The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

The Structure of Turbulent Shear Flow By Dr. A. A. Townsend. Pp. xii + 315. 8¾ in. × 5½ in. (Cambridge: At the University Press.) 40s.

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The Structure of Turbulent Shear Flow

Computer Methods in Applied Mechanics and Engineering

We study the velocity fields of unforced, high Reynolds number, subsonic jets, issuing from round nozzles with turbulent boundary layers. The objective of the study is to educe wavepackets in such flows and to explore their relationship with the radiated sound. The velocity field is measured using a hot-wire anemometer and a stereoscopic, time-resolved PIV system. The field can be decomposed into frequency and azimuthal Fourier modes. The low-angle sound radiation is measured synchronously with a microphone ring array. Consistent with previous observations, the azimuthal wavenumber spectra of the velocity and acoustic pressure fields are distinct. The velocity spectrum of the initial mixing layer exhibits a peak at azimuthal wavenumbers ranging from 4 to 11, and the peak is found to scale with the local momentum thickness of the mixing layer. The acoustic pressure field is, on the other hand, predominantly axisymmetric, suggesting an increased relative acoustic efficiency of the axisymmetric mode of the velocity field, a characteristic that can be shown theoretically to be caused by the radial compactness of the sound source. This is confirmed by significant correlations, as high as 10 %, between the axisymmetric modes of the velocity and acoustic pressure fields, these values being significantly higher than those reported for two-point flow–acoustic correlations in subsonic jets. The axisymmetric and first helical modes of the velocity field are then compared with solutions of linear parabolized stability equations (PSE) to ascertain if these modes correspond to linear wavepackets. For all but the lowest frequencies close agreement is obtained for the spatial amplification, up to the end of the potential core. The radial shapes of the linear PSE solutions also agree with the experimental results over the same region. The results suggests that, despite the broadband character of the turbulence, the evolution of Strouhal numbers 0.3 ≤ St ≤ 0.9 and azimuthal modes 0 and 1 can be modelled as linear wavepackets, and these are associated with the sound radiated to low polar angles.

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Wavepackets in the velocity field of turbulent jets

A novel approach to the development of a hybrid prediction methodology for jet noise is described Modeling details and numerical techniques are optimized for each of the three components of the model Far-field propagation is modeled by solution of a system of adjoint linear Euler equations, capturing convective and refraction effects using a spatially developing jet mean flow provided by a Reynolds-averaged Navier―Stokes computational fluid dynamics solution Sound generation is modeled following Goldstein's acoustic analogy, including a Gaussian function model for the two-point cross correlation of the fourth-order velocity fluctuations in the acoustic source Parameters in this model describing turbulent length and time scales are assumed to be proportional to turbulence information also taken from the Reynolds-averaged Navier―Stokes computational fluid dynamics prediction The constants of proportionality are, however, not determined empirically, but extracted by comparison with turbulence length and time scales obtained from a large eddy simulation prediction The large eddy simulation results are shown to be in good agreement with experimental data for the fourth-order two-point cross-correlation functions The large eddy simulation solution is then used to determine the amplitude parameter and also to examine which components of the cross correlation are largest, enabling inclusion of all identified dominant terms in the Gaussian source model The acoustic source description in the present approach is therefore determined with no direct input from experimental data This model is applied to the prediction of sound to the experimental configuration of the European Union JEAN project, and gives encouraging agreement with experimental data across a wide spectral range and for both sideline and peak noise angles This paper also examines the accuracy of various commonly made simplifications, for example: a locally parallel mean flow approximation rather than consideration of the spatially evolving mean jet flow and scattering from the nozzle; the assumption of small radial variation in Green function over the turbulence correlation length; the application of the far-field approximation in the Green function; and the impact of isotropic assumptions made in previous acoustic source models

Jet Noise: Acoustic Analogy informed by Large Eddy Simulation

Broadband shock-associated noise is a component of jet noise generated by supersonic jets operating offdesign. It is characterized by multiple broadband peaks and dominates the total noise at large angles to the jet downstream axis. A new model is introduced for the prediction of broadband shock-associated noise that uses the solution of the Reynolds-averaged Navier-Stokes equations. The noise model is an acoustic analogy based on the linearized Euler equations. The equivalent source terms depend on the product of the fluctuations associated with the jet's shock-cell structure and the turbulent velocity fluctuations in the jet shear layer. The former are deterministic and are obtained from the Reynolds-averaged Navier-Stokes solution. A statistical model is introduced to describe the properties of the turbulence. Only the geometry and operating conditions of the nozzle need to be known to make noise predictions. This overcomes the limitations and empiricism present in previous broadband shock-associated noise models. Results for various axisymmetric circular nozzles and a rectangular nozzle operating at various conditions are compared with experimental data to validate the model.

Prediction of Broadband Shock-Associated Noise Using Reynolds-Averaged Navier-Stokes Computational Fluid Dynamics

Heated high speed subsonic and supersonic jets operating on- or off-design are a source of noise that is not yet fully understood. Helium-air mixtures can be used in the correct ratio to simulate the total temperature ratio of heated air jets and hence have the potential to provide inexpensive and reliable flow and acoustic measurements. This study presents a combination of flow measurements of helium-air high speed jets and numerical simulations of similar helium-air mixture and heated air jets. Jets issuing from axisymmetric convergent and convergent-divergent nozzles are investigated, and the results show very strong similarity with heated air jet measurements found in the literature. This demonstrates the validity of simulating heated high speed jets with helium-air in the laboratory, together with the excellent agreement obtained in the presented data between the numerical predictions and the experiments. The very close match between the numerical and experimental data also validates the frozen chemistry model used in the numerical simulation.

/pdf/experimental-and-numerical-investigation-of-flow-properties-rvl0ls61c4.pdf

Experimental and Numerical Investigation of Flow Properties of Supersonic Helium-Air Jets

An acoustic analogy is developed based on the Euler equations for broadband shock-associated noise (BBSAN) that directly incorporates the vector Green's function of the linearized Euler equations and a steady Reynolds-Averaged Navier-Stokes solution (SRANS) to describe the mean flow. The vector Green's function allows the BBSAN propagation through the jet shear layer to be determined. The large-scale coherent turbulence is modeled by two-point second order velocity cross-correlations. Turbulent length and time scales are related to the turbulent kinetic energy and dissipation rate. An adjoint vector Green's function solver is implemented to determine the vector Green's function based on a locally parallel mean flow at different streamwise locations. The newly developed acoustic analogy can be simplified to one that uses the Green's function associated with the Helmholtz equation, which is consistent with a previous formulation by the authors. A large number of predictions are generated using three differe...

/pdf/the-prediction-of-broadband-shock-associated-noise-including-4i4z4i5bqa.pdf

The prediction of broadband shock-associated noise including propagation effects

An acoustic analogy is developed to predict the noise from jet flows. It contains two source models that independently predict the noise from turbulence and shock wave shear layer interactions. The acoustic analogy is based on the Euler equations and separates the sources from propagation. Propagation effects are taken into account by approximating the vector Green’s function of the linearized Euler equations with the use of a locally parallel mean flow assumption. A statistical model of the two-point cross correlation of the velocity fluctuations is used to describe the turbulence. The acoustic analogy attempts to take into account the correct scaling of the sources for a wide range of nozzle pressures and temperature ratios. It does not make assumptions regarding fine- or large-scale turbulent noise sources, self- or shear noise, or convective amplification. The acoustic analogy is partially informed by three-dimensional steady Reynolds-averaged Navier–Stokes solutions that include the nozzle geometry. ...

Toward a Comprehensive Model of Jet Noise Using an Acoustic Analogy

A method is presented to predict the far-eld sound pressure levels from the fan exhaust. The levels are found based on a known source distribution of the acoustic pressure and velocity perturbations inside the exhaust fan duct. This source distribution is propagated through the fan exhaust shear layer to a porous Ffowcs Williams-Hawkings surface using the linearized Euler equations. The linearized Euler equations are discretized in the frequency domain with the Streamline Upwind Petrov Galerkin method on an unstructured grid and are solved in parallel. This technique enables a stable numerical solution of the linearized Euler equations to be obtained. Results are shown for the Source Diagnostics Test which has a realistic engine geometry and a high speed fan-exit ow with fan tones at a relatively high frequency. Comparisons are made between the predicted and measured far-eld sound

Steven A. E. Miller

Papers

Prediction of Broadband Shock-Associated Noise Using Reynolds-Averaged Navier-Stokes Computational Fluid Dynamics

Experimental and Numerical Investigation of Flow Properties of Supersonic Helium-Air Jets

The prediction of broadband shock-associated noise including propagation effects

Toward a Comprehensive Model of Jet Noise Using an Acoustic Analogy

Prediction of Fan Exhaust Noise Propagation