Modal and nonmodal stability analysis of turbulent stratified channel flows

Unstable or optimally growing perturbations of turbulent flows are often representative of the energy-containing coherent structures populating the flow, as for streaks in a turbulent channel. Within this framework, this work aims at studying the modal and nonmodal stability of stably stratified turbulent channel flow, assessing the influence of stratification while changing the friction Richardson number, ${\text{Ri}}_{\tau }$, at fixed friction Reynolds number, ${\text{Re}}_{\tau }$. When increasing the stratification of the flow, the energy gain for streamwise-independent perturbations at the outer peak increases by two orders of magnitude, and the spanwise wavenumber for which the energy gain peaks reaches values comparable to those reported in the direct numerical simulations of Garcia-Villalba and Del Alamo. At the same time, the value of the optimal gain for the inner peak slightly changes, corroborating the observations made through direct numerical simulation (DNS) about the fact that the wall cycle is not altered by the presence of stratification. Moreover, for nonzero values of the streamwise and spanwise wavenumbers, $\alpha$ and $\beta$, the energy gain curve has two peaks, one for shorter target times and $\alpha >\beta$, leading to a center-channel temperature peak, and another occurring for $\alpha <\beta$ at larger target times. In the former case, energy production is mostly linked to velocity production, whereas, in the latter case, the strongest term is that of temperature production, indicating that this mechanism is driven by the increase of the potential energy rather than the kinetic one, and it is intimately linked to the presence of stratification. For strong stratification, the optimal energy gain considerably extends towards higher values of $\alpha$, leading to energy amplifications reaching three orders of magnitudes for values of $\alpha$ up to 2. The associated optimal perturbations are characterized by temperature patches at the center channel, phase lagged by $\pi /2$ with the wall-normal velocity, similarly to gravity waves recovered in the DNS for sufficiently large stratification. However, for large values of $\beta$, we observe an increasing asymmetry in the optimal perturbations, probably due to the shielding effect of the core of the channel, as also observed in the DNS of Garcia-Villalba and Del Alamo.

Figure 1

Scheme of the channel flow with its scales and a representation of the mean temperature $\overline{T}\left(y\right)$.

Figure 2

Streamwise velocity $\overline{u}$ (left) and temperature $\overline{T}$ (right) for ${\text{Re}}_{\tau }=550$ and ${\text{Ri}}_{\tau }=60, {\text{Ri}}_{\tau }=120$, and ${\text{Ri}}_{\tau }=480$.

Figure 3

Eddy viscosity ${\nu }_t$ (left) and thermal eddy diffusivity ${\kappa }_t$ (right) for ${\text{Re}}_{\tau }=550$ and ${\text{Ri}}_{\tau }=60, {\text{Ri}}_{\tau }=120$, and ${\text{Ri}}_{\tau }=480$.

Figure 4

Convergence study for optimal cases at ${\text{Ri}}_{\tau }=60, (\alpha ,\beta )=(0,2.2)$ (left) and ${\text{Ri}}_{\tau }=480, (\alpha ,\beta )=(0,6)$ (right).

Figure 5

Eigenspectra for ${\text{Ri}}_{\tau }=60 (\text{Ri}=0.139878)$ (left), ${\text{Ri}}_{\tau }=120 (\text{Ri}=0.2797757)$ (middle), and ${\text{Ri}}_{\tau }=480 (\text{Ri}=1.11910287)$ (right) for ${\text{Re}}_{\tau }=550 (\text{Re}=11390.6), \alpha =2$, and $\beta =2$.

Figure 6

Contours of $G_{\max }$ in function of $\alpha$ and $\beta$ for ${\text{Ri}}_{\tau }=60$ (left), ${\text{Ri}}_{\tau }=120$ (middle), and ${\text{Ri}}_{\tau }=480$ (right), where $G_{\max }$ refers to the max value of the gain in time. The values of $\beta$ used are [1, 1.25, 1.5, 1.75, 2, 2.5, 3, 3.5, 4, 5, 7.5, 10, 12.5, 15, 20, 25, 28, 30, 32, 35, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 250, 300, 350, 400, 450, 500].

Figure 7

$G_{\max }$ versus $\beta$ for different values of ${\text{Ri}}_{\tau }$ (left) and versus ${\text{Ri}}_{\tau }$ for the optimal value of $\beta$ (right) with $\alpha =0$ and ${\text{Re}}_{\tau }=550 (\text{Re}=11390.6)$. The black and red boxes represent the zoomed-in areas of the plot.

Figure 8

Module of the optimal perturbation at time $t=0$ for the inner peak at ${\text{Ri}}_{\tau }=0$ and ${\text{Ri}}_{\tau }=60$ (see Table 4).

Figure 9

Module of the optimal perturbation at time $t=T_{\mathrm{opt}}$ for the inner peak at ${\text{Ri}}_{\tau }=0$ and ${\text{Ri}}_{\tau }=60$ (see Table 4).

Figure 10

Optimal energy gain versus target time at ${\text{Ri}}_{\tau }=60$ for $\alpha =0, \beta =1.699$ (left), $\alpha =0.5, \beta =1$ (middle), and $\alpha =0.75, \beta =1$ (right).

Figure 11

Real part (upper frames) and imaginary part (lower frames) of the optimal perturbation at time $t=0$ (left) and at time $t=T_{\mathrm{opt}}=T_{\max }$ (right) for $\alpha =0, \beta =1.699$, and ${\text{Ri}}_{\tau }=60$.

Figure 12

Real part (upper frames) and imaginary part (lower frames) of the optimal perturbation at ${\text{Ri}}_{\tau }=60$ for cases $\alpha =0.5, \beta =1, t=T_{\mathrm{opt}}=25$ (left) and $\alpha =0.75, \beta =1, t=T_{\mathrm{opt}}=12$ (right).

Figure 13

Real part of the optimal perturbation at time $t=0$ (left) and $t=T_{\mathrm{opt}}$ (right) for $\alpha =0.5, \beta =1$, and ${\text{Ri}}_{\tau }=60$ in the plane $z\text{−}y$. Shaded contours of temperature and vectors of $w^{\prime }-v^{\prime }$.

Figure 14

Real part of the optimal perturbation at time $t=0$ (left) and $t=T_{\mathrm{opt}}$ (right) for $\alpha =0.5, \beta =1$, and ${\text{Ri}}_{\tau }=60$ in the plane $x\text{−}y$. Shaded contours of $u^{\prime }$ and vectors of $u^{\prime }-v^{\prime }$.

Figure 15

Real part of the optimal perturbation at time $t=0$ (left) and $t=T_{\mathrm{opt}}$ (right) for $\alpha =0.75, \beta =1$, and ${\text{Ri}}_{\tau }=60$ in the plane $z\text{−}y$. Shaded contours of temperature and vectors of $w^{\prime }-v^{\prime }$.

Figure 16

Real part of the optimal perturbation at time $t=0$ (left) and $t=T_{\mathrm{opt}}$ (right) for $\alpha =0.75, \beta =1$, and ${\text{Ri}}_{\tau }=60$ in the plane $x\text{−}y$. Shaded contours of temperature and vectors of $u^{\prime }-v^{\prime }$.

Figure 17

Values of the spatial wavenumbers for which the optimal energy gain mechanism changes for ${\text{Ri}}_{\tau }=60$. The map is obtained by representing the isovalues of the difference of the two energy peaks: $\mathrm{\Delta }G=G_{1\mathrm{st}-\mathrm{mech}}-G_{2\mathrm{nd}-\mathrm{mech}}$. When $\mathrm{\Delta }G>0$ the first mechanism prevails; when $\mathrm{\Delta }G<0$ the second one dominates. The boundary represents the isovalue at $\mathrm{\Delta }G=0$. The colored dots represent the two flow cases for which the optimal perturbations are discussed in detail.

Figure 18

Energy budget contributions for the optimal mechanism at $\alpha =0.5, \beta =1$ (left) and for that at $\alpha =0.75, \beta =1$ (right) for ${\text{Ri}}_{\tau }=60$.

Figure 19

Optimal energy gain versus target time (left frame) and module of optimal perturbation (right frame) obtained by setting the term $v^{\prime }\partial \overline{T}/\partial y$ to zero in the temperature equations, as well as its adjoint counterpart in the adjoint wall-normal momentum equation, for the case at ${\text{Ri}}_{\tau }=60, \alpha =0.75$, and $\beta =1$.

Figure 20

Eigenspectra for ${\text{Ri}}_{\tau }=60, \alpha =0.75, \beta =1$ (left) and for ${\text{Ri}}_{\tau }=120, \alpha =0, \beta =2.597$ (right).

Figure 21

Optimal energy gain versus $T_{\mathrm{opt}}$ for $\alpha =0, \beta =2.597$ and ${\text{Ri}}_{\tau }=120$ (left) and close-up of the first peak (right).

Figure 22

Real part (upper frames) and imaginary part (lower frames) of the initial optimal perturbations at $t=0$ for $\alpha =0, \beta =2.597$, and ${\text{Ri}}_{\tau }=120$ related to the first (left), second (middle), and third (right) energy peaks shown in Fig. 21.

Figure 23

Real part (upper frames) and imaginary part (lower frames) of the optimal perturbations at $t=T_{\mathrm{opt}}$ for $\alpha =0, \beta =2.597$, and ${\text{Ri}}_{\tau }=120$ related to the first (left, $T_{\mathrm{opt}}=6.5$), second (middle, $T_{\mathrm{opt}}=19.5$), and third (right, $T_{\mathrm{opt}}=23.75$) energy peaks shown in Fig. 21.

Figure 24

Energy budget (left) and flux and production (right) for ${\text{Ri}}_{\tau }=120, \alpha =0, \beta =2.597$, and $T_{\mathrm{opt}}=23.75$. The right plots refer to the time instants identified by the black symbols in the left frames and reported in the legend.

Figure 25

Time evolution of the optimal streamwise velocity $u^{\prime }$ and temperature $T^{\prime }$ perturbations for $\alpha =0$ and $\beta =2.597$, at ${\text{Ri}}_{\tau }=120$ at times $T=5.8$ (left), $T=11.6$ (left middle), $T=17.4$ (right middle), and $T=23.2$ (right).

Figure 26

Values of the spatial wavenumbers for which the optimal energy gain mechanism changes for ${\text{Ri}}_{\tau }=120$. This map is obtained by assigning each combination of $\alpha$ and $\beta$ a value denoting the dominant mechanism: 10 for the first (short timescale), 0 for the second, and $-10$ to the first (long timescale).

Figure 27

Optimal energy gain versus $T_{\mathrm{opt}}$ for ${\text{Ri}}_{\tau }=480, \alpha =0$, and $\beta =2.2$ (left), $\beta =6$ (right).

Figure 28

Real part (upper frames) and imaginary part (lower frames) of the optimal perturbation at ${\text{Ri}}_{\tau }=480$ for $(\alpha ,\beta )=(0,2.2)$ and $T_{\mathrm{opt}}=12.3$ (left), $(\alpha ,\beta )=(0,6)$ and $T_{\mathrm{opt}}=9.9$ (right).

Figure 29

Optimal energy gain versus $T_{\mathrm{opt}}$ for ${\text{Ri}}_{\tau }=480, \alpha =2.5$, and $\beta =0.5$ (left) and real (top right) and imaginary (bottom right) part of the optimal perturbation at ${\text{Ri}}_{\tau }=480$ for $(\alpha ,\beta )=(2.5,0.5)$.

Figure 30

Optimal perturbation at time $t=0$ (left) and $t=T_{\mathrm{opt}}$ (right) for $\alpha =2.5, \beta =0.5$, and ${\text{Ri}}_{\tau }=480$ in the plane $x\text{−}y$. Shaded contours of temperature and vectors of $u^{\prime }-v^{\prime }$.

Figure 31

Temperature $T^{\prime }$ and wall-normal velocity $v^{\prime }$ perturbations for the optimal disturbance at ${\text{Ri}}_{\tau }=480, \alpha =2.5$, and $\beta =0.5$.

Figure 32

$G_{\max }$ versus $\beta$ with $\alpha =0$ for different values of ${\text{Re}}_{\tau }$ in outer scale (left) and inner scale (right).

Figure 33

Comparison for ${\nu }_t$ at ${\text{Ri}}_{\tau }=60$ (left), ${\text{Ri}}_{\tau }=120$ (middle), and ${\text{Ri}}_{\tau }=480$ (right).

Figure 34

Comparison for ${\kappa }_t$ at ${\text{Ri}}_{\tau }=60$ (left), ${\text{Ri}}_{\tau }=120$ (middle), and ${\text{Ri}}_{\tau }=480$ (right).

Figure 35

Singular value decomposition for the case at ${\text{Ri}}_{\tau }=120, \alpha =0$, and $\beta =2.597$ with a constant prefactor for the potential energy.