On the basis of previous work by the author, equations are derived describing one-dimensional unsteady flow in bubble-fluid mixtures. Attention is subsequently focused on pressure waves of small and moderate amplitude propagating through the mixture. Four characteristic lengths occur, namely, wavelength, amplitude, bubble diameter and inter-bubble distance. The significance of their relative magnitudes for the theory is discussed. It appears that for high gas content the dispersion is weak and then the conservation of mass and momentum lead to equations similar to the Boussinesq equations, describing long dispersive waves of finite amplitude on a fluid of finite depth. For waves propagating in one direction only, the corresponding equation is similar to the Korteweg–de Vries equation.It is shown that for mixtures of low gas content the frequency dispersion is in most cases not small. Finally, solutions of the Korteweg–de Vries equation representing cnoidal and solitary waves in a bubble–liquid mixture are given explicitly.
We conduct experiments in a turbulent bubbly flow to study the nature of the
transition between the classical $-$5/3 energy spectrum scaling for a
single-phase turbulent flow and the $-$3 scaling for a swarm of bubbles rising
in a quiescent liquid and of bubble-dominated turbulence. The bubblance
parameter, which measures the ratio of the bubble-induced kinetic energy to the
kinetic energy induced by the turbulent liquid fluctuations before bubble
injection, is often used to characterise the bubbly flow. We vary the bubblance
parameter from $b = \infty$ (pseudo-turbulence) to $b = 0$ (single-phase flow)
over 2-3 orders of magnitude ($0.01 - 5$) to study its effect on the turbulent
energy spectrum and liquid velocity fluctuations. The probability density
functions (PDFs) of the liquid velocity fluctuations show deviations from the
Gaussian profile for $b > 0$, i.e. when bubbles are present in the system. The
PDFs are asymmetric with higher probability in the positive tails. The energy
spectra are found to follow the $-$3 scaling at length scales smaller than the
size of the bubbles for bubbly flows. This $-$3 spectrum scaling holds not only
in the well-established case of pseudo-turbulence, but surprisingly in all
cases where bubbles are present in the system ($b > 0$). Therefore, it is a
generic feature of turbulent bubbly flows, and the bubblance parameter is
probably not a suitable parameter to characterise the energy spectrum in bubbly
turbulent flows. The physical reason is that the energy input by the bubbles
passes over only to higher wave numbers, and the energy production due to the
bubbles can be directly balanced by the viscous dissipation in the bubble wakes
as suggested by Lance $\&$ Bataille (1991). In addition, we provide an
alternative explanation by balancing the energy production of the bubbles with
viscous dissipation in the Fourier space.Comment: J. Fluid Mech. (in press
Equations of motion correct to the first order of the gas concentration by volume are derived for a dispersion of gas bubbles in liquid through systematic averaging of the equations on the microlevel. First, by ensemble averaging, an expression for the average stress tensor is obtained, which is non-isotropic although the local stress tensors in the constituent phases are isotropic (viscosity is neglected). Next, by applying the same technique, the momentum-flux tensor of the entire mixture is obtained. An equation expressing the fact that the average force on a massless bubble is zero leads to a third relation. Complemented with mass-conservation equations for liquid and gas, these equations appear to constitute a completely hyperbolic system, unlike the systems with complex characteristics found previously. The characteristic speeds are calculated and shown to be related to the propagation speeds of acoustic waves and concentration waves.
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