Vibroacoustic Simulation. Alexander Peiffer
Чтение книги онлайн.
Читать онлайн книгу Vibroacoustic Simulation - Alexander Peiffer страница 32
Название: Vibroacoustic Simulation
Автор: Alexander Peiffer
Издательство: John Wiley & Sons Limited
Жанр: Отраслевые издания
isbn: 9781119849865
isbn:
2 Cyril M. Harris and Charles E. Crede. Shock and Vibration Handbook. McGraw-Hill, New York, NY, U.S.A., second edition, 1976. ISBN 0-07-026799-5.
3 P. A. Nelson and S. J. Elliott. Active Control of Sound. Academic, London, 1993. ISBN 0-12-515426-7.
Notes
1 1 In this book the convention ejωt for the complex harmonic function is used. Literature that deals with wave propagation often use e−jωt to have positive wavenumber for positive wave propagation. However, as in every textbook in acoustics I denote the used convention on the first page to avoid confusion.
2 Waves in Fluids
2.1 Introduction
The acoustic wave motion is described by the equations of aerodynamics that are linearized because of the small fluctuations that occur in acoustic waves compared to the static state variables. The fluid motion is described generally by three equations:
Continuity equation – conservation of mass
Newton’s law – conservation of momentum
State law – pressure volume relationship.
For lumped systems the velocity is simply the time derivative of the point mass position in space. The same approach can be used in fluid dynamics, but here the continuous fluid is subdivided into several cells and their movement is described by trajectories. This is called the Lagrange description of fluid dynamics. Even if the equation of motions are simpler in that formulation it is quite complicated to follow all coordinates of fluid volumes in a complex flow. Thus, the Euler description of fluid dynamics is used. In this description the conservation equations are performed for a control volume that is fixed in space and the flow passes through this volume.
In this chapter, the three dimensional space is given by the Cartesian coordinates x={x,y,z}T and the velocity of the fluid v={ vx, vy, vz }T.
2.2 Wave Equation for Fluids
2.2.1 Conservation of Mass
For simplicity we consider first the flow in the x-direction as in Figure 2.1. The mass flow balance contains the following quantities:
1 The elemental mass m=ρdV=ρA with A=dydz.
2 Mass flow into the volume (ρvxA)x.
3 Mass flow out of the volume (ρvxA)x+dx.
4 Mass input from external sources m˙.
Figure 2.1 Mass flow in x-direction through control volume. Source: Alexander Peiffer.
leading to equation
for mass conservation. Expanding the second term on the right hand side in a Taylor series gives
and finally
This one dimensional equation of mass conservation in x-direction can be extended to three dimensions:
The second term of (2.3) may be confusing, but it says that the change of density is not only determined by a gradient in the velocity field but also by a gradient of the density.
2.2.2 Newton’s law – Conservation of Momentum
The same procedure is applied to the momentum of the fluid. As shown in Figure 2.2 we get for flow in x-direction:
1 The momentum of the control volume is ρvxdV=ρvxAdx.
2 The momentum flow into the volume (ρvx2A)x.
3 mass flow out of the volume (ρvx2A)x+dx.
4 The force at position x is (PA)x.
5 The force at position x+dx is (PA)x+dx.
6 External volume force density fx.
Figure 2.2 Momentum flow in x-direction through control volume. Source: Alexander Peiffer.
Thus, the conservation of momentum in x reads
Using Taylor expansions for (ρvx2A)x+dx and (PA)x+dx gives