Outer-layer universality of the mean velocity profile in turbulent wall-bounded flows

We derive objective criteria to discern universality of the outer-layer mean velocity profile in wall-bounded flows, which yield a set of outer velocity and length scales. The new scaling generalizes the Rotta-Clauser scaling and includes the Zagarola-Smits scaling as a special case for internal flows, and it supports the friction velocity as the most plausible velocity scale in the asymptotic high-Reynolds-number regime. A composite logarithmic/parabolic formulation is proposed for the outer mean velocity profile, which yields simple and accurate approximations for boundary layers, pipes, and channels.

Figure 1

Boundary layers: (a) inner-scaled external velocity ($U_e^{+}={(2/C_f)}^{1/2}$), (b) shape factor ($H={\delta }^{*}/\theta$), and (c) boundary layer thickness ratio (${\delta }_{99}/{\delta }_{95}$) as a function of displacement thickness Reynolds number, ${\mathit{\text{Re}}}_{{\delta }^{*}}=U_e{\delta }^{*}/\nu$. See Table 1 for nomenclature of symbols. The dashed line in panel (a) corresponds to $U_e^{+}=1/kln{\mathit{\text{Re}}}_{{\delta }^{*}}+C$, with $k=0.38$, $C=3.0$. Symbols as in Table 1.

Figure 2

Boundary layers: mean defect velocity in classical wall scaling. (a) linear scaling; (b) semilog scaling; (c) premultiplied form; (d) statistical scatter around the mean. In panel (d) the shaded blue region marks unit standard deviation uncertainty from the mean distribution (solid blue), and the dash dotted red line denotes a compound logarithmic/parabolic fit given by Eqs. (14) and (15) (see Table 2). The inset reports the same data in linear scale. Lines denote DNS data, symbols as in Table 1.

Figure 3

Boundary layers: mean defect velocity in (a) classical scaling, (b) Rotta-Clauser scaling, (c) Zagarola-Smits scaling, (d) new scaling. Lines denote DNS data, symbols as in Table 1. The insets show separately DNS data at low Reynolds number from Sillero et al. [27] ($709\le {\mathit{\text{Re}}}_{{\delta }^{*}}\le 2474$, gray lines), and from Schlatter and Örlü [28] ($306\le {\mathit{\text{Re}}}_{{\delta }^{*}}\le 5045$, purple lines).

Figure 4

Boundary layers: fitting error for outer-layer defect velocity representations. (a) Wall-normal variation of standard deviations of individual velocity profiles from the averaged distributions $[\sigma$, see Eq. (12)], and (b) $y$ integrated standard deviations of individual profiles from the averaged distributions $[\mathrm{\Sigma }$, see Eq. (13)], as a function of ${\mathit{\text{Re}}}_{{\delta }^{*}}$. Baseline (blue), Zagarola-Smits (purple), Rotta-Clauser (red), new (black).

Figure 5

Boundary layers: mean defect velocity in (a) classical scaling, (b) Rotta-Clauser scaling, (c) Zagarola-Smits scaling, (d) new scaling. Data as in Fig. 3. The shaded blue regions mark unit standard deviation uncertainty from the mean distribution (solid blue), and the dash dotted red line denotes a compound logarithmic/parabolic fit given by Eqs. (14) and (15) (see Table 2). The insets report the same data in linear scale.

Figure 6

Boundary layers: defect velocity in expected logarithmic layer (defined as $y^{+}\ge 100$, $y/{\delta }_o\le 0.1$), in the classical (a) and new (d) scaling. The reference log layers are given by dashed lines (see Table 2).

Figure 7

Boundary layers: Reynolds number dependence of the Zagarola-Smits velocity scale (a), and of the newly introduced velocity and length scales (b)–(f), under different normalizations. To facilitate interpretation, percent deviations with respect to the overall mean values are reported on the axes on the right. Symbols as in Table 1.

Figure 8

Pipe flow: mean defect velocity in (a) classical scaling, (b) Zagarola-Smits scaling. Lines denote DNS data, symbols as in Table 3.

Figure 9

Pipe flow: standard deviation from average velocity profile. (a) $5500\le R{e}_{\delta }*\le 1,458000$; (b) $R{e}_{\delta }*<15,000$. Baseline (blue), Zagarola-Smits (purple).

Figure 10

Pipe flow: Reynolds number dependence of the Zagarola-Smits velocity scale, under inner (a) and outer (b) normalizations. Symbols as in Table 3.

Figure 11

Channel flow: mean defect velocity in (a) classical scaling, (b) Zagarola-Smits scaling. Lines denote DNS data, symbols as in Table 4.

Figure 12

Channel flow: standard deviation from average velocity profile ($R{e}_{\delta }*<15,000$). Baseline (blue), Zagarola-Smits (purple).

Figure 13

Channel flow: Reynolds number dependence of the Zagarola-Smits velocity scale, under inner (a) and outer (b) normalization. Symbols as in Table 4.

Figure 14

(a) Pipe mean defect velocity in classical scaling, (b) Pipe mean defect velocity in Zagarola-Smits scaling. (c) Channel mean defect velocity in classical scaling; (d) Channel mean defect velocity in Zagarola-Smits scaling. Data as in Figs. 8 and 11. The shaded blue regions mark unit standard deviation uncertainty from the mean distribution (solid blue), and the dash dotted red line denotes a compound logarithmic/parabolic fit given by Eqs. (14) and (15) (see Table 2). The insets report the same data in linear scale.