Modeling mode interactions in boundary layer flows via the parabolized Floquet equations

In this paper, we develop a model based on successive linearization to study interactions between different modes in boundary layer flows. Our method consists of two steps. First, we augment the Blasius boundary layer profile with a disturbance field resulting from the linear parabolized stability equations (PSE) to obtain the modified base flow; and, second, we draw on Floquet decomposition to capture the effect of mode interactions on the spatial evolution of flow fluctuations via a sequence of linear progressions. The resulting parabolized Floquet equations (PFE) can be conveniently advanced downstream to examine the interaction between different modes in slowly varying shear flows. We apply our framework to two canonical settings of transition in boundary layers; the H-type transition scenario that is initiated by exponential instabilities, and streamwise elongated laminar streaks that are triggered by the lift-up mechanism. We demonstrate that the PFE capture the growth of various harmonics and provide excellent agreement with the results obtained in direct numerical simulations and in experiments.

Figure 1

Geometry of a transitional boundary layer flow.

Figure 2

(a) The PFE are triggered with a primary disturbance ${\mathbf{u}}_{\mathrm{pr}}$ that results from linear PSE and modulates the base flow. The diagonal lines represent the base flows that enter as coefficients into the linear PSE and PFE, respectively. (b) When the secondary disturbances are of the same frequencies and spatial wave numbers as the primary disturbances, the dominant harmonics resulting from each PFE iteration (e.g., ${\widehat{\mathbf{q}}}_0$, ${\widehat{\mathbf{q}}}_{\pm 1}$, and ${\widehat{\mathbf{q}}}_{\pm 2}$ in the formation of streaks) can be used to update the modulation to the base flow, iterate the PFE, and compute an equilibrium configuration.

Figure 3

(a) The rms amplitudes of the streamwise velocity components of the fundamental and subharmonic modes resulting from experiments [27] ($▵$), DNS [28] ($--$), nonlinear PSE [36] ($\circ$), PFE (thick solid lines), and linear PSE (thin solid lines). The fundamental ($2,0$) mode, ($1,1$) subharmonic, and ($3,1$) subharmonic are represented by black, blue, and red colors, respectively. (b)–(d) The normalized amplitudes of the streamwise velocity components of the (2,0) (b), (1,1) (c), and (3,1) (d) modes at $\text{Re}=620$ resulting from PFE ($-$), and nonlinear PSE ($\circ$).

Figure 4

The rms amplitudes of the streamwise velocity components for various harmonics with $\omega =0$ and $\beta =0.4065$ resulting from DNS ($▵$), PFE ($-$), and linear PSE ($--$). The MFD, first, second, and third harmonics are shown in black, blue, red, and green, respectively.

Figure 5

Cross-plane contours of the streamwise velocity of the streaks comprised of all harmonics in the spanwise direction at $x=2400$ resulting from DNS (a), linear PSE (b), and PFE with (c) and without (d) the MFD component.

Figure 6

(a) The rms amplitudes of the streamwise velocity components for various harmonics with $\omega =0$ and $\beta =0.4065$ after the first ($\cdots$), third ($--$), and seventh ($-$) run of the PFE. The MFD, first, and second harmonics are shown in black, blue, and red, respectively. (b) The rms amplitudes resulting from DNS ($▵$), nonlinear PSE ($\circ$), and PFE ($-$). The evolution of the fundamental harmonic due to linear PSE is shown by the thin solid line.

Figure 7

Cross-plane contours of the streamwise velocity of the higher-amplitude streaks at $x=2400$, which is comprised of all harmonics in the spanwise direction; (a) DNS and (b) PFE.

Figure 8

(a) The streamwise velocity component of the MFD; and (b) the magnitude of the streamwise component of the first harmonic at $x=1700$ resulting from DNS ($▵$) and PFE ($-$).

Figure 9

(a) The streamwise velocity component of the MFD; and (b) the magnitude of the streamwise component of the first harmonic at $x=2400$ resulting from DNS ($▵$) and PFE ($-$).

Figure 10

Block diagram illustrating the inclusion of control into the PFE loop. The controller uses the fluctuation field resulting from previous PFE iterations to dictate the control signal $\mathbf{f}$ that perturbs the flow dynamics.