There are wide range of wave phenomena in fluids. These phenomena are extremely important in governing many of the characteristics of the flow; they also facilitate transfer of energy, mixing, and turbulization of fluid. In this article we will discuss two forms of such waves, namely gravity and capillary waves at free surfaces and pressure waves in compressible media.
Consider a plane free liquid surface in a state of equilibrium in the gravitational field. Deviation from equilibrium brings about an oscillatory motion due to gravity or surface tension (capillarity) forces. This motion propagates along the liquid surface in the form of waves known as gravity or capillary waves. The wavelength or the length between points on the wave with identical phases remains constant.
If the oscillation amplitude in the wave is small in relation to the wavelength, then the motion is potential and in the gravitational field ρgz it is described by the equation
where p is the pressure, ρ the density, g the acceleration due to gravity, φ the velocity potential, the z axis is perpendicular to the plane liquid surface (x, y). The solution of this equation has the form of a plane harmonic wave , where ω is the angular frequency, the wave vector ( , where is the unit vector, k the wave number, λ = 2π/k the wavelength).
Equal pressures on the liquid surface and in the ambient produce a boundary condition which relates the frequency and the wave vector by a dispersion equation ω^{2} = kg. Liquid particles describe circles, with the radius exponentially decreasing deep into the liquid, about an equilibrium point. With an arbitrary ω–k relation the rate of displacement of the amplitude distribution profile in a wave is U = ∂ω/∂k (the group velocity). If the dependence of ω on k is linear, the group velocity coincides with phase velocity U = ω/k. The group velocity of a short gravity wave in liquid depends on the wavelength and is given by . If account is taken of the surface tension forces σ, the dispersion equation takes the form . The surface tension forces play a predominant role for very short waves (ripples). Another limiting case is long gravity waves the length of which is great in relation to the liquid depth h_{0}. In a viscous liquid, transverse waves may appear in which liquid particles do not move in circles but at the right angles to the wave propagation. These waves arise when a body submerged in a viscous liquid oscillates.
A disturbance of the finite amplitude on the liquid surface propagates as a simple, or Rieman, wave. The velocity of displacement of points with equal phases on the wave depends on the amplitude and the sign of the disturbance, and the wave profile is continuously deformed, i.e., the point with a greater amplitude propagates faster, the wave crest catches up with its foot, the steepness of the front slope enhances and at a certain critical height the wave falls down to form capillary ripples or white caps (Figure 1). In the areas with an extremely steep profile, dispersion and dissipation have a pronounced effect. Dissipation facilitates smoothing of saw-tooth profile of the wave. Stationary waves that are a sequence of short pulses (Figure 2) may arise as a result of competing nonlinear effects, "overturning" of the wave profile, and dispersion effects, contributing to broadening of the profile (Figure 2). A particular case of a series of pulsating waves with an infinitely large period is a solitary wave known also as soliton.
Internal gravity waves may also arise in a stratified liquid. If two liquid layers move slipping one on the other, their interface is a tangential discontinuity because the liquid velocity tangential to the surface changes abruptly. The disturbance of the interface may bring about an interfacial instability ( Kelvin-Helmholtz Instability).
An incompressible fluid uniformly rotating as a whole may give rise to internal waves due to Coriolis forces (inertia! waves).
Pressure waves appear in a compressible liquid. Pressure perturbation in a compressible fluid involves perturbation of density, velocity, and other parameters. In this case the velocity potential φ satisfies the wave equation ∂^{2}φ/∂t^{2} − a^{2}Δφ = 0 which describes propagation of the wave with the velocity of sound (an acoustic wave). The phase velocity of an acoustic wave U = a does not depend on the disturbance amplitude and is determined by the thermodynamic properties of the liquid: a^{2} = (∂p/∂ρ)_{s} , where p is the pressure, ρ the density, and S the entropy. In a perfect gas, , where γ is the polytropic exponent, R the universal gas constant, T the temperature, and the molecular weight.
A disturbance of the finite amplitude propagates in compressible fluid as a simple, or Rieman, wave. In contrast to an acoustic wave, the velocity of each point of profile U in the finite-amplitude wave depends on the disturbance amplitude U = v ± a(v), where v is the velocity of gas particles in the wave, v in an oscillatory process takes a positive, zero, or a negative value. The density variation also changes sign (Figure 3). This circumstance results in the profile deformation as the wave propagates, the rarefaction points lagging behind the compression points, and an ambiguity arising, i.e., at the same point density, velocity, and other parameters must have three different values. Physically the ambiguity gives rise to a discontinuity, known as a shock wave, approaching which from the left and from the right the density is single-valued. The discontinuity displaces in space and attenuates if the velocity of gas flow behind it is not kept constant by an appropriate boundary condition. Propagating through the surface of a shock wave the gas parameters change sharply (see Compressible flows and Shock tubes). The values of gas parameters behind the shock wave front in a perfect gas can be conveniently related to the parameters before the front in terms of the Mach number of the shock wave Ma_{0} = u_{0}/a_{1}, where u_{0} is the velocity of the shock wave front with respect to an undisturbed gas and a_{1} the velocity of sound in the undisturbed gas
The above relations show that at high shock wave velocities (Ma_{0} → ∞), and is independent of γ, ρ_{2}/ρ_{1} ≈ (γ + 1)/(γ − 1) and does not depend on Ma_{0}. As Ma_{0} → ∞, T_{2}/T_{1} increases as square of the Mach number. However, the substantial temperature elevation behind the shock wave is not attained because in an imperfect gas, as the translational temperature grows, part of energy is expended on excitation of molecule vibrations and dissociation. In this case, heat capacity depends on temperature, and enthalpy is calculated by the physics methods of statistics. Figure 4 demonstrates the temperature T_{2} behind the shock wave versus the shock wave velocity u_{0} in air: for a perfect gas (curve 1) taking into account the excitation of molecule vibrations without dissociation (curve 2), and taking into account the dissociation at initial pressures of p_{1} = 1, 10^{−3}, 10^{–4}, and 10^{–5} atm, respectively (curves (3, 4, 5, 6).
The gas parameters behind the shock wave can be calculated by the formulas for a perfect gas if γ_{2} is assigned the value γ* such that the relation between enthalpy, pressure, and density takes the same form as that for a perfect gas: h_{2} = [γ*/(γ* – 1)](p_{2}/ρ_{2})·γ* is equal to the ratio of enthalpy to internal energy at a temperature T_{2}. The values of γ* and the molecular weight for air can be approximated by the formulas
In a medium, in which (∂^{2}V/∂p^{2}) is positive (where V = 1/ρ), the entropy grows in transition across the shock compression wave front and must have dropped in transition across the rarefaction wave front. This suggests that in most media rarefaction shocks do not exist and in the rarefaction wave entropy is constant. The head of the stationary rarefaction wave propagates with the velocity of sound, while the shock wave front may propagate with a velocity tens of times higher than the velocity of sound (depending on the boundary conditions).
With instantaneous release of energy a blast wave, that is a shock wave followed by the rarefaction wave, appears in a compressible fluid (Figure 5). The velocity and intensity of a blast wave fall with the distance from the point of energy release. The most rapid drop of the shock wave velocity and the pressure behind it is observed in a spherical blast wave.
A shock wave propagating in a medium capable of exothermic reaction is called a detonation wave. In contrast to the shock wave, the detonation wave propagates with a constant velocity due to release of the energy of chemical reaction of the front. The stationary regime of propagation of the detonation wave is established if its velocity with respect to the gas behind it is equal to the local velocity of sound (the Chapman-Jouget condition).
REFERENCES
Whitham, G. B. (1974) Linear and Nonlinear Waves, Wiley.
Korobeinikov, V. P. Ed. (1989) Unsteady Interactions of Shock and Detonation Waves in Gases, Hemisphere, New York.
Referências
- Whitham, G. B. (1974) Linear and Nonlinear Waves, Wiley.
- Korobeinikov, V. P. Ed. (1989) Unsteady Interactions of Shock and Detonation Waves in Gases, Hemisphere, New York.