scholarly journals The interaction of a contact discontinuity with sound waves according to the linearised Navier-Stokes equations

1970 ◽  
Vol 17 (1) ◽  
pp. 37-46
Author(s):  
W. M. Anderson
Keyword(s):  
Fourier Transform ◽  
Contact Surface ◽  
Stokes Equations ◽  
Asymptotic Series ◽  
Density Perturbation ◽  
Navier Stokes ◽  
Sound Waves ◽  
Navier Stokes Equations ◽  
Initial Discontinuity ◽  
The Fourier Transform

AbstractThe resolution of a small initial discontinuity in a gas is examined using the linearised Navier-Stokes equations. The smoothing of the resultant contact surface and sound waves due to dissipation results in small flows which interact. The problem is solved for arbitrary Prandtl number by using a Fourier transform in space and a Laplace transform in time. The Fourier transform is inverted exactly and the density perturbation is found as two asymptotic series valid for small dissipation near the contact surface and the sound waves respectively. The modifications to the structures of the contact surface and the sound waves are exhibited.

Boundary layer leading-edge receptivity to sound at incidence angles

2001 ◽  
Vol 444 ◽  
pp. 383-407 ◽  
Author(s):  
ERCAN ERTURK ◽  
THOMAS C. CORKE
Keyword(s):  
Acoustic Waves ◽  
Stokes Equations ◽  
Incidence Angle ◽  
Leading Edge ◽  
Quantitative Agreement ◽  
Navier Stokes ◽  
Angle Of Incidence ◽  
Sound Waves ◽  
Nose Radius ◽  
Navier Stokes Equations

The leading-edge receptivity to acoustic waves of two-dimensional parabolic bodies was investigated using a spatial solution of the Navier–Stokes equations in vorticity/streamfunction form in parabolic coordinates. The free stream is composed of a uniform flow with a superposed periodic velocity fluctuation of small amplitude. The method follows that of Haddad & Corke (1998) in which the solution for the basic flow and linearized perturbation flow are solved separately. We primarily investigated the effect of frequency and angle of incidence (−180° [les ] α2 [les ] 180°) of the acoustic waves on the leading-edge receptivity. The results at α2 = 0° were found to be in quantitative agreement with those of Haddad & Corke (1998), and substantiated the Strouhal number scaling based on the nose radius. The results with sound waves at angles of incidence agreed qualitatively with the analysis of Hammerton & Kerschen (1996). These included a maximum receptivity at α2 = 90°, and an asymmetric variation in the receptivity with sound incidence angle, with minima at angles which were slightly less than α2 = 0° and α2 = 180°.

scholarly journals The scattering of sound waves by a vortex: numerical simulations and analytical solutions

1994 ◽  
Vol 260 ◽  
pp. 271-298 ◽  
Author(s):  
Tim Colonius ◽  
Sanjiva K. Lele ◽  
Parviz Moin
Keyword(s):  
Boundary Conditions ◽  
Analytical Solutions ◽  
Acoustic Waves ◽  
Stokes Equations ◽  
Spatial Differentiation ◽  
Navier Stokes ◽  
Sound Waves ◽  
Navier Stokes Equations ◽  
Refraction Effect ◽  
Approximate Analytical Solutions

The scattering of plane sound waves by a vortex is investigated by solving the compressible Navier–-Stokes equations numerically, and analytically with asymptotic expansions. Numerical errors associated with discretization and boundary conditions are made small by using high-order-accurate spatial differentiation and time marching schemes along with accurate non-reflecting boundary conditions. The accuracy of computations of flow fields with acoustic waves of amplitude five orders of magnitude smaller than the hydrodynamic fluctuations is directly verified. The properties of the scattered field are examined in detail. The results reveal inadequacies in previous vortex scattering theories when the circulation of the vortex is non-zero and refraction by the slowly decaying vortex flow field is important. Approximate analytical solutions that account for the refraction effect are developed and found to be in good agreement with the computations and experiments.

INTRODUCING OF INTERNAL VISCOUS FRICTION IN EQUATIONS OF BEAM OSCILLATION AS LONG RECTANGULAR STRIP

2021 ◽  
pp. 54-58
Author(s):  
E.M. Zveriaev ◽  
Keyword(s):  
Stokes Equations ◽  
Asymptotic Series ◽  
Viscous Friction ◽  
Theory Of Elasticity ◽  
Navier Stokes ◽  
Dimensional Theory ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
The One ◽  
Beam Oscillation

Abstract. On the base of the method of simple iterations generalising methods of semi-inverse one of Saint-Venant, Reissner and Timoshenko the one-dimensional theory is constructed using the example of dynamic equations of a plane problem of elasticity theory for a long elastic strip. The resolving equation of that one-dimensional theory coincides with the equation of beam vibrations. The other problems with unknowns are determined without integration by direct calculations. In the initial equations of the theory of elasticity the terms corresponding to the viscous friction in the Navier-Stokes equations are introduced. The asymptotic characteristics of the unknowns obtained by the method of simple iterations allow to search for a solution in the form of expansions of the unknowns into asymptotic series. The resolving equation contains a term that depends on the coefficient of viscous friction.

scholarly journals Asymptotic solution of the low Reynolds-number flow between two co-axial cones of common apex

1984 ◽  
Vol 7 (4) ◽  
pp. 765-784 ◽  
Author(s):  
M. A. Serag-Eldin ◽  
Y. K. Gayed
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Asymptotic Series ◽  
Creeping Flow ◽  
Navier Stokes ◽  
Cross Sectional ◽  
Navier Stokes Equations ◽  
Spherical Coordinate ◽  
Flow Solution ◽  
Reynolds Number Flow

The paper is concerned with the axi-symmetrlc, incompressible, steady, laminar and Newtonian flow between two, stationary, conical-boundaries, which exhibit a common apex but may include arbitrary angles. The flow pattern and pressure field are obtained by solving the pertinent Navier-Stokes' equations in the spherical coordinate system. The solution is presented in the form of an asymptotic series, which converges towards the creeping flow solution as a cross-sectional Reynolds-number tends to zero. The first term in the series, namely the creeping flow solution, is given in closed form; whereas, higher order terms contain functions which generally could only be expressed in infinite series form, or else evaluated numerically. Some of the results obtained for converging and diverging flows are displayed and they are demonstrated to be plausible and informative.

On the Dynamics and Stability of Cylindrical Shells Conveying Inviscid or Viscous Fluid in Internal or Annular Flow

1991 ◽  
Vol 113 (3) ◽  
pp. 409-417 ◽  
Author(s):  
A. El Chebair ◽  
A. K. Misra
Keyword(s):  
Annular Flow ◽  
Stokes Equations ◽  
Dynamical Behavior ◽  
Internal Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Fluid Forces ◽  
Viscous Forces ◽  
The Fourier Transform ◽  
The Stability

This paper investigates theoretically for the first time the dynamical behavior and stability of a simply supported shell located coaxially in a rigid cylindrical conduit. The fluid flow is incompressible and the fluid forces consist of two parts: (i) steady viscous forces which represent the effects of upstream pressurization of the flow; (ii) unsteady forces which could be inviscid or viscous. The inviscid forces were derived by linearized potential flow theory, while the viscous ones were derived by means of the Navier-Stokes equations. Shell motion is described by the modified Flu¨gge’s shell equations. The Fourier transform technique is employed to formulate the problem. First, the system is subjected only to the unsteady inviscid forces. It is found that increasing either the internal or the annular flow velocity induces buckling, followed by coupled mode flutter. When both steady viscous and unsteady inviscid forces are applied, for internal flow, the system becomes stabilized; while for annular flow, the system loses stability at much lower velocities. Second, the system is only subjected to the unsteady viscous forces. Calculations are only performed for the internal flow case. The results are compared to those of inviscid theory. It is found that the effects of unsteady viscous forces on the stability of the system are very close to those of unsteady inviscid forces.

scholarly journals WEAK SOLUTIONS OF NAVIER–STOKES EQUATIONS CONSTRUCTED BY ARTIFICIAL COMPRESSIBILITY METHOD ARE SUITABLE

2011 ◽  
Vol 08 (01) ◽  
pp. 101-113 ◽  
Author(s):  
DONATELLA DONATELLI ◽  
STEFANO SPIRITO
Keyword(s):  
Fourier Transform ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Time Derivative ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Artificial Compressibility ◽  
Artificial Compressibility Method

We prove that weak solutions constructed by artificial compressibility method are suitable in the sense of Scheffer. Using Hilbertian setting and Fourier transform with respect to time, we obtain non-trivial estimates on the pressure and the time derivative which allow us to pass to the limit.

Linear internal waves generated by density and velocity perturbations in a linearly stratified fluid

1988 ◽  
Vol 186 ◽  
pp. 419-444 ◽  
Author(s):  
James C. S. Meng ◽  
James W. Rottman
Keyword(s):  
Stokes Equations ◽  
Rotational Velocity ◽  
Vortex Pair ◽  
Density Perturbation ◽  
Stratified Fluid ◽  
Swirl Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Flow Solution ◽  
Linear Internal Wave

A generalized theoretical analysis and finite-difference solutions of the Navier-Stokes equations of the initial-value problem are applied to obtain the linear internal wave fields generated by a density perturbation and two rotational velocity perturbations in an inviscid linearly stratified fluid. The velocity perturbations are those due to an axisymmetric swirl and a vortex pair. Solutions obtained correspond to the strong stratification limit.The theoretical results of the rotational perturbation cases show an oscillating non-propagating disturbance, which is absent in the density-perturbation case. The swirl-flow solution shows an oscillatory behaviour in both the angular momentum deposited in the fluid and in the torque exerted by the external gravitational force field. The vortex-flow solution shows a vertical ray pattern.The equi-partitioning of energy is reached at about 0.4 of a Brunt-Väisälä (B.V.) period. The potential energy-kinetic energy conversion, or vice versa, takes place between 0.15 and 0.3 B.V. periods.

Laminar flow in symmetrical channels with slightly curved walls, I. On the Jeffery-Hamel solutions for flow between plane walls

1962 ◽  
Vol 267 (1328) ◽  
pp. 119-138 ◽  
Keyword(s):  
Series Solution ◽  
Stokes Equations ◽  
Asymptotic Series ◽  
Navier Stokes ◽  
Physical Parameters ◽  
Navier Stokes Equations ◽  
Laminar Separation ◽  
Single Region ◽  
Leading Term ◽  
Asymptotic Series Solution

The Jeffery-Hamel solutions for plane, viscous, source or sink flow between straight walls are not unique. In this paper these solutions are regarded as providing the leading term of a series solution for a class of channels with walls that are nearly straight in a certain sense, but are such that the fluid is not required to emerge from, or converge on, a point. This approach suggests a further condition which the appropriate solution must satisfy, and hence leads to uniqueness in a limited domain of the physical parameters. The resulting velocity profiles include, at one extreme, the parabolic one of Poiseuille flow, and, at the other, profiles with a single region of flow reversal at each wall. The way is thus opened to an asymptotic series solution of the Navier-Stokes equations which shows laminar separation

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