Linear and nonlinear mechanisms within a forced plane wall jet

The plane wall jet (PWJ) was used as a model flow field to study the linear and nonlinear mechanisms within complex wall-bounded flows. Large-scale, large-amplitude forcing was used to introduce a “tracer” flow scale into the unforced base flow and the resulting embedded mechanisms were studied. The PWJ had a nominal Reynolds number ${\text{Re}}_j=b{V}_j/\nu =5965$ and was forced acoustically at a Strouhal number $\text{St}=f_{f}b/V_j=3.4\times {10}^{-3}$ and the resulting flow field was characterized at various downstream locations ($x/b\le 162$) using hot-wire anemometry. Here, $b$ is the PWJ exit slot width, $V_j$ is the nominal PWJ exit centerline velocity, $\nu$ is the kinematic viscosity, and $f_f$ is the forcing frequency. The forcing increased the outer length scale $\delta$ (increased spreading rate of the PWJ) along with a decrease in the outer velocity scale $U_m$ and an increase in the local Reynolds number. The self-similarity of the mean streamwise velocity profile of the forced and unforced flow, respectively, was unaltered by the forcing, but the momentum in the wall region and the friction velocity were reduced along with a large increase in the streamwise turbulence in the wall region. The linear response of the PWJ to the forcing resulted in flow structures that have strong similarities to the naturally occurring structures of the unforced flow, both in the inner boundary layer and in the outer free-shear layer. A comparison of the energy spectra of the forced and unforced flow showed that the recipient scales, of the excess energy from forcing, were primarily the large scales in the wall region, with the specific scale of these recipient structures being that of the outer free-shear structures. At the upstream locations this energy transfer was primarily in the form of a forward cascade, while at downstream locations the transfer was in the manner of an inverse cascade. The nonlinear interaction between the large scales in the flow and finer scales was also found to be enhanced by the forcing. The spectra also suggested that the forcing isolated the energy transfer mechanism within the flow where the input energy was supplied at the forcing frequency. These observations led to the interpretation that the natural energy transfer mechanism within a PWJ is to transfer energy from the boundary layer coherent structures to the larger outer free-shear layer structures. The implications of these observations to modeling such complex wall-bounded flow fields, particularly considering the on-going work on the linear dynamics of the Navier-Stokes equation, were also discussed.

Figure 1

A schematic highlighting the salient features of a PWJ velocity profile.

Figure 2

A schematic highlighting the key features of the experimental facility and setup (not to scale).

Figure 3

Instantaneous velocity of the forced flow at the PWJ exit ($x/b=1$) centerline (top). The corresponding input signal to drive the speaker, in the settling chamber, is also shown (bottom).

Figure 4

(a) Comparison of the streamwise mean velocity profiles nondimensionalized using outer scales at $x/b=75$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$), 110 ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$), and 162 ($\mathbf{———}$) respectively. The shaded region highlights the nominal logarithmic region of the unforced flow at $x/b=110$. (b) Outer length scale $\delta$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{⁕}\mathbf{·}\mathbf{·}\mathbf{·}$) with values represented on the right $y$ axis and velocity scale $U_m$ ($\sout{○}$) with values represented on the left $y$ axis, as a function of streamwise distance. Black lines and symbols are used to represent unforced quantities whereas red lines and symbols are used to show forced quantities.

Figure 5

Development of the streamwise mean velocity ($\mathbf{———}$) and turbulence intensity ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) profiles of unforced and forced flow. Also shown are the respective developments of the shear-layer thickness $\delta$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$). Black lines are used to represent unforced quantities whereas red lines are used to show forced quantities. Figures 20 and 21 in Appendix pp1 show zoomed-in profiles of the mean streamwise velocity and turbulence intensity, respectively.

Figure 6

(a) Streamwise turbulence intensity (TI) profile at $x/b=137$. The vertical lines indicate wall-normal locations $z^{+}\approx 15$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$), $z^{+}\approx 60$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$), $z\approx z_m$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$), and $z\approx 0.6{\delta }^0$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$), respectively. The total TI ($\mathbf{———}$) along with its large-scale component ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) and small-scale component ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) are shown. The filled symbols mark the inner peaks of the total TI ($⚫$), large-scale TI ($\blacksquare$), the small-scale TI ($♦$), respectively. The unfilled symbols mark the outer peaks of the total TI ($○$), large-scale TI ($\square$), and small-scale TI ($\diamond$), respectively. (b)–(d) The variation of the inner and outer peaks as a function of streamwise distance corresponding to (b) the total TI, (c) the large-scale TI, and (d) the small-scale TI peaks, respectively. While comparing forced and unforced quantities black lines and symbols are used to represent the unforced quantities whereas red lines and symbols are used to show forced quantities.

Figure 7

(a) Variation of the skin friction coefficient $C_{f_j}=2U_{\tau }^2/V_j^2$ ($\sout{○}$) as a function of streamwise distance. While comparing forced and unforced flows black lines and symbols are used to represent the unforced quantities whereas red lines and symbols are used to show forced quantities. (b) Variation of the PWJ momentum as a function of streamwise distance decomposed into its contributions from the wall ($z<z_m$) and the outer ($z\ge z_m$) regions. These are nondimensionalized with respect to the total unforced momentum $M_j^0$ as per Eq. (1). The outline around the bars is used to differentiate between the unforced (black) and the forced (red) flows.

Figure 8

Spectrogram at $x/b=50$ for the (a) unforced and (b) forced flows as well as the (c) difference between the two spectra. The vertical lines show the wall-normal locations $z^{+}\approx 15$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) and 60 ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and $z=0.6{\delta }^0$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$). (d)–(f) Profiles of the difference in energy spectra at the three highlighted wall-normal locations. Shown also for reference is the forcing wavelength $V_j/f_{f}b\approx 295$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$).

Figure 9

Variation of the phase-locked velocity $U_f$ [see Eq. (2)] over two forcing cycles shown in logarithmic (top) and linear (bottom) wall-normal coordinates. The streamwise distance increases going from left to right. The color contours are the magnitude of $U_f$ as indicated by the color bar. The contour level corresponding to $U_f=0$ is also shown. The horizontal lines show wall-normal locations $z^{+}\approx 15$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) and 60 ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and $z=0.6{\delta }^0$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$), respectively.

Figure 10

Variation of the phase-locked velocity $U_f$ [see Eq. (2)] over two forcing cycles in linear wall-normal coordinates at $x/b=110$. The color levels represent the magnitude of $U_f$ as indicated by the color bar. The contour level corresponding to $U_f=0$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) is also shown. The horizontal lines show the wall-normal locations $z^{+}\approx$ 60 ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and $z=0.6{\delta }^0$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$), respectively. The arrows highlight the three distinct flow features discussed in the text.

Figure 11

Spectrograms of the unforced (top) and the forced (bottom) flows. The streamwise development is shown from left to right. The vertical lines show the wall-normal locations $z^{+}\approx 15$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) and 60 ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and $z\approx 0.6{\delta }^0$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$), respectively. The horizontal line shows the forcing wavelength $V_j/f_{f}b$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$).

Figure 12

Comparison of the streamwise, one-dimensional, premultiplied energy spectra $f{\varphi }_{uu}/V_j^2$ ($\mathbf{———}$) at a near-wall location $z^{+}=15$ (top), a logarithmic location $z^{+}=60$ (middle), and an outer location $z=0.6\delta$ (bottom) (corresponds to the locations highlighted in Fig. 11). At the wall layer locations $z^{+}\approx 15$ and 60, the outer unforced spectrum from $z\approx 0.6{\delta }^0$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) is also plotted for a comparison. The black color represents unforced quantities while the red color represents forced quantities. The vertical lines show the cutoff wavelength ${\lambda }_x=2{\delta }^0$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and the forcing wavelength $V_j/f_{f}b$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$), respectively. Also shown are the peaks of the recipient scales ${\lambda }_{x-\text{peak}}^r$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) in the wall layer when $x/b\ge 110$.

Figure 13

(a) Spectrogram of the unforced flow at $x/b=137$. The circled regions indicate wavelength extents of the naturally energetic large-scale structures in the inner (${\lambda }_{xi}^n$) and the outer (${\lambda }_{xo}^n$) regions, respectively. The vertical lines show the wall-normal locations $z^{+}\approx 60$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$), $z\approx z_m$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$), and $z\approx 0.6{\delta }^0$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$), respectively. (b) Premultiplied energy spectrum for the wavelength $V_j/fb\approx 600$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) highlighted in figure (a). The shaded regions illustrate the nominal wall-normal extents of ${\lambda }_{xi}^n$ (blue) and ${\lambda }_{xo}^n$ (red), respectively. The vertical lines show the wall-normal locations of the corresponding inner ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and outer ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) peak energy values, respectively. The vertical line at $z\approx z_m$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) nominally differentiates the wall region from the outer region. (c) Comparison of the energy peaks in the wall region at $z^{+}\approx 60$ (blue) and the outer region at $z\approx 0.6{\delta }^0$ (red). The vertical lines indicate the peak values for inner ${\lambda }_{xi-\text{peak}}^n$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and outer ${\lambda }_{xo-\text{peak}}^n$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$), respectively.

Figure 14

Downstream development of the difference in the energy spectra between the forced and unforced flow [$f\left({\varphi }_{uu}^{*}-{\varphi }_{uu}^0\right)/V_j^2$]. The vertical lines show wall-normal locations $z^{+}\approx 15$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) and 60 ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and $z\approx 0.6{\delta }^0$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$), respectively. The horizontal lines show the cutoff wavelength ${\lambda }_x=2{\delta }^0$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and the forcing wavelength $V_j/f_{f}b$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$), respectively. The numbers on the top left corner of each figure show the forcing wavelength nondimensionalized with respect to the outer variables at the respective streamwise location.

Figure 15

Difference of premultiplied energy spectra between a streamwise location and the immediately preceding downstream location corresponding to both the unforced (top) and forced (bottom) flow. The vertical lines show the wall-normal locations $z^{+}\approx 15$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) and 60 ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and $z\approx 0.6{\delta }^0$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$), respectively, at the upstream location. The horizontal lines show the forcing wavelength $V_j/f_{f}b$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) and the cutoff frequency ${\lambda }_x=2{\delta }^0$ ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$). Here, the axes are nondimensionalized by the respective local shear-layer thickness $\delta$.

Figure 16

Line plots of the energy difference shown in Fig. 15 at three wall-normal locations $z^{+}\approx 15,60$, and $0.6{\delta }^0$. Black lines ($\mathbf{———}$) are used to represent the unforced flow, whereas the red lines ($\mathbf{———}$) represent the forced flow. Here, the axes are nondimensionalized by the respective to the local shear-layer thickness $\delta$.

Figure 17

Variation of the amplitude modulation coefficient $R_{\text{AM}}$ ($\mathbf{———}$) as the flow develops downstream. The black color represents the unforced flow and the red color represents the forced flow. The horizontal lines highlight the wall-normal locations $z^{+}\approx 15$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) and 60 ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and $z\approx 0.6{\delta }^0$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$), respectively.

Figure 18

The profiles of the amplitude modulation coefficient ${\left(R_{\text{AM}}\right)}_{\text{Band}}$ at $x/b=137$ showing only the contribution at the forcing frequency $f_f$. The black color represents the unforced flow and the red color represents the forced flow. The vertical lines show the wall-normal locations $z^{+}\approx 15$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) and 60 ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and $z\approx 0.6{\delta }^0$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$), respectively.

Figure 19

A schematic highlighting the key mechanisms in the forced plane wall jet. Shown are the linear response, the direction of energy transfer, and the recipient scales.

Figure 20

Zoomed-in mean streamwise velocity profiles of the unforced ($\mathbf{———}$) and forced ($\mathbf{———}$) flow shown in Fig. 5.

Figure 21

Zoomed-in streamwise turbulence intensity profiles of the unforced ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) and forced ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) flow shown in Fig. 5.

Figure 22

Variation of the phase-locked velocity $U_f$ [see Eq. (2)] over two forcing cycles shown in logarithmic (top) and linear (bottom) wall-normal coordinates. The streamwise distance increases going from left to right. The color contours are the magnitude of $U_f$ as indicated by the color bar. The contour level corresponding to $U_f=0$ is also shown. The horizontal lines show wall-normal locations $z^{+}\approx 15$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) and 60 ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and $z=0.6{\delta }^0$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$), respectively. This figure is identical to Fig. 9, except that the velocity field was filtered to isolate the response associated with the fundamental forcing frequency, filtering out the higher harmonics. $\mathrm{\Psi }$ shows the relative phase between the near-wall response feature and the middle response feature.

Figure 23

Contour maps of the streamwise, one-dimensional, premultiplied energy spectra $k_x{\varphi }_{uu}/V_j^2$ for the unforced (top) and the forced (bottom) flow conditions. Here, the local mean streamwise velocity is used as the convective velocity in Taylor's hypothesis to convert a flow frequency into a wavelength. The streamwise development is shown from left to right. The vertical lines show the wall-normal locations $z^{+}\approx 15$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) and 60 ($\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}$) and $z\approx 0.6{\delta }^0$ ($\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}$), respectively. The forcing wavelength $\overline{U}/f_{f}b$ ($\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}\mathbf{·}\mathbf{-}$) is highlighted for reference.