Resolvent-based modeling of coherent wave packets in a turbulent jet

Coherent turbulent wave-packet structures in a jet at Reynolds number $460000$ and Mach number 0.4 are extracted from experimental measurements and are modeled as linear fluctuations around the mean flow. The linear model is based on harmonic optimal forcing structures and their associated flow response at individual Strouhal numbers, obtained from analysis of the global linear resolvent operator. These forcing-response wave packets (“resolvent modes”) are first discussed with regard to relevant physical mechanisms that provide energy gain of flow perturbations in the jet. Modal shear instability and the nonmodal Orr mechanism are identified as dominant elements, cleanly separated between the optimal and suboptimal forcing-response pairs. A theoretical development in the framework of spectral covariance dynamics then explicates the link between linear harmonic forcing-response structures and the cross-spectral density (CSD) of stochastic turbulent fluctuations. A low-rank model of the CSD at given Strouhal number is formulated from a truncated set of linear resolvent modes. Corresponding experimental CSD matrices are constructed from extensive two-point velocity measurements. Their eigenmodes (spectral proper orthogonal or SPOD modes) represent coherent wave-packet structures, and these are compared to their counterparts obtained from the linear model. Close agreement is demonstrated in the range of “preferred mode” Strouhal numbers, around a value of 0.4, between the leading coherent wave-packet structures as educed from the experiment and from the linear resolvent-based model.

Figure 1

Axial velocity of the mean flow, as used throughout this study [33]. The distribution has been modeled such as to closely reproduce the experimental measurements [13]. The pipe wall is represented as a white line, and only a portion of the numerical domain is shown. The rasterization of the color plot corresponds to the standard numerical grid resolution (Sec. 3d).

Figure 2

Comparison of axial velocity profiles along $r$, at various $x$ locations, between the numerically modeled mean flow (red lines) and the experimental measurements [13] (circles).

Figure 3

Spacing of mesh points (a) in the radial direction and (b) in the axial direction. The mesh is orthogonal.

Figure 4

The five leading resolvent gain values ($•{\sigma }_1$, $\blacksquare {\sigma }_2$, $♦{\sigma }_3$, $▴{\sigma }_4$, $\circ {\sigma }_5$) as functions of Strouhal number.

Figure 5

Optimal forcing modes at various Strouhal numbers, associated with the response modes in Fig. 6. The real part of axial velocity forcing is represented. [(a), (c), (e), (g)] Optimal forcing, plotted with aspect ratio 2. [(b), (d), (f), (h)] Closeup of the pipe boundary layer, where the forcing is localized, at the same St values as in the left column. The rasterization corresponds to the numerical mesh; each field is normalized with respect to its maximum amplitude.

Figure 6

Optimal response modes at various Strouhal numbers, associated with the forcing modes in Fig. 5. The real part of axial velocity perturbations is represented, with aspect ratio 1. The pipe wall is shown as a black line. Each field is normalized with respect to its maximum amplitude.

Figure 7

Suboptimal forcing and response modes for $\text{St}=0.7$. The real part of axial velocity perturbations is represented. [(a), (c), (e), (g)] Forcing modes and [(b), (d), (f), (h)] associated response modes. The pipe wall is shown as a black line. Each field is normalized with respect to its maximum amplitude. The aspect ratio is 2, and strong magnification is required in order to visualize the fine-scale radial variations.

Figure 8

Growth rate $-k_i$ and real phase velocity $c_r$ of the local shear instability mode, and their downstream variations in the jet mean flow. The phase velocity is scaled with the local mean centerline velocity.

Figure 9

Local shear instability contribution to the optimal response mode at $\text{St}=0.7$. (a) Optimal response mode from global resolvent analysis; (b) its reconstruction from projection onto the $k^{+}$ local shear instability mode. Axial velocity fluctuations are shown over the interval in $x$ where the local mode can be identified numerically. Both fields are normalized with respect to their amplitude maxima, but the color scale is saturated in order to make small-amplitude fluctuations visible. The zero contour is traced in black.

Figure 10

Forcing mode structures in a parallel incompressible jet with Gaussian base flow profile. Kinetic energy at $x=10$ is maximized. Streamwise velocity forcing is shown in linear color scale. Black lines: contours that are convected into vertical lines at $x=10$ after $\mathrm{\Delta }t=(2.5,5,7.5,10)$. (a) Forcing mode 1; (b) forcing mode 2.

Figure 11

(a) Sketch of the experimental setup, viewed from the top. The green lines represent the laser light sheets $S_1$ and $S_2$, placed at $x_1$ and $x_2$, respectively; these planes are shifted during the experiment in the range $x=[1,8]$ (see Ref. [31]). (b) Front view of the experiment during the PIV acquisition.

Figure 12

Correlation coefficients $\alpha$ [Eq. (15)], for a quantification of the statistical convergence of experimental SPOD modes. Two subsets of data are taken from the experimental acquisitions, and for each of these the SPOD modes are computed and compared: ($•$) ${\alpha }_{1,k}$ and ($\circ$) ${\alpha }_{2,k}$. Satisfactory convergence is observed at least for mode 1 (throughout) and for mode 2 (at $St<0.8$). (a) St = 0.2, (b) St = 0.4, (c) St = 0.6, and (d) St = 0.8.

Figure 13

Modulus of the first SPOD mode at $\text{St}=0.2,0.4,0.6,$ and 0.7, as obtained from the experimental data (left column) and from the resolvent-based model (right column). The resolution of the color plots corresponds to the spatial grid where the CSD is defined, without interpolation.

Figure 14

Modulus of the second SPOD mode at $\text{St}=0.2,0.4,0.6,$ and 0.7, as obtained from the experimental data (left column) and from the resolvent-based model (right column). The resolution of the color plots corresponds to the spatial grid where the CSD is defined, without interpolation.

Figure 15

The leading five CSD eigenvalue branches as functions of Strouhal number (a) from the experiment and (b) from the resolvent-based linear model.

Figure 16

Interpolated SPOD wave packets at $\text{St}=0.4$ (a) from the experiment and (b) from the linear model based on five resolvent modes. Amplitude and phase are interpolated between the available data points, and the resulting real part is represented. (c) The corresponding optimal response mode alone.

Figure 17

Comparison between experimental (red) and model (black) SPOD wave packets at various Strouhal numbers. Amplitude and phase variations on the centerline are interpolated between the available data points, indicated by markers. Both the amplitude envelopes and the oscillating real parts are shown. (a) St = 0.2, (b) St = 0.3, (c) St = 0.4, (d) St = 0.5, (e) St = 0.6, and (f) St = 0.7.

Figure 18

Power-spectral density (PSD) along the jet centreline as a function of $x$ at different Strouhal numbers. ($\circ$) Hot-wire measurements [13]; ($\text{–}$) PSE model [13]; ($͟$) optimal response mode (present calculations). (a) St = 0.2, (b) St = 0.3, (c) St = 0.4, (d) St = 0.5, (e) St = 0.6, and (f) St = 0.7.