Role of long waves in the stability of the plane wake

This work is directed toward investigating the fate of three-dimensional long perturbation waves in a plane incompressible wake. The analysis is posed as an initial-value problem in space. More specifically, input is made at an initial location in the downstream direction and then tracing the resulting behavior further downstream subject to the restriction of finite kinetic energy. This presentation follows the outline given by Criminale and Drazin [W. O. Criminale and P. G. Drazin, Stud. Appl. Math. 83, 123 (1990)] that describes the system in terms of perturbation vorticity and velocity. The analysis is based on large scale waves and expansions using multiscales and multitimes for the partial differential equations. The multiscaling is based on an approach where the small parameter is linked to the perturbation property independently from the flow control parameter. Solutions of the perturbative equations are determined numerically after the introduction of a regular perturbation scheme analytically deduced up to the second order. Numerically, the complete linear system is also integrated. Since the results relevant to the complete problem are in very good agreement with the results of the first-order analysis, the numerical solution at the second order was deemed not necessary. The use for an arbitrary initial-value problem will be shown to contain a wealth of information for the different transient behaviors associated to the symmetry, angle of obliquity, and spatial decay of the long waves. The amplification factor of transversal perturbations never presents the trend—a growth followed by a long damping—usually seen in waves with wave number of order one or less. Asymptotical instability is always observed.

Figure 1

(Color online) Base flow sketch. The base flow has been chosen in order to be an acceptable representation of the intermediate-far field. To this aim we build a homogeneous field in $x,z$ by using the information associated to a section, $x_0$, placed in the intermediate region, $x∊\left[5,30\right]$. In the sketch the longitudinal and transversal profiles at $\text{Re}=100$ are frozen at $x_0=10$ (note that the transversal velocity $V$ is multiplied by a factor of ten). The base flow $\left[U\left(y;x_0,\text{Re}\right),V\left(y;x_0,\text{Re}\right)\right]$ is thus a slightly non parallel flow homogeneous in $x,z$, which makes it possible to Laplace transform the perturbative equations in $x$ and to Fourier transform them in $z$.

Figure 2

Example of velocity profiles in the intermediate wake at the downstream station $x=x_0=20$. (a) Longitudinal velocity $U$ and (b) transversal velocity $V$. Continuous curves: analytical solutions ($\text{Re}=20$, 60, and 100) by Tordella and Belan (2003) [13], triangles: numerical results $\left(\text{Re}=34\right)$ by Berrone (2001) [14], and circles: laboratory data $\left(\text{Re}=34\right)$ by Kovasznay (1948) [15].

Figure 3

(Color online) Perturbation geometry scheme.

Figure 4

(Color online) The wave spatial evolution in the $x$ direction for $k={\alpha }_r=0.05$ and ${\alpha }_i=0.1,0.01,0.001$.

Figure 5

Effects of the spatial damping rate ${\alpha }_i$. (a) The amplification factor $G$ and (b) the temporal growth rate $r$ as function of time. Comparison between multiscale $O\left(1\right)$ (thick curves) and full problem (thin curves). $\text{Re}=50$, $k=0.03$, $\varphi =\pi /4$, $x_0=12$, asymmetric initial condition, and ${\alpha }_i=0.04,0.4$.

Figure 6

Transversal long waves. Effect of the symmetry of the perturbation. Comparison between multiscale $O\left(1\right)$ (thick curves) and full problem (thin curves). (a) The amplification factor $G$ and (b) the temporal growth rate $r$ as function of time. $\text{Re}=100$, $k=0.02$, $\varphi =\pi /2$, $x_0=10$, ${\alpha }_i=0.08$, symmetric and asymmetric initial conditions.

Figure 7

Effects of the polar wave number $k$ used as small parameter. (a) The amplification factor $G$ and (b) the temporal growth rate $r$ as function of time. Comparison between multiscale $O\left(1\right)$ (thick curves) and full problem (thin curves). $\text{Re}=100$, $\varphi =0$, $x_0=27$, ${\alpha }_i=0.2$, symmetric initial condition, $k=0.1,0.01,0.001$.

Figure 8

Transversal long waves. Effect of the spatial decay, ${\alpha }_i$. Comparison between multiscale $O\left(1\right)$ (thin curves) in the limit for ${\alpha }_i\to 0$ and full problem (thick curves) with ${\alpha }_i=0$. (a) The amplification factor $G$ and (b) the temporal growth rate $r$ as function of time. $\text{Re}=50$, $\varphi =\pi /2$, $x_0=10$, asymmetric initial condition, $k=0.04$, ${\alpha }_i=0.005,0.01,0.05$ [multiscale $O\left(1\right)$], ${\alpha }_i=0$ (full problem).

Figure 9

Temporal asymptotic values of (a) the temporal growth rate and (b) the angular frequency. Comparison between multiscale $O\left(1\right)$ (squares: symmetric inputs and dots: asymmetric inputs) and full linear problem (circles: symmetric inputs, triangles: asymmetric inputs). $k=0.01$, $\text{Re}=100$, $\varphi =\pi /4$, $x_0=10$, and $dr/dt<ϵ$ with $ϵ\sim {10}^{-4}$.