Simple model for mean stress in turbulent boundary layers

The mean stress is one of the most important quantities of interest in turbulent boundary layers. The governing equations for the mean flow are used to derive a relation between the mean total stress and the mean velocity in a zero pressure gradient turbulent boundary layer, allowing the mean shear stress to be written as a function of wall-normal distance. The relation contains an unknown term, which is modeled using a linear function of the wall-normal distance, inspired by existing data sets. The model for the mean total stress requires the wall-normal mean velocity profile, which requires modeling if not available. The existing data sets and scaling arguments are used to obtain a simple and compact fit for the mean wall-normal velocity, which is subsequently used to obtain a simple model for the mean total stress. The model shows good agreement with the available simulation and experimental data for a large range of Reynolds number.

Figure 1

Profiles of (a) $T^{+}$ and (b) $-{\overline{u^{\prime }{v}^{\prime }}}^{+}$ are shown in zero pressure gradient turbulent boundary layers using the DNS data of Schlatter and Örlü [12] (see Table 1).

Figure 2

Profiles of $V/V_e$ are shown in zero pressure gradient turbulent boundary layers using the DNS data of Schlatter and Örlü [12] (see Table 1).

Figure 3

$I/(H-1)$ profiles are shown in zero pressure gradient turbulent boundary layers using the simulation data of Schlatter and Örlü [12].

Figure 4

The model for $V/V_e$ is compared to the simulation data [12] (see Table 1). Note that the profiles are shifted upwards by 0.5 for clarity.

Figure 5

The model for $T^{+}$ is compared to the simulation data [12] (see Table 1). Note that the profiles are shifted upwards by 0.5 for clarity.

Figure 6

Models for (a) $V/V_e$ and (b) $T^{+}$ are compared to the simulation data at higher Re [13, 18] (see Table 1). Note that the profiles are shifted upwards by 0.5 for clarity.

Figure 7

The model for viscous stress ($d{U}^{+}/d{y}^{+}$) is compared to the simulation data [13, 18] (see Table 1). Note that the profiles are shifted upwards by 0.5 for clarity.

Figure 8

The model for $T^{+}$ is compared to the Reynolds shear stress experimental data of (a) De Graaff and Eaton [14] and (b) Baidya et al. [15] (see Table 1). Note that the profiles are shifted upwards by 0.5 for clarity.

Figure 9

Sensitivity of model constants for the $V/V_e$ model: $a$ (solid) and $b$ (dashed).