Modal and nonmodal stability of a stably stratified boundary layer flow

The modal and nonmodal linear stability of a stably stratified Blasius boundary layer flow, composed of a velocity and a thermal boundary layer, is investigated. The temporal and spatial linear stability of such flow is investigated for several Richardson, Reynolds, and Prandtl numbers. While increasing the Richardson number stabilizes the flow, a more complex behavior is found when changing the Prandtl number, leading to a stabilization of the flow up to $\text{Pr}=7$, followed by a destabilization. The nonmodal linear stability of the same flow is then investigated using a direct-adjoint procedure optimizing four different approximations of the energy norm based on a weighted sum of the kinetic and the potential energies. No matter the norm approximation, for short target times an increase of the Richardson number induces a decrease of the optimal energy gain and time at which it is obtained and an increase of the optimal streamwise wave number, which considerably departs from zero. Moreover, the dependence of the energy growth on the Reynolds number transitions from quadratic to linear, whereas the optimal time, which varies linearly with Re in the nonstratified case, remains constant. This suggests that the optimal energy growth mechanism arises from the joint effect of the lift-up and the Orr mechanism, that simultaneously act to increase the shear production term on a rather short timescale, counterbalancing the stabilizing effect of the buoyancy production term. Although these short-time mechanisms are found to be robust with respect to the chosen norm, a different amplification mechanism is observed for long target times for three of the proposed norms. This strong energy growth, due to the coupling between velocity and temperature perturbations in the free stream, disappears when the variation of the stratification strength with height is accurately taken into account in the definition of the norm.

Figure 1

Stratified Blasius base-flow sketch. (a) Streamwise velocity and temperature profiles of the analytical Blasius solution with $\text{Pr}=0.7$. (b) Ratio between the temperature and the velocity boundary layer thicknesses versus the Prandtl number.

Figure 2

Values of $\varphi \left(y\right)$ for the different approximations of the energy norm.

Figure 3

Richardson and Prandtl number influence on the Blasius flow. Contours of largest growth rate ${\alpha }_i$ (continuous lines) and corresponding angular frequency $\omega$ of largest growth (dashed lines) in the (Re, Ri) plane for $\text{Pr}=0.7$ (a) and in the (Re, Pr) plane for $\text{Ri}={10}^{-2}$ (b). The continuous black line is the neutral stability curve.

Figure 4

Absolute value of the five components of the unstable eigenvectors for the stratified Blasius flow at critical conditions (indicated in Table 1) for different Richardson and Prandtl numbers.

Figure 5

Temperature component of the unstable eigenvalues (continuous lines in both subfigures) and corresponding (a) temperature and (b) velocity profile of the base flow minus the phase speed of the perturbation (dashed lines) at $\text{Ri}={10}^{-2}$ and $\text{Re}={\text{Re}}_c$ [see Fig. 3] for four different values of the Prandtl number (see legend). In (b) the dots indicate the peak location of the temperature component of the eigenvectors, whereas the critical layer position corresponds to the $y$ location for which the dashed curves have zero value.

Figure 6

Isocontours of the spatial growth rate (blue for negative, red for positive values) and neutral curves (solid lines) for different Richardson and Prandtl numbers for the stratified (continuous lines) and unstratified (dashed lines) Blasius flow. The critical conditions are indicated in Table 1.

Figure 7

Optimal gain contours $G_{\mathrm{opt}}$ as a function of the streamwise wave number $\alpha$ and the spanwise wave numbers $\beta$, at fixed Reynolds number $\text{Re}=500$ and Prandtl number $\text{Pr}=0.7$ for different Richardson numbers ($\text{Ri}={10}^{-5}, {10}^{-4}, {10}^{-3}, {10}^{-2}$). The red cross symbols indicate the conditions of optimal growth.

Figure 8

Optimal time contours $t_{\mathrm{opt}}$ as a function of the streamwise $\alpha$ and the spanwise $\beta$ wave numbers at fixed Reynolds number $\text{Re}=500$ and Prandtl number $\text{Pr}=0.7$ for different Richardson numbers ($\text{Ri}={10}^{-5}, {10}^{-4}, {10}^{-3}, {10}^{-2}$). The red cross symbol indicates the conditions of optimal growth.

Figure 9

Optimal gain $G_{\mathrm{opt}}$ (a), optimal time $t_{\mathrm{opt}}$ (b), optimal streamwise wave number ${\alpha }_{\mathrm{opt}}$ (c), and optimal spanwise wave number ${\beta }_{\mathrm{opt}}$ (d) trends with the Richardson number Ri for different values of Reynolds number Re at fixed Prandtl number $\text{Pr}=0.7$.

Figure 10

Optimal gain $G_{\mathrm{opt}}$ (a) and optimal time $t_{\mathrm{opt}}$ (b) trends with the Reynolds number Re for different values of Richardson number Ri at fixed Prandtl number $\text{Pr}=0.7$.

Figure 11

Optimal gain $G_{\mathrm{opt}}$ (a) and optimal time $t_{\mathrm{opt}}$ (b) trends with the Prandtl number Pr for different values of Richardson number Ri at fixed Reynolds number $\text{Re}=500$.

Figure 12

(a) Envelope of the optimal energy gain versus time (continuous line) and time evolution of the gain for three optimal perturbations obtained for target times $\overline{t}=20,50,150$ (dashed lines from left to right, the black dots indicate the optimal gain obtained for each of the given target times). (b) Three-dimensional view of the streamwise velocity perturbation at the optimal time $\overline{t}=t_{\mathrm{opt}}=88.62$ [red dot in (a)]. The optimizations have been performed at fixed Richardson number $\text{Ri}={10}^{-3}$, Reynolds number $\text{Re}=500$, Prandtl number $\text{Pr}=0.7$, streamwise wave number $\alpha =0.15$, and spanwise wave number $\beta =0.8$.

Figure 13

Wall-normal profiles of the optimal disturbance at (a) $t=0$ and (b) at the optimal time $t=t_{\mathrm{opt}}$, for Richardson number $\text{Ri}={10}^{-3}$, Reynolds number $\text{Re}=500$, Prandtl number $\text{Pr}=0.7$, streamwise wave number $\alpha =0.15$, and spanwise wave number $\beta =0.8$.

Figure 14

Streamwise velocity evolution in $xy$ plane of the optimal perturbation at four different times: $t=0, t=30, t=t_{\mathrm{opt}}=88.62$, and $t=150$. At fixed Richardson number $\text{Ri}={10}^{-3}$, Reynolds number $\text{Re}=500$, Prandtl number $\text{Pr}=0.7$, streamwise wave number $\alpha =0.15$, and spanwise wave number $\beta =0.8$.

Figure 15

Richardson number influence on the optimal gain (a) and optimal time (b) for two-dimensional perturbations with (triangles) $\beta =0, \alpha ={{\alpha }_{\mathrm{opt}}}_{3\text{D}}$, and (circles) $\alpha =0, \beta ={{\beta }_{\mathrm{opt}}}_{3\text{D}}$ for $\text{Re}=500, \text{Pr}=0.7$.

Figure 16

Shear production (a) and vertical buoyancy flux production (b) for different Richardson numbers: $\text{Ri}={10}^{-5}, \text{Ri}={10}^{-4}, \text{Ri}={10}^{-3}, \text{Ri}={10}^{-2}$ at fixed Reynolds number $\text{Re}=500$ and Prandtl number $\text{Pr}=0.7$.

Figure 17

Envelope of the optimal energy gain $G$ versus the target time $\overline{t}$ for (a) $\text{Ri}={10}^{-2}, \text{Re}=500, \text{Pr}=0.7, \alpha =0.35, \beta =0.75$, and domain size $L_y=100$ for all the norm approximations considered (see legend within the figure) and for (b) (case a, black line) $\text{Ri}={10}^{-3}, \text{Re}=500, \text{Pr}=0.7, \alpha =0.15, \beta =0.8$; (case b, red line) $\text{Ri}={10}^{-3}, \text{Re}=500, \text{Pr}=0.7, \alpha =0.35, \beta =0.75$; (case c, gray line), $\text{Ri}={10}^{-3}, \text{Re}=750, \text{Pr}=0.7, \alpha =0.15, \beta =0.775$; (case d, blue line) $\text{Re}=250, \text{Ri}={10}^{-4}, \text{Pr}=0.7, \alpha =0, \beta =0.7$.

Figure 18

Shape of the optimal disturbance at (a) $t=0$ and (b) at the optimal time associated with the second peak $t=t_{\mathrm{opt}}^l=889.16$. The solution is obtained for $\text{Ri}={10}^{-2}, \text{Re}=500, \text{Pr}=0.7, \alpha =0.35, \beta =0.75$, and $L_y=100$.

Figure 19

Contours of temperature, with the black arrows denoting the velocity field in the $x-y$ plane of the optimal disturbance at $t_{\mathrm{opt}}^l=889.16$ on a $x-y$ plane for $\mathrm{Ri}={10}^{-2}, \mathrm{Re}=500, \mathrm{Pr}=0.7, \alpha =0.35, \beta =0.75$, and $L_y=100$.

Figure 20

Energy budget comparison for the optimal perturbations associated to the long-time peak (a) and short-time one (b) at $\text{Ri}={10}^{-2}, \text{Re}=500, \text{Pr}=0.7, \alpha =0.35, \beta =0.75$: shear production $P_s$, buoyancy flux production $P_b$, and temperature production $P_T$ evolution in time.

Figure 21

Time evolution of the wall-normal velocity and temperature components of the long-time optimal disturbance extracted at $y=48.8$ (corresponding to their peak position, see Fig. 19), for the norm approximations $E_1$ (a), $E_2$ (b), and $E_3^{{\delta }^{*}}$ (c). The solutions are obtained for $\text{Ri}={10}^{-2}, \text{Re}=500, \text{Pr}=0.7, \alpha =0.35, \beta =0.75$, and $L_y=100$.

Figure 22

Phase of the five components of the unstable eigenvectors for the stratified Blasius flow at critical conditions (indicated in Table 1) for different Richardson and Prandtl numbers.

Figure 23

Spatial structure of the unstable eigenvectors for the stratified Blasius flow at critical conditions (indicated in Table 1) for different Richardson and Prandtl numbers: contours of the temperature component and vectors of the velocity component on a $x\text{−}y$ plane.