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Motivated by the problem of jet-flap interaction noise, we study the tonal dynamics that occur when a sharp edge is placed in the hydrodynamic nearfield of an isothermal turbulent jet. We perform hydrodynamic and acoustic pressure measurements in order to characterise the tones as a function of Mach number and streamwise edge position. The distribution of spectral peaks observed, as a function of Mach number, cannot be explained using the usual edge-tone scenario, in which resonance is underpinned by coupling between downstream-travelling Kelvin-Helmholtz wavepackets and upstream-travelling sound waves. We show, rather, that the strongest tones are due to coupling between the former and upstream-travelling jet modes recently studied by Towne et al. (2017) and Schmidt et al. (2017). We also study the band-limited nature of the resonance, showing a high-frequency cut-off to be due to the frequency dependence of the upstream-travelling waves. At high Mach number these become evanescent above a certain frequency, whereas at low Mach number they become progressively trapped with increasing frequency, a consequence of which is their not being reflected in the nozzle plane. Additionally, a weaker, low-frequency, forced-resonance regime is identified that involves the same upstream travelling jet modes but that couple, in this instance, with downstream-travelling sound waves. It is suggested that the existence of two resonance regimes may be due to the non-modal nature of wavepacket dynamics at low-frequency.
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Direct numerical simulations are carried out to investigate the flow features responsible for secondary tones arising in trailing-edge noise at moderate Reynolds numbers. Simulations are performed for a NACA 0012 airfoil at freestream Mach numbers 0.1, 0.2 and 0.3 for angle of incidence 0 deg. and for Mach number 0.3 at 3 deg. angle of incidence. The Reynolds number based on the airfoil chord is fixed at $Re_c=10^5$. Flow configurations are investigated where noise generation arises from the scattering of boundary layer instabilities at the trailing edge. Results show that noise emission has a main tone with equidistant secondary tones, as discussed in literature. An interesting feature of the present flows at zero incidence is shown; despite the geometric symmetry, the flows become non-symmetric with a separation bubble only on one side of the airfoil. A separation bubble is also observed for the non-zero incidence flow. For both angles of incidence analyzed, it is shown that low-frequency motion of the separation bubbles induce a frequency modulation of the flow instabilities developed along the airfoil boundary layer. When the airfoil is at 0 deg. angle of attack an intense amplitude modulation is also observed in the flow quantities, resulting in a complex vortex interaction mechanism at the trailing edge. Both amplitude and frequency modulations directly affect the velocity and pressure fluctuations that are scattered at the trailing edge, what leads to secondary tones in the acoustic radiation.
A wall-resolved large eddy simulation is performed to study secondary tones generated by a NACA0012 airfoil at $alpha = 3^{circ}$ with freestream Mach number $M_{infty} = 0.3$ and Reynolds number $Re = 5 times 10^4$. Laminar separation bubbles are observed over the suction side and near the trailing edge on the pressure side. Flow visualization and spectral analysis are employed to investigate vortex shedding aft of the suction side separation bubble. Vortex interaction results in merging or bursting such that coherent structures or turbulent packets are advected towards the trailing edge leading to different levels of noise emission. Despite the intermittent occurrence of laminar-turbulent transition, the noise spectrum depicts a main tone with multiple equidistant secondary tones. To understand the role of flow instabilities on the tones, the linearized Navier-Stokes equations are examined in its operator form through bi-global stability and resolvent analyses, and by time evolution of disturbances using a matrix-free method. These linear global analyses reveal amplification of disturbances over the suction side separation bubble. Non-normality of the linear operator leads to further transient amplification due to modal interaction among eigenvectors. Two-point correlations of pressure along the spanwise direction elucidate aspects of the acoustic feedback loop mechanism in both the linear and non-linear solutions. This feedback process is self-sustained by acoustic waves radiated from the trailing edge, which reach the most sensitive flow location between 10 and 18% of the airfoil chord as identified by the resolvent analysis.
Nonlinear dynamics of the free surface of finite depth non-conducting fluid with high dielectric constant subjected to a strong horizontal electric field is considered. Using the conformal transformation of the region occupied by the fluid into a strip, the process of counter-propagating waves interaction is numerically simulated. The nonlinear solitary waves on the surface can separately propagate along or against the direction of electric field without distortion. At the same time, the shape of the oppositely traveling waves can be distorted as the result of their interaction. In the problem under study, the nonlinearity leads to increasing the waves amplitudes and the duration of their interaction. This effect is inversely proportional to the fluid depth. In the shallow water limit, the tendency to the formation of a vertical liquid jet is observed.
A circular disc impacting on a water surface creates a remarkably vigorous jet. Upon impact an axisymmetric air cavity forms and eventually pinches off in a single point halfway down the cavity. Immediately after closure two fast sharp-pointed jets are observed shooting up- and downwards from the closure location, which by then has turned into a stagnation point surrounded by a locally hyperbolic flow pattern. This flow, however, is {it not} the mechanism feeding the two jets. Using high-speed imaging and numerical simulations we show that jetting is fed by the local flow around the base of the jet, which is forced by the colliding cavity walls. Based on this insight, we then show how the analytical description of a collapsing void (using a line of sinks along the axis of symmetry) can be continued beyond the time of pinch-off to obtain a quantitative model for jet formation which is in good agreement with the numerical and experimental data.
The similarity of the two-point correlation tensor along the streamwise direction in the axi-symmetric jet far-field is analyzed, herein its utility in spectral theory. A separable two-point correlation coefficient has been the basis for the argument that the energy-optimized basis functions along the streamwise direction are Fourier modes (from the approach of equilibrium similarity theory). This would naturally be highly desirable both from a computational and an analytical perspective. The present work, however, shows that the two-point correlation tensor multiplied by the Jacobian is not displacement invariant even in logarithmically stretched coordinates. This result directly impacts the motivation for a Fourier-based representation of the correlation function in spectral space in relation to the Proper Orthogonal Decomposition (POD) of the field. It is demonstrated that a displacement invariant form of the kernel is impossible to achieve using the suggested coordinate transformations from earlier works. This inability is shown to be related to the fundamental differences between the turbulent flow at hand and the ideal case of homogeneous turbulence.
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