Experimental and numerical investigation of separated flows in subsonic diffusers

The results of a special investigation of the diffuser flowfield are presented for two models of curvilinear diffuser channels with annular and rectangular cross-sections. The flow is visualized and the total pressure fields are measured by means of low-inertia transducers. At the same time, the flows are numerically calculated using commercial programs, together with codes developed by the authors. In these calculations the stationary and time-dependent Reynolds equations closed by different turbulence models, as well as the time-dependent Navier-Stokes equations, were integrated. A considerable difference between the measured data and the results of the numerical calculations in the stationary formulation is found to exist. At the same time, it has been possible to describe the occurrence of spatial inhomogeneities, the flow pattern, and the level of the experimentally observed aerodynamic losses on the basis of the solution of time-dependent problems.     

A hybrid approach for nonlinear computational aeroacoustics predictions

In many aeroacoustics applications involving nonlinear waves and obstructions in the far-field, approaches based on the classical acoustic analogy theory or the linearised Euler equations are unable to fully characterise the acoustic field. Therefore, computational aeroacoustics hybrid methods that incorporate nonlinear wave propagation have to be constructed. In this study, a hybrid approach coupling Navier–Stokes equations in the acoustic source region with nonlinear Euler equations in the acoustic propagation region is introduced and tested. The full Navier–Stokes equations are solved in the source region to identify the acoustic sources. The flow variables of interest are then transferred from the source region to the acoustic propagation region, where the full nonlinear Euler equations with source terms are solved. The transition between the two regions is made through a buffer zone where the flow variables are penalised via a source term added to the Euler equations. Tests were conducted on simple acoustic and vorticity disturbances, two-dimensional jets (Mach 0.9 and 2), and a three-dimensional jet (Mach 1.5), impinging on a wall. The method is proven to be effective and accurate in predicting sound pressure levels associated with the propagation of linear and nonlinear waves in the near- and far-field regions.     

Solving the incompressible fluid flows by a high-order mesh-free approach

In this paper, we propose for the first time to extend the application field of the high-order mesh-free approach to the stationary incompressible Navier-Stokes equations. This approach is based on a high-order algorithm, which combines a Taylor series expansion, a continuation technique, and a moving least squares (MLS) method. The Taylor series expansion permits to transform the nonlinear problem into a succession of continuous linear ones with the same tangent operator. The MLS method is used to transform the succession of continuous linear problems into discrete ones. The continuation technique allows to compute step-by-step the whole solution of the discrete problems. This mesh-free approach is tested on three examples: a flow around a cylindrical obstacle, a flow in a sudden expansion, and the standard benchmark lid-driven cavity flow. A comparison of the obtained results with those computed by the Newton-Raphson method with MLS, the high-order continuation with finite element method, and those of literature is presented.     

Near-resonance highly nonlinear gas oscillations in tubes

Near-resonance highly nonlinear ideal perfect gas oscillations in tubes are studied numerically for boundary conditions of various types. The oscillations are initiated by weak periodic perturbations at one end of the tube. As distinct from earlier studies [1–10], the oscillation amplitudes were not assumed to be small and the entropy increase at the shock waves formed was taken into account. Periodic flow regimes result as a limit of the solution of a Cauchy problem for one-dimensional time-dependent gasdynamic equations. The frequency responses of the oscillations under consideration are determined for boundary conditions of various types. It is shown that in specific cases the attainment of a periodic regime is accompanied by the appearance of long-wave modulations. The “repeated resonance” effect is revealed. This is due to the change in the tube's natural acoustic frequency, which takes place during the heating of the gas in the tube by the shock waves traveling in it. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 150–157, July–August, 1994.     

PML absorbing boundary conditions for the linearized and nonlinear Euler equations in the case of oblique mean flow

For the case of uniform mean flow in an arbitrary direction, perfectly matched layer (PML) absorbing boundary conditions are presented for both the linearized and nonlinear Euler equations. Although linear perfectly matched side layers with an oblique mean flow have been studied in previous works, we propose in the present paper a construction of corner layer equations that are dynamically stable. Stability issues are investigated by examining the dispersion relations of linear waves supported by the corner layer equations. For increased efficiency, a pseudo mean flow is included in the derivation of the PML equations for the nonlinear case. Numerical examples are given to support the validity of the proposed equations. Specifically, the linear PML formulation is tested for the case of acoustic, vorticity, and entropy waves traveling with an oblique mean flow. The nonlinear formulation is tested with an isentropic vortex moving diagonally with a constant velocity. Copyright © 2008 John Wiley & Sons, Ltd.     

Derivation of a linear theory for nonequilibrium and equilibrium flows

Conventional linear theory of nonequilibrium and equilibrium gas flows yields correct results only for very small deviations of the stream parameters from the unperturbed values. Moreover, if in linearization we take the coordinates in planar flow as independent variables, then the flow past concave and convex corners is described in exactly the same fashion. In this case the characteristic emanating from the corner is (depending on the type of corner) a compression or rarefaction shock. In the case of a break in the wall of an axisymmetric channel the shock intensity approaches infinity with approach to the centerline, which indicates a deficiency of this type of linear theory. In the following we use a modification which eliminates the deficiencies noted above. This involves conversion to new independent and dependent variables such that the coefficients of the exact equations being linearized become weakly varying functions of the unknown parameters, the linearized boundary conditions coincide with the exact conditions at all or part of the boundaries, and the rarefaction shocks become rarefaction wave bundles of finite width. The last condition is achieved as a result of the fact that, in accordance with the Lighthill method of deformable coordinates [1], we take as one of the independent variables a quantity which maintains a constant value on each characteristic of the bundle of characteristics emanating from the break point [for equilibrium flows the semicharacteristic (or characteristic) independent variables were used in deriving the linear theory, for example, in [2–4]]. The study was based on the example of two-dimensional stationary nonequilibrium flow of an inviscid and nonheatconducting gas. In this case we find that boththelinear equations at a finite distance from the walls and the boundary conditions for determining the potential and nonequilibrium parameters outside the rarefaction wave bundles coincide with the equations and the conditions of conventional linear theory [5], while the relations associating the values of the parameters on the closing characteristics of each bundle (outside the bundles the same value of the characteristic variable corresponds to these characteristics) at some distance from the axis or from some reflecting surface are identical to the conditions on the rarefaction shocks. This fact makes it possible to use several results of conventional linear theory.     

Development of magnetohydrodynamic boundary layers associated with the sudden initiation or deceleration of a supersonic flow at the boundary of a half-space

The problem of nonstationary magnetohydrodynamic flow of a viscous fluid in a half-space resulting from the motion of an infinite plate has received much attention. In [1], for example, solutions are presented for the case of isotropic conductivity, while in [2] a solution of the Rayleigh problem is offered for the case of anisotropic conductivity. In these instances the fluid was assumed incompressible and uniform, and the system of equations was found to be linear. In problems involving nonstationary flow of a gas in a transverse magnetic field resulting from the deceleration of a high-velocity gas flow at the boundary of a half-space or the motion of an infinite plate at supersonic speed relative to a stationary gas it becomes necessary to take into account the compressibility of the gas and the temperature dependence of the conductivity. It is then possible to have flows in which the gas becomes electrically conducting and begins to interact with the magnetic field solely as a result of the increase in temperature due to viscous dissipation of energy. The magnetic field, interacting with the conducting gas, exerts an effect on the drag and heat transfer to the surface of the plate. At sufficiently low gas pressures and strong magnetic fields a Hall effect may be observed. The system of equations describing the motion of a compressible gas with variable conductivity is essentially nonlinear. The present article is devoted to a study of such motions.     

The region of nonlinear effects for intensive sound pulses in the ocean

The formation of a stationary shock wave is studied in media with an arbitrary power dependence of the damping coefficient on the frequency. The conditions for existence of a stationary shock wave are defined and it is shown that when acoustic signals propagate in the ocean the region of nonlinear effects is limited. For acoustic waves generated by explosive sources a calculation is given of the location of the transition point of the nonlinear wave into a linear one, and the dependence of this point on the charge weight is defined.     

Multicomponent fluid flow analysis using a new set of conservation equations

In this work hydrodynamics of multicomponent ideal gas mixtures have been studied. Starting from the kinetic equations, the Eulerian approach is used to derive a new set of conservation equations for the multicomponent system where each component may have different velocity and kinetic temperature. The equations are based on the Grad's method of moment derived from the kinetic model in a relaxation time approximation (RTA). Based on this model which contains separate equation sets for each component of the system, a computer code has been developed for numerical computation of compressible flows of binary gas mixture in generalized curvilinear boundary conforming coordinates. Since these equations are similar to the Navier-Stokes equations for the single fluid systems, the same numerical methods are applied to these new equations. The Roe's numerical scheme is used to discretize the convective terms of governing fluid flow equations. The prepared algorithm and the computer code are capable of computing and presenting flow fields of each component of the system separately as well as the average flow field of the multicomponent gas system as a whole. Comparison of the present code results with those of a more common algorithm based on the mixture theory in a supersonic converging-diverging nozzle provides the validation of the present formulation. Afterwards, a more involved nozzle cooling problem with a binary ideal gas (helium-xenon) is chosen to compare the present results with those of the ordinary mixture theory. The present model provides the details of the flow fields of each component separately which is not available otherwise. It is also shown that the separate fluids treatment, such as the present study, is crucial when considering time scales on the order of (or shorter than) the intercollisions relaxation times.     

Exact Analytic Solutions of the Nonlinear Long-Wave Equations in the Case of Axisymmetric Fluid Vibrations in a Parabolic Rotating Vessel

A class of exact analytic solutions of the system of nonlinear long-wave equations is found. This class corresponds to the axisymmetric vibrations of an ideal incompressible homogeneous fluid in a rotating vessel in the shape of a paraboloid of revolution. The radial velocity of these motions is a linear function, and the azimuthal velocity and free surface displacements are polynomials in the radial coordinate with time-dependent coefficients. The nonlinear vibration frequency is equal to the frequency of the lowest mode of linear axisymmetric standing waves in the parabolic vessel.     

Nonlinear fluid flow laws for orthotropic porous media

Nonlinear fluid flow laws for orthotropic porous media are written in invariant tensor form. As usual in the theory of fluid flow through porous media [1, 2], the equations contain the flow velocity up to the second power. Expressions that determine the nonlinear resistances to fluid flow are presented and it is shown that, on going over from linear to nonlinear flow laws, the asymmetry effect may manifest itself, that is, the fluid flow characteristics may differ along the same straight line in the positive and negative directions. It is shown that, as compared with the linear fluid flow law for orthotropic media when for three symmetry groups a single flow law is sufficient, in nonlinear laws the anisotropy manifestations are much more variable and each symmetry group must be described by specific equations. A system of laboratory measurements for finding the nonlinear flow characteristics for orthotropic porous media is considered.     

The jump phenomenon effect on the sound absorption of a nonlinear panel absorber and sound transmission loss of a nonlinear panel backed by a cavity

Theoretical analysis of the nonlinear vibration effects on the sound absorption of a panel absorber and sound transmission loss of a panel backed by a rectangular cavity is herein presented. The harmonic balance method is employed to derive a structural acoustic formulation from two-coupled partial differential equations representing the nonlinear structural forced vibration and induced acoustic pressure; one is the well-known von Karman??s plate equation and the other is the homogeneous wave equation. This method has been used in a previous study of nonlinear structural vibration, in which its results agreed well with the elliptic solution. To date, very few classical solutions for this nonlinear structural-acoustic problem have been developed, although there are many for nonlinear plate or linear structural-acoustic problems. Thus, for verification purposes, an approach based on the numerical integration method is also developed to solve the nonlinear structural-acoustic problem. The solutions obtained with the two methods agree well with each other. In the parametric study, the panel displacement amplitude converges with increases in the number of harmonic terms and acoustic and structural modes. The effects of excitation level, cavity depth, boundary condition, and damping factor are also examined. The main findings include the following: (1)?the well-known ??jump phenomenon?? in nonlinear vibration is seen in the sound absorption and transmission loss curves; (2)?the absorption peak and transmission loss dip due to the nonlinear resonance are significantly wider than those in the linear case because of the wider resonant bandwidth; and (3)?nonlinear vibration has the positive effect of widening the absorption bandwidth, but it also degrades the transmission loss at the resonant frequency.     

Weakly nonlinear acoustic instabilities

With the aim of eventually improving numerical solutions of small-scale phenomena, the Hunter-Keller theory of weakly nonlinear high-frequency waves is applied to the study of short wavelength instabilities in inviscid fluids driven by a heat or pressure source. A nonlinear damping effects is found which, for acoustic perturbations of a stationary, homogeneous state, reduces the growth rate to half the linear estimate. This is due primarily to the interactions of the expansion fan and the weak shock generated by the cumulative effect of the nonlinear convective term. For acoustic perturbations driven by an unbalanced heat source, the nonlinear damping actually stabilizes some modes which are unstable according to the linear theory. For the isentropic compression of a spherical shell of material obeying a γ-law equation of state, it is shown that the nonlinear damping again reduces the acoustic growth rate to the half the value predicted by conventional linear stability analyses.     

Assessment of a Two-Way Coupling Methodology Between a Flow and a High-Order Nonlinear Acoustic Unstructured Solvers

A two-way coupling on unstructured meshes between a flow and a high-order acoustic solvers for jet noise prediction is considered. The flow simulation aims at generating acoustic sources in the near field while the acoustic simulation solves the full Euler equations, thanks to a discontinuous Galerkin method, in order to take into account nonlinear acoustic propagation effects. This methodology is firstly validated on academic cases involving nonlinear sound propagation, shock waves and convection of aerodynamic perturbations. The results are compared to analytical solutions and direct computations. A good behaviour of the coupling is found regarding the targeted space applications. An application on a launch pad model is then simulated to demonstrate the robustness and reliability of the present approach.     

Finite Dimensional State Representation of Linear and Nonlinear Delay Systems

We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear autonomous systems and linear systems with time-dependent input we give necessary and sufficient conditions and in the nonlinear case we give sufficient conditions. Most of our results for linear renewal and delay differential equations are known in different guises. The novelty lies in the approach which is tailored for applications to models of physiologically structured populations. Our results on linear systems with input and nonlinear systems are new.     

Two-equation (k, ϵ) turbulence computations by the use of a finite element model

A finite element technique is presented and applied to some one- and two-dimensional turbulent flow problems. The basic equations are the Reynolds averaged momentum equations in conjunction with a two-equation (k, ?) turbulence model. The equations are written in time-dependent form and stationary problems are solved by a time iteration procedure. The advection parts of the equations are treated by the use of a method of characteristics, while the continuity requirement is satisfied by a penalty function approach. The general numerical formulation is based on Galerkin's method. Computational results are presented for one-dimensional steady-state and oscillatory channel flow problems and for steady-state flow over a two-dimensional backward-facing step.     

Run-up of long waves on background shear currents

Long waves in shallow water propagating over a background shear current towards a sloping beach are investigated, and exact solutions are found using a hodograph transform and separation of variables. Inspired by the work of Carrier and Greenspan on steady waves over a uniform beach profile in the irrotational setting, we study waves which propagate over a background shear current. The shallow-water equations are obtained from the nonlinear Benney equations, and exact solutions are found with help of the hodograph transformation in conjunction with several further changes of variables. The hodograph transformation is effected by finding the Riemann invariants after the equations are written in the standard form of barotropic gas dynamics. In the current work, the background flow features zero mass flux, as would be required by a real flow at a beach. Moreover, in contrast with previous work, the present approach allows separate study of the influence of the strength of the shear current and the slope of the bottom profile. This enables us to provide an estimate of the run-up as a function of the shear flow while keeping the bottom slope constant.     

Linearized and non‐linear acoustic/viscous splitting techniques for low Mach number flows

Computation of the acoustic disturbances generated by unsteady low‐speed flow fields including vortices and shear layers is considered. The equations governing the generation and propagation of acoustic fluctuations are derived from a two‐step acoustic/viscous splitting technique. An optimized high order dispersion–relation–preserving scheme is used for the solution of the acoustic field. The acoustic field generated by a corotating vortex pair is obtained using the above technique. The computed sound field is compared with the existing analytic solution. Results are in good agreement with the analytic solution except near the centre of the vortices where the acoustic pressure becomes singular. The governing equations for acoustic fluctuations are then linearized and solved for the same model problem. The difference between non‐linear and linearized solutions falls below the numerical error of the simulation. However, a considerable saving in CPU time usage is achieved in solving the linearized equations. The results indicate that the linearized acoustic/viscous splitting technique for the simulation of acoustic fluctuations generation and propagation by low Mach number flow fields seems to be very promising for three‐dimensional problems involving complex geometries. Copyright © 2003 John Wiley & Sons, Ltd.     

Droplet break-up through an oblique shock wave

The interaction of a two-phase flow with a wedge where a stationary shock wave is initially settled is studied in a two-dimensional configuration. Before the introduction of the dispersed phase, the flow around the wedge is a supersonic one phase flow such as an attached stationary shock wave is present. Then, the dispersed phase is introduced upstream the initial position of the stationary shock wave. The purpose of this study is to point out two-phase and droplets break-up effects on the oblique shock wave. The two-dimensional equations are solved by a TVD scheme where fluxes are computed by using Riemann solver for the gas phase equations and also for the dispersed phase equations wich is an original approach due to the authors (Saurel et al. 1994). In addition to drag forces and heat and mass transfers, the process of droplets fragmentation based on the particle oscillation is considered. Accepted April 28, 1995