Steady Mean Flow Research Articles

Computational AeroAcoustics of Realistic Co‐Axial Engines

This study, that is relevant from the turbofan engines noise prediction/reduction, aims at CAA‐computing the aft fan noise propagation/radiation of a realistic full‐3D exhaust (with pylon and internal bifurcations), the latter being affected of (i) typical in‐flight (take‐off) thermodynamic conditions and of (ii) a representative fan noise modal content. As for previous studies conducted over baseline geometries, this CAA computation is conducted following the usual hybrid process, where a preliminary aerodynamic calculation provides a heterogeneous steady mean flow on which an acoustic calculation is then conducted A RANS computation is first performed, delivering the stationary jet mean flow characterizing the 3D exhaust in its typical 'take‐off flight' (M∞ = 0.25). A CAA grid (22 blocks, 28 millions cells) is then derived from the CFD one, before the RANS steady jet mean‐flow is interpolated on it. After what the CAA computation is computed, a fan noise mode (26, 1) being emitted at a reduced frequency of kR = 30 (1 BPF) in the upstream of the engine's secondary exhaust, and numerically propagated along and outside the latter. Finally, a Kirchhoff post‐treatment provides the far‐field radiation characterizing these engine geometry, modal content and thermodynamic conditions.

On the one-dimensional acoustic propagation in conical ducts with stationary mean flow

This paper proposes a direct time-domain calculation of the time-domain responses of anechoic conical tubes with steady weak mean flow. The starting point is the approximated linear one-dimensional wave equation governing the velocity potential for the case of steady flow with low Mach number. A traveling solution with general space-dependent propagation velocity is then proposed from which the inward and outward pressure and velocity impulse responses can be obtained. The results include the well-known responses of conical and cylindrical ducts with zero mean flow.

On the use of a high order overlapping grid method for coupling in CFD/CAA

This paper presents a theoretical analysis and two applications of a high-order overlapping grid method for coupling Cartesian and curvilinear grids, developed in order to simulate aerodynamic noise. First, the overlapping grid method based on Lagrange interpolating polynomials is described and a theoretical analysis of the interpolation operator is then carried out. It shows that the interpolation generates spurious modes that depend on the wavenumbers of the signal. Besides it also gives the optimal conditions in which interpolation can be applied. Then an application to the simulation of the aeroacoustic noise generated by the vortex shedding behind a cylinder is presented. During this simulation, it appears that interpolation can create some spurious acoustic modes in regions where hydrodynamic fluctuations are significant, as predicted by the theoretical analysis. It is shown that these spurious modes disappear when a refined Cartesian grid is used (26 points per wavelength of the vortex shedding were found to be adequate in this study). At last, the simulation of the aerodynamic noise of a three element high-lift wing profile has then been carried out. For this application, the main acoustic source at the slat trailing edge is represented analytically. The propagation of the generated acoustic wave is simulated with a mean flow at rest and with a steady turbulent mean flow computed by RANS. The first application allows us to assess the method by comparing the results to a reference solution. The second one shows that the influence of a non-uniform mean flow on the directivity of an acoustic source can be observed in complex geometries. This application therefore shows that the proposed coupling method is well adapted to complex geometries that are usually met in industrial applications.

Sound Produced When a Vortex Pair is Swallowed by a Jet

An analysis is made of the sound generated during the high-Reynolds-number convection of a vortex pair in a jet of water exhausting from a large vessel through a slit aperture. The equations of motion are linearized about the classical free-streamline solution describing steady flow through the aperture. It is assumed that the vortex pair is swept through the aperture into the jet by the steady mean flow, with no account taken of the nonlinear influence on the motion of ‘images’ in the boundaries. Additional vorticity is shed from the edges of the aperture in order that the flow should remain smooth and continuous (the Kutta condition). This vorticity is convected away within a sheet of ‘bound’ vorticity on the mean free streamlines of the jet. A strong peak in the bound vorticity is established when the vortex pair enters the aperture. Both the incident and the shed vorticity generate sound, but their respective contributions to the acoustic pressure are of opposite phase. The dominant radiation in the water above the aperture is produced as the vortex enters the jet, and has the form of a pressure pulse of width ~h/M and monopole strength \(\sim \rho_o v^2/\sqrt{M}\), where h is the width of the aperture, ρo the density of the water, v a typical flow velocity, and M is the jet Mach number.

Nonmodal energy growth and optimal perturbations in compressible plane Couette flow

Nonmodal transient growth studies and an estimation of optimal perturbations have been made for the compressible plane Couette flow with three-dimensional disturbances. The steady mean flow is characterized by a nonuniform shear rate and a varying temperature across the wall-normal direction for an appropriate perfect gas model. The maximum amplification of perturbation energy over time, Gmax, is found to increase with increasing Reynolds number Re, but decreases with increasing Mach number M. More specifically, the optimal energy amplification Gopt (the supremum of Gmax over both the streamwise and spanwise wave numbers) is maximum in the incompressible limit and decreases monotonically as M increases. The corresponding optimal streamwise wave number, αopt, is nonzero at M=0, increases with increasing M, reaching a maximum for some value of M and then decreases, eventually becoming zero at high Mach numbers. While the pure streamwise vortices are the optimal patterns at high Mach numbers (in contrast to incompressible Couette flow), the modulated streamwise vortices are the optimal patterns for low-to-moderate values of the Mach number. Unlike in incompressible shear flows, the streamwise-independent modes in the present flow do not follow the scaling law G(t∕Re)∼Re2, the reasons for which are shown to be tied to the dominance of some terms (related to density and temperature fluctuations) in the linear stability operator. Based on a detailed nonmodal energy analysis, we show that the transient energy growth occurs due to the transfer of energy from the mean flow to perturbations via an inviscid algebraic instability. The decrease of transient growth with an increasing Mach number is also shown to be tied to the decrease in the energy transferred from the mean flow (Ė1) in the same limit. The sharp decay of the viscous eigenfunctions with increasing Mach number is responsible for the decrease of Ė1 for the present mean flow.

On the Rayleigh conductivity of a bias-flow aperture

Abstract An analysis is made of the attenuation of sound by vorticity production in a bias flow aperture. A modified form of the Cummings equation describing unsteady flow through a small aperture is used to extend the linear theory of the bias flow conductivity of a circular aperture in a thin wall of Howe (1979. On the theory of unsteady high Reynolds number flow through a circular aperture. Proceedings of the Royal Society of London A 366, 205–233) to a wall of arbitrary thickness and to examine the influence acoustic nonlinearity within the aperture. Numerical results are compared with existing analytic predictions of linear theory. It is shown that attenuations predicted by both linear and nonlinear theories agree over a wide range of incident acoustic pressures, approaching in amplitude that required to maintain the steady mean flow through the aperture. The dominant nonlinear effect is a small reduction (less than about 5%) in the mean bias flow velocity. Application of the Cummings equation in the linear regime leads to a new, simple formula for the bias flow conductivity for a screen of finite thickness.

Travel time and trajectory moments of conservative solutes in two-dimensional convergent flows

We address advective transport of a solute traveling toward a single pumping well in a two-dimensional randomly heterogeneous aquifer. The two random variables of interest are the trajectory followed by an individual particle from the injection point to the well location and the particle travel time under steady-state conditions. Our main objective is to derive the predictors of trajectory and travel time and the associated uncertainty, in terms of their first two statistical moments (mean and variance). We consider a solute that undergoes mass transfer between a mobile and an immobile zone. Based on Lawrence et al. [Lawrence, A.E., Sánchez-Vila, X., Rubin, Y., 2002. Conditional moments of the breakthrough curves of kinetically sorbing solute in heterogeneous porous media using multirate mass transfer models for sorption and desorption. Water Resour. Res. 38 (11), 1248, doi:10.1029/2001WR001006.], travel time moments can be written in terms of those of a conservative solute times a deterministic quantity. Moreover, the moments of solute particles trajectory do not depend on mass transfer processes. The resulting mean and variance of travel time and trajectory for a conservative species can be written as functions of the first, second moments and cross-moments of trajectory and velocity components. The equations are developed from a consistent second order expansion in σ Y (standard deviation of the natural logarithm of hydraulic conductivity). Our solution can be completely integrated with the moment equations of groundwater flow of Guadagnini and Neuman [Guadagnini, A., Neuman, S.P., 1999a. Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly non uniform domains 1. Theory and computational approach. Water Resour. Res. 35(10), 2999–3018.,Guadagnini, A., Neuman, S.P., 1999b. Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly non uniform domains 2. Computational examples. Water Resour. Res. 35(10), 3019–3039.], it is free of distributional assumptions regarding the log conductivity field, and formally includes conditioning. We present analytical expressions for the unconditional case by making use of the results of Riva et al. [Riva, M., Guadagnini, A., Neuman, S.P., Franzetti, S., 2001. Radial flow in a bounded randomly heterogeneous aquifer. Transport in Porous Media 45, 139–193.]. The quality of the solution is supported by numerical Monte Carlo simulations. Potential uses of this work include the determination of aquifer reclamation time by means of a single pumping well, and the demarcation of the region potentially affected by the presence of a contaminant in the proximity of a well, whenever the aquifer is very thin and Dupuit–Forchheimer assumption holds.

CFD/CAA Hybrid 기법을 이용한 뒷전에서 음향파의 산란모사

In this study, the main focus is the simulation of acoustic wave scattering in trailing edge and the analysis of the generation mechanism of instability wave by the interaction of trailing edge, shear flow and initial disturbance. The numerical algorithm is based on CFD/CAA hybrid method with high-order computational aeroacoustic method. It is found that steady mean flow gradient terms play a crucial role on the generation of instability wave through the comparison of simulations of Simple Linearized Euler Equation and Full Linearized Euler Equation. Through the comparison with the results of Full Navier-Stokes Equation, it is reasonable and efficient to use the Full Linearized Euler Equation in the initial generation mechanism of the instability wave near the trailing edge.

Stability of oscillatory flows past compliant surfaces

The stability of oscillatory flows over compliant surfaces is studied analytically and numerically. The types of compliant surfaces studied are the spring backed wall model, which permits tangential motion of the surface, and the incompressible viscoelastic gel model. The stability is determined using the Floquet analysis, where amplitude of perturbations at time intervals separated by one time period is examined to determine whether perturbations grow or decay. The oscillatory flows past both the spring backed wall model and the viscoelastic gel model exhibit an instability in the limit of zero Reynolds number, and the transition amplitude of the oscillatory velocity increases with the frequency of oscillations. The transition amplitude has a minimum at zero wave number for the spring backed plate model, whereas the minimum occurs at .nite wavenumber for the viscoelastic gel model. For the spring backed plate model, it is shown that the instability due to steady mean flow and the purely oscillatory instability reinforce each other, and the regions of instability are mapped in the $(\Lambda - A)$ plane, where \Lambda is the steady strain rate and A is the oscillatory strain rate. For the viscoelastic gel model, the instability is found to depend strongly on the gel viscosity $\eta_g$, and the effect of oscillations on the continuation of viscous modes at intermediate Reynolds number shows a complicated dependence on the oscillation frequency.

Echogenicity variations from porcine blood II: the “bright ring” under oscillatory flow

Echogenicity variations from porcine blood were observed in a mock flow loop under pulsatile flow in a series of experiments (Paeng et al. 2004). In this paper, oscillatory flow was generated to further investigate the cyclic and radial variation of blood echogenicity and its origin and mechanisms by several parameters, including stroke volume, stroke rate, mean steady flow and transducer angle, using a GE LOGIQ 700 Expert system. The echogenicity at the center of the tube was enhanced during acceleration and lower during deceleration, and the expansion and collapse of the “bright ring” was observed twice per cycle. The “black hole,” a central echo-poor zone surrounded by a hyperechoic zone, was barely observable under oscillatory flow, and these patterns differed from those under pulsatile flow. The cyclic and radial variation of echogenicity under oscillatory flow was affected by such hemodynamic parameters as stroke volume, stroke rate and mean steady flow. It was suggested that rouleaux might be aligned at an angle of about 25° relative to the tube axis during the acceleration phase, based on the experimental results reaching a maximum of the echogenicity variation at a transducer angle of 25°. Radial distribution of rouleaux alignments was proposed to be another important factor to blood echogenicity variation, in addition to combined effects of shear rate and flow acceleration on erythrocyte aggregation and blood echogenicity. The weak cyclic variation of echogenicity was also observed from the porcine erythrocyte suspensions under pure oscillatory flow, but not under pulsatile flow. It is postulated that the echogenicity variations from erythrocyte suspensions are from red cell deformation. (E-mail: paeng@cheju.ac.kr)

Conditioning mean steady state flow on hydraulic head and conductivity through geostatistical inversion

Nonlocal moment equations allow one to render deterministically optimum predictions of flow in randomly heterogeneous media and to assess predictive uncertainty conditional on measured values of medium properties. We present a geostatistical inverse algorithm for steady-state flow that makes it possible to further condition such predictions and assessments on measured values of hydraulic head (and/or flux). Our algorithm is based on recursive finite-element approximations of exact first and second conditional moment equations. Hydraulic conductivity is parameterized via universal kriging based on unknown values at pilot points and (optionally) measured values at other discrete locations. Optimum unbiased inverse estimates of natural log hydraulic conductivity, head and flux are obtained by minimizing a residual criterion using the Levenberg-Marquardt algorithm. We illustrate the method for superimposed mean uniform and convergent flows in a bounded two-dimensional domain. Our examples illustrate how conductivity and head data act separately or jointly to reduce parameter estimation errors and model predictive uncertainty.

An unsteady wave driver for narrowbanded waves: modeling nearshore circulation driven by wave groups

In this paper, we derive an unsteady refraction–diffraction model for narrowbanded water waves for use in computing coupled wave–current motion in the nearshore. The end result is a variable coefficient, nonlinear Schrödinger-type wave driver (describing the envelope of narrow-banded incident waves) coupled to forced nonlinear shallow water equations (describing steady or unsteady mean flows driven by the short-wave field). Comparisons with experimental data show that good accuracy can be obtained for cases of nonbreaking wave transformation. Numerical simulations show that the interaction of wave groups with longshore topographic nonuniformities generates strong edge wave resonances, providing a generating mechanism for low-order edge waves.

Stability of Periodically Unsteady Axisymmetric Jet

To date, the majority of studies on stability of axisymmetric jets have been completed under the assumption of steady mean flow. Yet, many of the natural and man-made flows that are modeled by these jets can have an inherent unsteadiness; the effects of which on the stability and transition have not been determined. Moreover, controlled unsteadiness can be used to control stability and possibly the transition to turbulence. In this note, the effects of periodic variations of the mean flow on the stability of axisymmetric jets are examined. The problem is treated analytically. The results show that the governing equations and dispersion relation for the unsteady jet can be reduced to those governing the steady jet with a time transformation. It is shown that the periodic variations in the mean flow cause amplitude and phase modulations of the unstable modes. The implications of the modulations on the subsequent transition stages are discussed.

Particle deposition in low-speed, high-turbulence flows

The experimental and numerical study considers the concentration of airborne particulate contaminants, such as spores of spoilage fungi, and their deposition on a surface, in a Petri dish, and on a warm box-shaped product placed in a food-processing environment. Field measurements by standard, active and passive samplers provide typical values of airborne concentrations and specific deposition fluxes. Velocity and turbulence data from field studies are used as input in large eddy simulations of the process, and estimates of deposition fluxes are of the same order of magnitude as those deduced from field measurements. Particle deposition is shown to be associated with near-wall coherent structures. Flow reversal, simulated by impulsive start, is shown to give higher deposition rates than steady mean flows.

The use of pulse responses and system reduction for 2D unsteady flows using the Euler equations

AbstractThis paper describes the application of the theory of pulse responses to a linearised Euler scheme on a moving grid. The impulse responses of such a linear system are memory functions or temporal representations of the manner in which, and the time over which a perturbation remains active in the response of the system. The exact response of the linear system to an arbitrary input can then be predicted via convolution. The linearised Euler scheme is derived from a full nonlinear Euler method based on Jameson’s cell-centred scheme which has been modified to make it time-accurate and has the necessary terms to account for grid motion. The discrete form of the full Euler equations are linearised about the corresponding nonlinear steady mean flow assuming the unsteadiness in the flow and grid are small. The resulting set of differential equations for the flow perturbations are solved at each time step using a dual-time scheme. These equations can be easily solved to obtain pulse responses, which capture the entire frequency content of the linear system. These pulse responses then provide a simple and efficient method for calculating the system response to many different inputs via convolution. A further use of the pulse responses is their potential to allow a reduced-order linear model to be constructed, which preserves some of the accuracyof the original system. Preliminary work carried out in this area is presented. Results of calculations for heave, pitch and ramp test cases are presented.

A two-dimensional linearized unsteady Euler scheme for pulse response calculations

An unsteady linearized Euler scheme for use on moving meshes is presented. This is derived from a scheme for the full non-linear unsteady Euler equations. This scheme is based on the Jameson cell-centred scheme, but is time-accurate and includes the necessary terms to account for grid motion. It is assumed that the unsteadiness in the flow and mesh is small. Using this assumption the discrete unsteady Euler equations are linearized about the full, non-linear, steady mean flow. The resulting equations are solved in the work presented here using a dual-time scheme. In the basic scheme no assumptions are made about the form of the perturbations other than that they are small. This permits the direct calculation of non-periodic flows, e.g. pulse responses. Linear pulse responses are a useful tool as they can be used to calculate the flow due to general inputs. The equations that would result from the assumption of harmonic flow are also derived. Results are presented for heave, pitch and ramp test cases and compared to full non-linear Euler results calculated using a dual-time scheme.

Higher order stability effects in a natural convection boundary layer over a vertical heated wall

The stability of a laminar boundary layer flow under natural convection on a vertical isothermally heated wall is studied analytically. The analysis is performed by using two different two-dimensional linear models: (1) The non-parallel flow model in which the steady mean flow as well as the disturbance amplitude functions can change in the streamwise direction; (2) The parallel flow model in which the effects of the mean flow and disturbance changes in the streamwise direction are neglected. The linear non-parallel stability analysis is based on the so-called parabolised stability equations (PSEs) which have been successfully applied to the stability analysis of forced convection boundary layers. In this study the PSE equations are applied to natural convection boundary layers in order to show the difference between parallel and non-parallel stability analysis. A second part of this study deals with the effects of variable properties, which are always present in natural convection flows. They are analysed by an extended version of the Orr–Sommerfeld equation (EOSE).

SE—Structures and Environment: Effect of Ventilator Configuration on Wind-driven Ventilation in a Crop Protection Structure for the Tropics

A laboratory method for studying wind-driven natural ventilation was applied to a single-span greenhouse with side-wall and roof ventilation openings, a design common to crop protection structures used in tropical countries. A 1:15 scale model replica of the greenhouse, made from clear Perspex, was immersed in a flume tank filled with water. A steady mean flow generated by the flume simulated wind flow past the model. The flow inside the model greenhouse was examined for four different ventilator configurations and ventilation flow rates compared in each case. Neutrally-buoyant particles seeded in the flow allowed for clear flow visualization and these particles were tracked using a digital image analysis system from which detailed velocity measurements were obtained. Flow patterns observed in the scale model agree qualitatively well with full-scale observations. Predictions of the flow at full scale inferred from measurements of pressure and flow speed made at the side-wall openings of the model differed by less than 10% and 28%, respectively, from measurements made in the full-size greenhouse. For all ventilator configurations using side-wall openings an incoming jet-like flow crossed the greenhouse and established a circulating flow. The greenhouse with roof ventilation only had the lowest ventilation rate and the maximum rate was achieved with combined side-wall and roof openings. The results of these studies are detailed and design recommendations to improve the ventilation of tropical greenhouses are suggested.

THE BOUNDARY CONDITION AT AN IMPEDANCE WALL IN A NON-UNIFORM DUCT WITH POTENTIAL MEAN FLOW

The boundary condition at an impedance wall in a duct with a steady mean flow requiring the specification of the normal component of acoustic particle velocity is examined. It is found that when implemented in the weak formulation of the finite element method it can be considerably simplified. The boundary condition would appear to require data which includes the tangential derivative of the tangential mean flow velocity, the normal derivative of the normal component of mean flow velocity, and the derivatives of the mean flow density and the boundary admittance along the boundary. It is shown that with suitable rearrangement the normal and tangential velocity derivatives can be eliminated, as can the derivatives of the mean flow density and admittance. The boundary condition becomes only slightly more complicated than the corresponding boundary condition when mean flow is absent, and is no more difficult to implement, requiring only local values of tangential mean flow velocity, density, and admittance which are already required as data for the weak formulation of the field equation.

ON THE BEHAVIOUR OF FLUID-LOADED SANDWICH PANELS WITH MEAN FLOW

This paper considers a fluid-loaded sandwich panel, made up of layers of relatively rigid skin material surrounding a layer of less dense core material, with an unsteady point driver in steady mean flow. For a simple plate Brazier-Smith and Scott, and Crighton and Oswell predicted a number of unusual phenomena, including absolute instability, a neutral mode with its group velocity pointing towards the driver, and negative energy waves. The possible occurrence of these phenomena for realistic parameter values is considered. It is shown that absolute instability is unlikely to be found at normal speeds, but in contrast the anomalous propagation mode is present, although over a limited frequency range. It is shown that the negative energy waves could also well occur in practice. Study of both the full dispersion relation and asymptotic analysis demonstrate that at very low frequencies it is the ratio of core to skin stiffnesses and the flow speed which control the dynamics, but at higher frequencies shear coupling and rotational effects become important as well. Consideration of the relative magnitudes of the energy fluxes leads to the unexpected conclusion that for predominantly shear waves more energy is carried in the fluid than is carried in the plate.