In this study, we consider the effects of linear and nonlinear instability waves on the near-field dynamics and aeroacoustics of two-dimensional laminar compressible mixing layers. Through a combination of direct computations, linear and nonlinear stability calculations, we demonstrate the significant role of nonlinear mechanisms in accurately describing the behaviour of instability waves. In turn, these processes have a major impact on sound generation mechanisms such as Mach wave radiation and vortex pairing sound. Our simulations show that the mean flow correction, which is required in order to accurately describe the dynamics of large-scale vortical structures, is intrinsically tied to the nonlinear modal interactions and accurate prediction of saturation amplitudes of instability waves. In addition, nonlinear interactions are largely responsible for the excitation and development of higher harmonics in the flow which contribute to the acoustic radiation. Two flow regimes are considered: In supersonic shear layers, where the far-field sound is determined by the instability wave solution at sufficiently high Mach numbers, it is shown that these nonlinear effects directly impact the Mach wave radiation. In subsonic shear layers, correctly capturing the near-field vortical structures and the interactions of the subharmonic and fundamental modes become critical due to the vortex pairing sound generation process. In this regime, a method is proposed to combine the instability wave solution with the Lilley–Goldstein acoustic analogy in order to predict far-field sound.
Two-phase laminar mixing layers are susceptible to shear-flow and interfacial instabilities, which originate from infinitesimal disturbances. Linear stability theory has successfully described the early stages of instability. In particular, parallel-flow linear analyses have demonstrated the presence of mode competition, where the dominant unstable mode can vary between internal and interfacial modes, depending on the flow parameters. However, the dynamics of two-phase mixing layers can be sensitive to additional factors, such as the spreading of the mean flow. In addition, beyond the early linear stage, the amplitude of the instability waves becomes finite and nonlinear effects become appreciable. As a result, an accurate description of the evolution of the mixing layer must account for nonlinear interactions including the generation of higher harmonics of the instability waves and the modification of the mean flow. These effects are investigated herein using the framework of the nonlinear parabolized stability equations. The formulation includes nonparallel effects, nonlinear modal interactions, a coupled mean flow correction, and finite amplitude deformation of the interface. Mode competition between liquid and interfacial modes is investigated. We demonstrate that nonparallelism and streamwise evolution of the flow can significantly alter the predictions of locally parallel, linear stability analyses. This is followed by a discussion on nonlinear interactions of two-and three-dimensional instability waves. It is shown that nonlinear effects can serve dual purposes. On one hand, they can be a limiting mechanism for the growth of instability waves. On the other hand, they can destabilize high frequency, linearly stable modes, and thus lead to the generation of smaller scale features in the flow.
The linear instability of compressible axisymmetric unheated jets is investigated numerically. Solutions to the linear parabolized stability equations with Reynolds-averaged Navier-Stokes mean flows are used to describe the streamwise evolution of instability waves. For transonic jets, helical ͑azimuthal mode number m =1͒ instability waves tend to exhibit the largest gain over the jet potential core followed by the m = 2 and axisymmetric modes. At higher frequencies, the disparity in energy growth between the different azimuthal modes decreases, and there is a progressive reduction in energy growth as azimuthal mode number is increased from two to four. The entropic and pressure components of the total energy are compared to the kinetic energy. We find that the pressure term tends to be small, while the importance of the entropic component increases with frequency and decreases with azimuthal mode number. It is shown that the helical mode growth rates exhibit similarity when a local "mixing layer" scaling is adopted. This similarity is then used to concisely illustrate the stabilizing effect of compressibility over a broad range of frequencies and Mach numbers. Computed phase velocities are compared to measured convective velocities for unforced turbulent jets with Mach numbers M j = 0.51 and M j = 1.41. Good agreement is observed provided that the appropriate azimuthal mode number is selected.
Case Presentation and EvolutionSix months after returning from a trip to Hawaii, a previously healthy 63-year-old man presented with 5 months of right upper-quadrant abdominal pain and non-bloody diarrhea, 3 months of nausea and non-bloody vomiting, and 1 month of fevers, chills, night sweats, and anorexia, culminating in a loss of 20 lbs. Comprehensive laboratory tests were notable for leukocytosis with a white blood cell (WBC) count of 11,300/mm 3 with 88% neutrophils, elevated liver chemistry tests (total bilirubin 0.6 mg/dl, aspartate aminotransferase [AST] 64 U/l, alanine aminotransferase [ALT] 92 U/l, and alkaline phosphatase 674 U/l), and renal failure (blood urea nitrogen 57 mg/dl, creatinine 2.3 mg/dl). Due to his renal insufficiency, non-contrast imaging of the abdomen was obtained, which revealed a 12-cm, thick-walled, multiloculated cystic lesion in the right lobe of the liver (Fig. 1a). Cultures obtained from the patient's blood, stool, and fluid aspirated from the liver abscess remained sterile. However, serum antibodies to Entamoeba histolytica were positive at a titer of 1:128.The infection was reported to the Department of Health who corroborated the diagnosis with three other cases of Entamoeba histolytica infection that were traced back to a restaurant worker in Hawaii. The patient was treated with oral metronidazole 750 mg three times daily for 10 days, followed by oral paromomycin 1 g three times daily for 10 days. His abdominal pain and diarrhea resolved. A contrast-enhanced abdominal computed tomography (CT) scan, obtained 2 months after finishing his course of antibiotics, documented complete resolution of the liver abscess (Fig. 1b).One year later, the patient again presented with fevers, chills, fatigue, abdominal pain, and diarrhea for 1 month, but this time also reported melena and maroon-colored stools for 1 week. His abdominal pain originated in the epigastrium, radiated to the right upper quadrant, and worsened when lying on his right side. On retrospection, he noted that he had suffered from diarrhea intermittently since his admission the previous year, but each episode had resolved with loperamide. He denied any weight loss and had actually regained 30 lbs. since discharge. He had not traveled anywhere since returning from Hawaii and could not recall any sick contacts.On physical examination, he had a low-grade fever to 100.2°F, but his vitals were otherwise normal. In general, he was well-appearing and without jaundice. He had normal bowel sounds, but his abdomen was distended and mild tenderness could be elicited upon palpation of the epigastrium, right upper-quadrant and right lower-quadrant of the abdomen. He had heme-positive, black-colored stool on rectal examination.Laboratory testing again revealed leukocytosis (WBC 9,500 with 80% neutrophils) and abnormal liver chemistry tests (total bilirubin \0.5 mg/dl, AST 33 U/l, ALT 39 U/l, and alkaline phosphatase 212 U/l). A contrast-enhanced CT of the abdomen showed a 6.0 9 5.2-cm hypo-attenuating mass in segment 4 of the liver wi...
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