Low Mach Number Limit of the Full Navier-Stokes Equations

The low Mach number limit for classical solutions of the full Navier-Stokes equations is here studied. The combined effects of large temperature variations and thermal conduction are taken into account. In particular, we consider general initial data. The equations lead to a singular problem, depending on a small scaling parameter, whose linearized system is not uniformly well-posed. Yet, it is proved that solutions exist and they are uniformly bounded for a time interval which is independent of the Mach number Ma ∈ (0,1], the Reynolds number Re ∈ [1,+∞] and the Péclet number Pe ∈ [1,+∞]. Based on uniform estimates in Sobolev spaces, and using a theorem of G. Métivier & S. Schochet [30], we next prove that the penalized terms converge strongly to zero. This allows us to rigorously justify, at least in the whole space case, the well-known computations given in the introduction of P.-L. Lions' book [26].     

The Low Mach Number Limit for the Full Navier–Stokes–Fourier System

We study the low Mach number asymptotic limit for solutions to the full Navier–Stokes–Fourier system, supplemented with ill-prepared data and considered on an arbitrary time interval. Convergencetowards the incompressible Navier–Stokes equations is shown.     

Efficient Large-eddy Simulations of Low Mach Number Flows Using Preconditioning and Multigrid

In the present paper, an implicit time accurate approach combined with multigrid, preconditioning and residual smoothing is used for the large-eddy simulation (LES) of low Mach number flow. In general, due to the restriction imposed on the time step by the physics of the flow, the advantage of an implicit method over an explicit one for LES is not obvious. It is shown that for the test cases considered in this paper, the present approach allows an efficiency gain of a factor 4–7 compared to the use of a purely explicit approach. The efficiency varies according to the test case, grid clustering, physical time step and requested residual drop. Numerical difficulties are catalogued and mitigatory procedures are introduced. Several problems with available experimental and DNS data are employed to verify the efficiency of the method.     

On the Accuracy of Upwind and Symmetric TVD Schemes in Simulating Low Mach Number Flow

The accuracy of MUSCL upwind and Yee-Roe-Davis symmetric TVD schemes for simulating low Mach number flow is studied through a numerical experiment of the 2-D lid driven cavity problem. The steady slate solution is reached by using a marching approach based on the pseudocompressibilty method in conjunction with implicit approximate factorization. A finite volume discretization of the conservation equations is used with a four level multigrid method to accelerate the convergence. The tests performed which were in the range of 100 ≤ Re ≤ 5000, show that the Yee-Roe-Davis symmetric scheme generates results in very good agreement with the benchmark results over this range of Re. The MUSCL upwind scheme accuracy deteriorates with the increasing Re.     

Continuity of Drag and Domain Stability in the Low Mach Number Limits

We consider a mathematical model of a rigid body immersed in a viscous, compressible fluid moving with a velocity prescribed on the boundary of a large channel containing the body. We assume that the Mach number is proportional to a small parameter ε and that the general boundary of the body contains small asperities of amplitude proportional to ε ^α for a certain α?>?0 and suppose the Navier’s slip condition on this rough boundary. We show that time averages of the drag functional converge, as ε → 0, to the corresponding time averages of the drag for the limit system, whereas the limit system is turning out to be the incompressible Navier–Stokes system with no-slip condition on the smooth limit body.     

The Effect of the Mach Number on a Turbulent Backward-Facing Step Flow

The flow around a backward-facing step in the sub-, trans- and supersonic regimes was investigated at the Trisonic Wind Tunnel Munich with particle image velocimetry and dynamic pressure measurements. These two techniques were combined to simultaneously measure and correlate the velocity fluctuations in a streamwise vertical plane with the pressure fluctuations on the reattachment surface. The results show that the dynamic loads on the reattachment surface increase from subsonic up to the transonic regime while the mean reattachment location moves downstream. As soon as the flow becomes locally supersonic aft of the backward-facing step, the mean reattachment location suddenly moves upstream while the normalized dynamic loads drastically decrease. By correlating the velocity and the dynamic pressure data, it was shown that a clear separation between outer flow and the flow close to the surface aft of the step is responsible for the drastic load reduction. Due to the large difference in pressure/density, the disturbances from the locally supersonic flow do not have an effect on the flow close to the surface. This is also reflected in the power spectral densities of the pressure fluctuations on the surface, showing that at supersonic free-stream Mach numbers a low-frequency pumping motion of the locally subsonic flow is the dominant mode, while in sub-/transonic flow Kelvin-Helmholtz instabilities and a cross-pumping motion of the shear layer dominate the dynamic loads.     

On the Lower Critical Reynolds Number for Flow in a Circular Pipe

Flow in a circular pipe is investigated experimentally at Reynolds numbers higher than that at which the resistance coefficients calculated from the Blasius formula for laminar flow and from the Prandtl formula for turbulent flow are equal. The corresponding Reynolds number based on the mean-flow velocity and the pipe diameter is about 1000. The experiments were performed at a high level of inlet pulsations produced by feeding gas into the pipe through a hole with a diameter several times smaller than the pipe diameter. In our experiments the critical Reynolds number was determined as the value, independent of the distance from the inlet, at which the ratio of the axial to the mean-flow velocity as a function of the Reynolds number deviated from 2. At the maximum ratio of the pipe cross-sectional area to the area of the hole through which the gas entered the pipe, equal to 26, the critical Reynolds number was about 2300. After a fivefold increase in the hole area the critical Reynolds number increased by approximately 4%.At Reynolds numbers below 2000, after at a high level of the inlet pulsations an almost laminar flow had developed in the pipe, a perturbation was introduced by inserting a diametrically oriented cylindrical rod with a diameter 10–20 times smaller than the pipe diameter. In these experiments, at Reynolds numbers higher than 1000, at a distance from the rod equal to 50 pipe diameters the axial to mean-flow velocity ratio was less than 2, approaching this value again at large distances from the rod. The insertion of the rod led to a decrease in the critical Reynolds number by approximately 12%.     

Numerical Analysis of the Impact of the Interior Nozzle Geometry on Low Mach Number Jet Acoustics

The flow and acoustic fields of subsonic turbulent hot jets exhausting from three divergent nozzles at a Mach number M=0.12 based on the nozzle exit velocity are conducted using a hybrid CFD-CAA method. The flow field is computed by highly resolved large-eddy simulations (LES) and the acoustic field is computed by solving the acoustic perturbation equations (APE) whose acoustic source terms are determined by the LES. The LES of the computational domain includes the interior of the nozzle geometry. Synthetic turbulence is prescribed at the inlet of the nozzle to mimic the exit conditions downstream of the last turbine stage. The LES is based on hierarchically refined Cartesian meshes, where the nozzle wall boundaries are resolved by a conservative cut-cell method. The APE solution is determined on a block structured mesh. Three nozzle geometries of increasing complexity are considered, i.e., the flow and acoustic fields of a clean geometry without any built-in components, a nozzle with a centerbody, and a nozzle with a centerbody plus struts are computed. Spectral distributions of the LES based turbulent fluctuated quantities inside the nozzle and further downstream are analyzed in detail. The noise sources in the near field are noticeably influenced by the nozzle built-in components. The centerbody nozzle increases the overall sound pressure level (OASPL) in the near field with respect to the clean nozzle and the centerbody-plus-strut nozzle reduces it compared to the centerbody nozzle due to the increased turbulent mixing. The centerbody perturbed nozzle configurations generate a remarkable spectral peak at S t=0.56 which also occurs in the APE findings in the near field region. This tone is generated by large scale vortical structures shed from the centerbody. The analysis of the individual noise sources shows that the entropy term possesses the highest acoustic contribution in the sideline direction whereas the vortex sound source dominates the downstream acoustics.     

Continuous Dependence of Entropy Solutions to the Euler Equations on the Adiabatic Exponent and Mach Number

We establish the L ¹-estimates for continuous dependence of entropy solutions to the full Euler equations away from the vacuum on two physical parameters: the adiabatic exponent γ → 1 that passes from the non-isentropic to isothermal Euler equations and the Mach number that passes from the compressible to incompressible Euler equations. Our analysis involves the effective approach developed in our earlier work and additional new techniques that generalize this approach to the setting of the full Euler equations.     

On the Application of the Critical Excitation Method to Aseismic Design

ABSTRACT

The problem of critical excitation of linear structures subjected to earthquake support motion is investigated. For a given time duration and peak ground acceleration, the critical time history which maximizes the absolute acceleration response is obtained. Similar considerations are given to the critical ground acceleration which maximizes the relative displacement and velocity responses. Some bounds on the response spectra curves are obtained and discussed.     

On the Von Neumann Paradox of Weak Mach Reflection*

Recent experimental and numerical studies of weak Mach reflections are examined. It is shown that the fundamental reason for the von Neumann paradox is that his theory of Mach reflection is based on the assumption that the flow downstream of the reflected wave and the Mach shock near the wave triple point is uniform. The assumption is shown to be valid for strong Mach reflection which agrees with experiment, but invalid for weak Mach reflection which does not agree with experiment. It is also shown that viscous effects are dominant when the incident shock is within about 100 mean free path lengths of the corner, but not otherwise. The analytical theory of the entire subsonic region supports these conclusions.     

Numerical Study of Turbulence and Noise in Chevrons Nozzles at the Mach Number Equal to 0.25

Fluid Dynamics - Turbulent jet flows at the Mach number equal to 0.25 from an industrial nozzle are studied using the large eddy simulation (LES). The feasibility of controlling the jet flow and...     

Potential Method in the Linear Theory of Viscoelastic Materials with Voids

In the present paper the linear theory of viscoelasticity for Kelvin–Voigt materials with voids is considered and some basic results of the classical theory of elasticity are generalized. Indeed, the basic properties of plane harmonic waves are established. The explicit expression of fundamental solution of the system of equations of steady vibrations is constructed by means of elementary functions. The Green’s formulas in the considered theory are obtained. The uniqueness theorems of the internal and external basic boundary value problems (BVPs) are proved. The representation of Galerkin type solution is obtained and the completeness of this solution is established. The formulas of integral representations of Somigliana type of regular vector and regular (classical) solution are obtained. The Sommerfeld-Kupradze type radiation conditions are established. The basic properties of elastopotentials and singular integral operators are given. Finally, the existence theorems for classical solutions of the internal and external basic BVPs of steady vibrations are proved by using of the potential method (boundary integral method) and the theory of singular integral equations.     

Design of a Laminated Plate Possessing the Required Stiffnesses Using the Minimum Number of Materials and Layers

This paper deals with the problem of the design of a laminated plate possessing the required set of stiffnesses under the condition of using the minimum number of layers and minimum number of materials. It is shown that the minimum number of layers is not more than four and the minimum number of materials is not more than two. We consider the case when three types of stiffness (bending, in-plane and out-of-plane) are prescribed and the case when two types of stiffness (physical bending and in-plane) are prescribed. It is proved that for both cases the sets of the possible values of physical stiffnesses are the same but the sets of designs can be different. A design algorithm is developed.      

On the role of turbulence in detonation induced by Mach stem reflection

A series of experiments conducted by Chan has shown that while some shock waves may not be strong enough to induce detonation when they collide with an obstacle the resulting Mach stem will induce detonation if it collides with a subsequent obstruction. A series of numerical simulations, however, failed to demonstrate the expected results if either the Euler or laminar Navier-Stokes equations are solved. On the other hand, calculations using the Favre averaged Navier-Stokes equations with a k--F turbulence model are able to reproduce the experimental results, indicating that turbulent effects may play an important role in the ignition process. A detailed examination of the results shows that turbulence causes the formation of activated kernels in a similar process to that observed in deflagration-detonation transition. The simulations in this paper have been undertaken using a modern high resolution hydrocode and a reduced kinetics mechanism for hydrogen combustion. The paper describes the reduced mechanism, the solution methods employed in the hydrocode and discusses the results of the simulations and their implications. Received 28 October 1997 / Accepted 30 April 1998     

Studies of Store-Induced Limit-Cycle Oscillations Using a Model with Full System Nonlinearities

The authors investigate limit-cycle oscillations of a wing/store configuration. Unlike typical aeroelastic studies that are based upon a linearized form of the governing equations, herein full system nonlinearities are retained, and include transonic flow effects, coupled responses from the structure, and store-related kinematics and dynamics. Unsteady aerodynamic loads are modeled with the equations from transonic small disturbance theory. The structural dynamics for the cantilevered wing are modeled by the nonlinear equations of motion for a beam. The effects of general store-placement are modeled by the nonlinear equations of motion related to the position-induced nonlinear kinematics. Chordwise deformations of the wing surface, as well as pylon and store flexibility, are assumed negligible. Nonlinear responses are studied by examining bifurcation and related response characteristics using direct simulation. Particular attention is given to cases for which large-time, time-dependent behavior is dependent on initial conditions, as observed for some configurations in flight test. Comparisons of results in which selective nonlinearities are excluded indicate that the accurate prediction of nonlinear responses such as limit cycle oscillations (LCOs) may depend upon consideration of all nonlinearities related to the full system.     

On Limit Design of Frames Using Linear Programming

Abstract

This paper deals with a broad class of optimum frame design problems amenable to the mathematical model of linear programming: allowance is made for self-weight (design dependent loads) and technological constraints “assigned minimum for yield moments, prescribed variation laws of yield moments along members”. Two alternative “static” formulations and the corresponding dual “kinematic” formulations are discussed and compared to each other. The main limit design theorems, generalized to the present broader context, are derived on the basis of duality theory of linear programming. Numerical examples, worked out by means of standard LP computer codes, are given.     

Noise Investigation of a High Subsonic, Moderate Reynolds Number Jet Using a Compressible Large Eddy Simulation

This study investigates the noise radiated by a subsonic circular jet with a Mach number of 0.9 and a Reynolds number of 65000 computed by a compressible Large Eddy Simulation (LES). First, it demonstrates the feasibility of using LES to predict accurately both the flow field and the sound radiation on a domain including the acoustic field. Mean flow parameters, turbulence intensities, velocity spectra and integral length scales are in very good agreement with experimental data. The noise generated by the jet, provided directly by the simulation, is also consistent with measurements in terms of sound pressure spectra, levels and directivity. The apparent location of the sound sources is at the end of the potential core in accordance with some experimental observations at similar Reynolds numbers and Mach numbers. Second, the noise generation mechanisms are discussed in an attempt to connect the flow field with the acoustic field. This study shows that for the simulated moderate Reynolds number jet, the predominant sound radiation in the downstream direction is associated with the breakdown of the shear layers in the central jet zone. Received 24 January 2002 and accepted 16 July 2002 Published online 3 December 2002 RID="*" ID="*" A preliminary version of some of the results presented here was reported in AIAA Paper 2000–2009 presented at the 6th AIAA/CEAS Aeroacoustics Conference in Lahaina, Hawaii, June 2000. Computing time was supplied by the Institut du Développement et des Ressources en Informatique Scientifique (IDRIS – CNRS). Communicated by T.B. Gatski