Linear stability of buffer layer streaks in turbulent channels with variable density and viscosity

We investigate the stability of streaks in the buffer layer of turbulent channel flows with temperature-dependent density and viscosity by means of linear theory. The adopted framework consists of an extended set of the Orr-Sommerfeld-Squire equations that accounts for density and viscosity nonuniformity in the direction normal to the walls. The base flow profiles for density, viscosity, and velocity are averaged from direct numerical simulations (DNSs) of fully developed turbulent channel flows. We find that the inner scaling based on semilocal quantities provides an effective parametrization of the effect of variable properties on the linearized flow. The spanwise spacing of optimal buffer layer streaks scales to ${\lambda }_{z,\mathrm{opt}}^{★}\approx 90$ for all cases considered and the maximum energy amplification decreases, compared to the one for a flow with constant properties, if the semilocal Reynolds number ${\text{Re}}_{\tau }^{★}$ increases away from the walls, consistently with less energetic streaks observed in DNSs of turbulent channels. A secondary stability analysis of the two-dimensional velocity profile formed by the mean turbulent velocity and the nonlinearly saturated optimal streaks predicts a streamwise instability mode with wavelength ${\lambda }_{x,\mathrm{cr}}^{★}\approx 230$ in semilocal units, regardless of the fluid property distribution across the channel. The threshold for the onset of the secondary instability is reduced, compared to a constant property flow, if ${\text{Re}}_{\tau }^{★}$ increases away from the walls, which explains the more intense ejection events reported in DNSs. The opposite behavior is predicted by the linear theory for decreasing ${\text{Re}}_{\tau }^{★}$, in accord with DNS observations. We finally show that the phase velocity of the critical mode of secondary instability agrees well with the convection velocity calculated by DNSs in the near-wall region for both constant and variable viscosity flows.

Figure 1

Distribution of the semilocal Reynolds number ${\text{Re}}_{\tau }^{★}$ for the flow cases of Table 1. Lines indicate constant density flows and symbols are $◯,{\mathrm{CRe}}_{\tau }^{★}$; $\square ,{\mathrm{C}}_{\nu }$; $△$, GL; and $▽,{\mathrm{SRe}}_{\tau }^{★}\mathrm{LL}$. The wall is located at $y/h=0$ and the centerline at $y/h=1$. Only the half channel is shown.

Figure 2

Reynolds-averaged distributions of (a) viscosity and (b) density for the flow cases of Table 1. The wall is located at $y/h=0$ and the centerline at $y/h=1$. Only the half channel is shown.

Figure 3

Joint PDFs of the streamwise and wall-normal Favre-averaged velocity fluctuations. Values are nondimensionalized by the instantaneous density and wall shear stress. Contours indicate the probability-weighted Reynolds shear stress $\rho uv/{\tau }_{w}P(\sqrt{\rho }u/\sqrt{{\tau }_w},\sqrt{\rho }v/\sqrt{{\tau }_w})$. Dotted lines refer to case CP in each image; solid lines indicate (a) ${\mathrm{CRe}}_{\tau }^{★}$, (b) GL, and (c) LL. For each case displayed, lines are spaced by 0.02 and range from $-0.02$ (most outer) to $-0.12$.

Figure 4

Effective viscosity profiles ${\mu }_e=\mu +{\mu }_t$ interpolated from the data set described in Sec. 2 and summarized in Table 1. Results are organized into (a) constant and (b) temperature-dependent density cases. The molecular viscosity is a function of temperature for both.

Figure 5

Base velocity profiles calculated using Eq. (5) and the interpolated viscosity of Fig. 4. Results are organized into (a) constant and (b) temperature-dependent density cases. Mean velocity profiles extracted from DNSs are also reported for comparison and are indicated by symbols that correspond to $◯$, CP and ${\mathrm{CRe}}_{\tau }^{★}$; $\square ,{\mathrm{C}}_{\nu }$ and ${\mathrm{SRe}}_{\tau }^{★}{\mathrm{C}}_{\nu }$; $△$, GL and ${\mathrm{SRe}}_{\tau }^{★}\mathrm{GL}$; and $▽$, LL and ${\mathrm{SRe}}_{\tau }^{★}\mathrm{LL}$.

Figure 6

Maximum optimal growth function for the flow cases of Table 1. Results for constant property flow at ${\text{Re}}_{\tau }=550,950$ are also included. Data for ${\text{Re}}_{\tau }=950$ are taken from the available database by Hoyas and Jiménez [37]. Lines and symbols (same as in Fig. 1) correspond to constant and variable density cases, respectively. The spanwise wavelength is scaled in semilocal units as ${\lambda }_z^{★}=2\pi \overline{\text{Re}}{}_{\tau }^{★}/\beta$. The shaded area demarcates the range of maximum and minimum values of ${\lambda }_{z,\mathrm{opt}}^{★}$ for the investigated flow cases. Panels on the right magnify the regions near the peaks of $G_{\max }$ and show the difference between semilocal (top two) and classical inner scaling (bottom two).

Figure 7

Optimal initial velocity profiles calculated for ${\lambda }_{z,\mathrm{opt}}^{★}$. Profiles are normalized by their peak values.

Figure 8

Streamwise wavelength dependence of the imaginary part of the critical mode of secondary instability. Lengths are expressed in semilocal units using the reference ${\text{Re}}_{\tau }^{★}$ at $y^{★}=12$ (top) and in classical inner units (bottom). The critical conditions reported in Table 3 correspond to the maximum of each curve. The shaded area indicates the range of values encompassing ${\lambda }_{x,\mathrm{cr}}^{★}$ for all cases considered.

Figure 9

Wall-normal profiles of mean turbulent velocity $\overline{U}{}^{+}$ (solid) and of streamwise convection velocity $C_u^{+}$ (dashed) in wall units. Here $C_u^{+}$ is defined as in [43]. The horizontal dotted lines indicate the phase velocity of the critical mode.