Effect of mean flow on mutual radiation resistances of a rectangular plate

. Very few investigations have considered the effects of fluid convection on the mutual-radiation resistance, which could be important in stationary fluids. In this work, we quantified the effects of a convected fluid on the radiation resistances including self- and mutual radiation resistances. We derived the non-dimensional modal impedance in the presence of mean flow based on an expansion in the panel mode of a simply supported plate. Then we compared the self-radiation resistances with the previous reports and analyzed the effects of flow on mutual radiation resistances. The results show that the radiation efficiency increases with the flow velocity, which compare very well with the previous reports. The flow introduces a strong coupling between modes that weakly coupled in no-flow case and with the increase of flow velocity. This coupling effect increases until Mach number (M) reaches 0.5 and then decreases. For the modes with the same parity indices, the flow shifts coupling lobes toward lower frequency with the increase of flow velocity due to the shift of critical frequency of higher mode radiation resistance, and the degree of coupling change is consistent with the change of critical frequency of the higher mode self-radiation resistance. effects of flow on the mutual radiation resistances.


Introduction
Radiation resistance of a plate is important in vibroacoustic. It determines the plate velocity, and further determines the radiated acoustic power. Radiation from vibrating structures (particularly for simply geometries) into a static fluid is well understood, and the radiation efficiencies [1][2][3][4][5] as well as the cross-modal coupling [6][7][8][9][10][11][12] are well considered. As is reported, the magnitude of the radiation efficiency depends on whether the mode is supersonic or subsonic, and the modal coupling is not negligible both at low frequency and under off-resonant excitation.
However, the sound radiation behaviors of convective fluid-loaded plates could be very different. Maestrello et al. [13] reported one of the earliest acoustic response of a panel in convected fluid and derived the radiation sound pressure in the form of extended Green function. Dowell [14] summarized earlier investigations into the aeroelastic behavior of plates and shells, and demonstrated the significant effects of particular high subsonic and supersonic flow on plate dynamics. Sgard et al. [15] presented that the flow could increase the radiation acoustic power, which is more notable in water than in air, and the coupling with different modal indices parity can become important in the presence of mean flow. Wu et al. [16] also showed that the radiated acoustic pressure are directly related to turbulent boundary layer.
All of these previous works mainly focused on the response of the panel excited by flow and radiation characters, while the radiation impedance of a convected fluid-loaded plates has not been well discussed. Chang et al. [17] derived the modal radiation impedance, including self-and cross-modal coupling impedance of a simply supported rectangular panel in the presence of a uniform subsonic flow. They claimed that a modal impedance will be zero only if co-directed wavenumber indices of the panel modes in the direction perpendicular to the flow have different parity. Graham [18] and Kou et al. [19] have discussed the averaged self-radiation resistance (the so-called radiation efficiency) of a rectangular plate subjected to turbulent boundary layer fluctuations. They summarized that mean flow results in a significant increase in the modal radiation efficiency. Frampton [20] investigated the influence of mean flow on the frequency-dependent radiation efficiency and obtained the same conclusion. However, the mutual radiation resistances that could be meaningful for no-flow condition [6,8,9] have got little attention, due to the belief that the mutual-radiation impedance (coupling terms) can be ignored [18] or the misgivings that the modes of the structure are no longer orthogonal where the fluid is convected [20].
In this work, we focused on the effects of a convected fluid on the mutual radiation resistances based on modal expansion method. We derived the nondimensional modal impedances for subsonic flow case based on an expansion in the panel mode of a simply supported panel. Then we compared the self-radiation resistances with previous reports and discussed the effects of flow on the mutual radiation resistances.

Derivation of the impedance
Consider a finite, flexible, rectangular plate, which is embedded in an infinite baffle as depicted in Fig. 1. The plate is subjected to a semi-infinite idealization uniform steady fluid, flowing parallel to the x axis on one side and a vacuum on the other. The fluid is inviscid, with mean density and sound speed . When flow velocity U is not too close to the speed of sound c, the sound pressure satisfies the convected wave equation, given by As aforementioned, for convenience, the mean flow is assumed to move along the x-direction. As a sequence, the material derivative can be expressed as   which measures rate of change following the motion of a fluid particle. The sound pressure is assumed to satisfy the boundary condition in the plane of the panel where ( , ; ) w x y t is the normal displacement of the panel.
Applying the Fourier transform on the (x, y) coordinates to the convected wave equation, and assuming that the system is undergoing harmonic motion, Eq.(1) becomes which is similar to the classic Rayleigh's integral, and here we will define For a simply supported plate, the relevant panel mode shape is Expanding the sound pressure and panel vibration displacement in the panel modal, the sound pressure can be expanded as where the non-dimensional modal impedance often referred as the non-dimensional modal radiation resistance represents the radiation damping, and the non-dimensional modal radiation reactance mnpq  represents added mass.

Derivation of the Green function
Here, we focus on subsonic mean flow (M<1). Applying the following changes in variables Expanding the square terms, Eq.(11) can be expressed as

Transformation of the resistance
We can rewrite the non-dimensional modal impedance as

Numerical results and discussions
The example of a thin elastic plate with isotropic properties will serve to illustrate the effects of flow on the radiation resistances. The self-and mutual radiation resistances for some lower order modes are plotted. The acoustic wave number have all been normalized by the structural wave number for a simply supported mode shape, where 22 = ( / ) ( / ) mn k m a n b   The structural parameters used here are listed in table 1.    For the sake of accuracy, the self-radiation resistance calculations are established through comparison with results previously published by Graham [18] and Frampton [20]. The effects of mean flow on the radiation efficiencies for several modes are demonstrated in Fig.2-4. It can be simply pointed out that good agreements are found for the subsonic flow with those of Frampton. The effects of flow are negligible for small Mach numbers (M<2). As for the (1,1) plate mode shown in Fig.2, the notable change is that the critical frequency shifts toward low frequency with the increase of flow velocity. The resultant effect is a significant increase in the total radiation efficiency. Compared Fig.3 with Fig.4, the asymmetry between (1,3) and (3,1) plate modes in the radiation resistance also have been observed. The increase in radiation resistance for (3,1) mode is more prominent than that for (1,3) mode. Note that, there exists minor difference in (3,1) modal radiation resistance at critical frequency with Frampton's results. The increase of the peak at critical frequency is more significant shown in Fig.3, which may be more accurate because of no approximation in this paper.

Mutual-radiation resistance
The effects of flow on the mutual-radiation resistances are plotted in Fig.5-9 for some possible cross-couplings of the modes. It should be noted that all the curves have been normalized with the self-radiation resistance of the lowest mode in that group. In the no-flow case, it is well known that the cross-modal coupling occurs only between a pair of modes having the same parity indices. While in the presence of low, a strong coupling appears between different parity indices of panel modes in the flow direction as shown in Fig.5-7. As the frequency extends to near the coincidence frequency, the coupling disappears. As the increase of flow velocity, the effect of coupling increases in going from M=0.2 to M=0.5 and then decreases. That is because the supersonic wave number region gradually envelops the spectrum peak and then deviates from it. Compared Fig.5 and Fig.6, the effects of flow on the (1,1) (2,1)  and (2,1) (1,1)  mode mutual radiation resistance is antisymmetric. This coupling between higher modes is stronger but decreases rapidly, for example, the coupling between (3, 3) and (2, 3) modes in Fig.7.    In stationary fluids, the mutual-radiation resistance is quite significant in a quite wide frequency that well extends to the coincidence frequency of the higher mode. After the coincidence frequency, the mutual-radiation resistance oscillatory decays with frequency [9]. In flow case, the coupling lobes move toward lower frequency with the increase of flow velocity due to the shift in coincidences frequency. The influence of flow on the mutual radiation resistance depends on the effect of flow on the higher mode self-radiation resistance. In Fig.8, for the mutual radiation resistance of (1,1) (1,3)  mode, the coupling lobes exhibit a slightly shift to lower frequency and the mutual-radiation resistance oscillatory decays slightly with the increase of flow. While, in Fig.9, it is easy to see that the shift of coupling lobes in the (1,1) (3,1)  modal radiation resistance is more signi-ficant than that on the mutual-radiation resistance of the (1,1) (1,3)  mode. The oscillatory peaks increase remarkably due to the notable effect of flow on (1,3) selfradiation resistance.

Conclusions
Based on the modal expansion approach, the effect of subsonic flow on the self-and mutual-radiation resistance is investigated. The flow causes a decrease of the critical frequency when the modal becomes an efficient radiator, which results in a notable increase in the mid-wave number domain, and further increases the self-modal radiation resistance, which is in consistent with the previous reports. The flow introduces a strong coupling even between modes that were weakly coupled in the no-flow case. This coupling increases with Mach number(M) until M reaches 0.5, and then decreases with the increase of flow velocity. For the modes with the same parity indices, the flow shifts coupling lobes toward lower frequency with the increase of flow velocity due to the shift of critical frequency of higher order modal radiation resistance, and the degree of coupling change is consistent with the change of critical frequency of the higher mode self-radiation resistance.