
Summary of computational parameters.
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