Transition of a temporally developing three-dimensional turbulent boundary layer

A temporally developing three-dimensional turbulent boundary layer is investigated using direct numerical simulation. The flow is initiated by subjecting a statistically stationary turbulent channel flow to a constant transverse pressure gradient while maintaining the streamwise pressure gradient unchanged. It is shown that this nonequilibrium three-dimensional boundary layer can be described as a turbulent-turbulent transition that is characterized by the development of a laminar boundary layer in an initial turbulent environment followed by a transition to turbulence. Both transient energy growth and crossflow instability may influence the transition, though the former is likely to have a stronger effect when the initial Reynolds number is lower and the transverse pressure gradient is stronger. The transient developments of both the mean flow and turbulence are understood by relating them to the process of transition. The rotation of streaks and the damping effect of the spanwise boundary layer work together to suppress the streamwise and wall-normal turbulence. This effect is stronger than the energy growth in the spanwise direction, causing the overall turbulent kinetic energy and the structure parameter to decrease, explaining the observations made of such flows.

Figure 1

Schematic diagrams of (a) the computational setup with sketches of the near-wall velocities (not to scale) and (b) the streamwise ($U$) and spanwise ($W$) velocities and the principal and crossflow directions in a three-dimensional boundary layer. Here $U_c$ and $W_c$ are centerline velocities. The flow is initially in the streamwise direction only, driven by $\partial P/\partial x$. At the initial time, a constant spanwise pressure $\partial P/\partial z$ is applied while maintaining $\partial P/\partial x$, which causes a three-dimensional boundary layer to form and develop over time. Hence the ratio between the two pressure gradients is a constant $\mathrm{\Pi }$, i.e., $(\partial P/\partial z)/(\partial P/\partial x)=\mathrm{\Pi }$.

Figure 2

Overall flow development of all flow cases studied: (a) the streamwise bulk velocity, (b) the spanwise bulk velocity, and (c) the angle in degrees of the wall shear ${\alpha }_w$ (red lines) and that of the centerline velocity ${\alpha }_c$ to the streamwise flow (black lines).

Figure 3

Spanwise fluctuating velocity $w^{\prime }$ in an $xz$ plane with $y^{+}=3.71$ in case R180P60A. The range of the color bar was selected to highlight the new streaks created during the transition.

Figure 4

Variation of the (a) streamwise and (b) spanwise velocities along a horizontal line along the channel at $y^{+}=8$ (case R180P60A).

Figure 5

Formation of a single turbulent spot illustrated with isosurfaces: blue (dark gray), $u^{\prime }=-0.3$; green (light gray), $u^{\prime }=0.3$; and red (deep dark gray), ${\lambda }_2=-5$ (case R180P60A). The time instants from left to right are $t^{+}=220,242,255,262,266,279$.

Figure 6

Spanwise velocity profile development in R180P60 and comparison with the solution of the extended Stokes first problem for an equivalent laminar acceleration. Here $W_c$ is the centerline velocity.

Figure 7

Friction coefficient in the (a)–(c) streamwise and (d) spanwise directions. The triangle markers show the onset of the transition detected using minimum $C_{fx}$. Also shown in (d) is the extended Stokes flow solution for R180P60 as an example and solutions for other cases are only slightly different due to the small differences in their acceleration rates.

Figure 8

(a) Displacement and (b) momentum spanwise boundary layer thickness. The red lines are trend lines for the pretransition stage in the form of ${\delta }^{+},{\theta }^{+}\propto {\left(t^{+}\right)}^{1/2}$. The proportion factors are (a) 0.75 and (b) 0.36, chosen to fit the data.

Figure 9

Boundary layer shape factor. The onset of transition detected using minimum $C_{fx}$ is shown triangle markers.

Figure 10

Critical Reynolds number versus freestream turbulence intensity. The dashed line is a best-fit curve of the data.

Figure 11

Development of the velocity profiles (a) along the centerline velocity direction $U_P$ (principal flow) and (b) perpendicular to it $W_P$ (crossflow) in case R180P60. Dashed lines show the initial flow and solid lines profiles at $t$. (c) Relative crossflow strength ${\left(W_P\right)}_{\text{max}}/{\left(U_P\right)}_{\text{max}}$ in all cases.

Figure 12

Directional velocity profiles at several time instants in cases R180P60 and R550P60. These directional velocities are on a plane at an angle $\alpha$ to the $x$ axis, where $\alpha =0$ (red lines), $\alpha =0.5\pi$ (blue lines), $\alpha =\pi$ (green lines), $\alpha =0.1\pi ,0.2\pi ,0.3\pi ,0.4\pi$ (black solid lines), and $\alpha =0.6\pi ,0.7\pi ,0.8\pi ,0.9\pi$ (black dashed lines).

Figure 13

Fluctuating velocity vectors $w^{\prime }$ and $v^{\prime }$ in a $zy$ plane at $t^{+}=100$ in case R180P60A with corotating vortices identified.

Figure 14

Root mean square of the fluctuating velocities (a) $u_{\text{rms}}^{\prime }$, (b) $v_{\text{rms}}^{\prime }$, and (c) $w_{\text{rms}}^{\prime }$ in case R180P60. The black solid lines are profiles at various $t^{+}$ and the dashed lines are the initial profiles at $t^{+}=0$ for comparison. The blue and red lines show the locations of the peak crossflow and the displacement boundary layer thickness, respectively.

Figure 15

Turbulent shear stresses (a) $-\overline{u^{\prime }{v}^{\prime }}$ and (b) $-\overline{w^{\prime }{v}^{\prime }}$ and (c) structure parameter $a_1$ in case R180P60. The black solid lines are profiles at various $t^{+}$ and the dashed lines are the initial profiles at $t^{+}=0$ for comparison.

Figure 16

Time development of (a) the maximum streamwise fluctuations and (b) the radial integral of the turbulence kinetic energy.

Figure 17

Energy growth. (a) Spanwise energy growth $\mathrm{\Delta }\overline{w^{\prime 2}}$ normalized by the developing bulk velocity squared $W_b^2$. Also shown is the energy growth in the form of velocity normalized by the initial friction velocity and corrected by the acceleration parameter ${\left(\mathrm{\Delta }\overline{w^{\prime 2}}\right)}^{1/2}/u_{\tau 0}a$, where $a={(d{W}_b/dt)}_0(1/U_{b0})(\nu /u_{\tau 0}^2)$ [81], as a function of (b) time $t^{+}$ and (c) ${\text{Re}}_{\tau \delta }=u_{\tau 0}\delta /\nu$.

Figure 18

Wall-normal location of the maximum growth of the spanwise fluctuating velocity $\mathrm{\Delta }\overline{w^{\prime 2}}$ relative to the displacement boundary thickness.