Representing rectangular jet dynamics through azimuthal Fourier modes

Rectangular propulsion nozzles offer thrust-vectoring and air-frame-integration advantages over their more commonly studied circular counterparts. However, they display many distinguishing features which violate assumptions, such as azimuthal homogeneity, typically used in prediction tools for circular jets. In the present paper, we examine the utility of an azimuthal Fourier decomposition for rectangular Mach 1.3 jets of aspect ratio (AR) 1, 4, and 8 using large-eddy simulations, with a circular jet of the same equivalent diameter for reference. The simulations manifest key features of rectangular jets, including higher spreading rates and shorter potential cores with increasing AR, axis switching (AR=4), and azimuthal variation in peak acoustic intensity (AR=8). We show that, after projection on a cylindrical frame, a sine-cosine ansatz for the azimuthal Fourier series affords a more convenient representation of nonaxisymmetric flow features than the commonly used complex exponential ansatz. Fluctuation magnitudes of the higher azimuthal modes show rapid reduction in amplitude, similar to those observed in circular jets, especially if an acoustic fluctuation field based on momentum potential theory is chosen instead of pressure fluctuations. The leading modes differ, however, from those of a circular jet in two important aspects, namely, the mechanisms represented by the sine and cosine coefficients of the first azimuthal mode and the rate of streamwise decay of all modes with increasing AR. These differences are traced to the near- and far-field rectangular jet asymmetry by examining azimuthal inhomogeneity, the implications of which are assessed with a generalized expression for acoustic intensity based on energies of leading modes. The significant simplicity of circular plumes is recovered as a special case of the analysis. Invocation of the twofold mirror symmetry property of rectangular jets eases the prediction procedure so that only two extra terms, representing mechanisms unique to rectangular jets, specifically preferential flapping in the minor axis direction and coupling of axisymmetric and second azimuthal modes, are sufficient to recover the advantages of azimuthal decomposition.

Figure 1

Characteristics of inflow and spatial discretization.

Figure 2

Isosurfaces of the $Q$ criterion ($Q=1$) colored by streamwise velocity.

Figure 3

Contours of mean streamwise velocity at several axial stations showing the spatial development of the jets.

Figure 4

Effect of the AR on the length of the potential core shown using streamwise mean velocity.

Figure 5

Variation of far-field OASPL with polar angle for jets of various AR.

Figure 6

Isosurfaces of three-dimensional snapshots of the pressure fluctuations (first row) from the LES compared with those of the acoustic (second row) and hydrodynamic fluctuations (third row) extracted from the simulated flow fields.

Figure 7

Polar plots depicting spatial distributions of the two leading azimuthally varying Fourier modes, with nozzle outline for orientation.

Figure 8

Contours of pressure perturbations ($p^{\prime }$, left column) and acoustic fluctuations ($\frac{\partial {\psi }_A^{\prime }}{\partial x}$, right column) of the circular jet on a 2D cross section along with their leading azimuthal Fourier modes.

Figure 9

The streamwise acoustic fluctuations ($\frac{\partial {\psi }_A^{\prime }}{\partial x}$) of the $\text{AR}=1$ jet on the symmetry plane and its leading azimuthal modes.

Figure 10

Contours of streamwise acoustic fluctuations ($\frac{\partial {\psi }_A^{\prime }}{\partial x}$) of the $\text{AR}=4$ (left column) and $\text{AR}=8$ (right column) jets shown on the major and minor axis planes along with the leading azimuthal Fourier modes.

Figure 11

Isosurfaces ($\frac{\partial {\psi }_A^{\prime }}{\partial x}=\pm 0.007$) showing the reconstructed 3D azimuthal mode shapes of an instantaneous snapshot of the acoustic fluctuations in the near field of the various AR jets.

Figure 12

A comparison of partial reconstructions using a few leading azimuthal modes to the original instantaneous snapshot of the acoustic fluctuations in the $\text{AR}=8$ rectangular jet. The reconstruction error for each of the partial reconstructions is also shown.

Figure 13

Comparison of the rates of convergence for the circular jet (a) and the $\text{AR}=4$ (b) and $\text{AR}=8$ (c) rectangular jets.

Figure 14

Modal energies of the three leading azimuthal modes of the acoustic fluctuations in the nonaxisymmetric jets. The dashed line in the figure represents the lipline of the circular jet ($y/D_e=0.5$) for scale.

Figure 15

Streamwise variation of the radially integrated modal energies of five leading azimuthal modes for the circular jet.

Figure 16

Streamwise variation of the radially integrated RMS (iRMS) of the five leading azimuthal modes of the acoustic (row 1) and pressure fluctuations (row 2) for the $\text{AR}=1$ (a, d), $\text{AR}=4$ (b, e), and $\text{AR}=8$ (c, f) jets.

Figure 17

Contours showing the leading $m=0$ azimuthal mode and spatial distributions of the dominant terms associated with azimuthal inhomogeneity and acoustic radiation asymmetry for all simulated rectangular jets.

Figure 18

Azimuthal distributions of the acoustic intensity on the FWH surface at the streamwise location of the maximal jet noise shown using polar plots for the circular and $\text{AR}=8$ rectangular jets. Reconstructions of the acoustic intensity using the cumulative sum [Eq. (27)] of the $m$ leading modal energies (indicated by the $\sum m$ value) are shown for both jets.

Figure 19

Comparison of the near-field acoustic intensity reconstruction without [Eq. (27)] and with [Eq. (29)] a consideration of the terms representing the inhomogeneity of the nonaxisymmetric jets.

Figure 20

Comparison of the relative importance of preferential flapping motions in rectangular jets along the direction of peak acoustic radiation.