1. Preliminaries 2. Conservation equations 3. Rigid particle heat transfer at Re 4. Translation motion at Re 5. Shape deformations 6. Volume pulsations 7. Thermodynamics of suspensions 8. The two phase model 9. Sound propagation in suspensions 10. Application and extensions.

Suspension Acoustics: An Introduction to the Physics of Suspensions

A study of the structure of stationary, fully and partly dispersed, normal shock waves in steady vapour–droplet, two-phase flow is presented. Pure substances only are considered, but, unlike most previous work, the droplet population is allowed to be polydispersed. It is shown how the effects of thermal relaxation for such a mixture can be elegantly incorporated into the analysis.Three types of fully dispersed wave are identified. Type I waves are dominated by thermal relaxation and an approximate analytical solution is presented which gives results in close agreement with accurate numerical solutions of the governing equations. The analysis predicts some unexpected behaviour of the thermodynamic variables and demonstrates the correct scaling parameters for such flows. An approximate analysis is also presented for Type II waves, dominated by both velocity and thermal relaxation. Type III waves, where all three relaxation processes are important, are of little practical significance and are only briefly discussed. Partly dispersed waves are also considered and the results of a numerical simulation of the relaxation zone are presented. A linearized solution of this problem is possible but, unlike other relaxing gas flows, does not give good agreement with the more exact numerical calculations.The apparent discontinuity in the speed of sound in a vapour–droplet mixture as the wetness fraction tends to zero has been responsible for some confusion in the literature. This problem is reconsidered and it is shown that the transition from the two-phase equilibrium to the single-phase frozen shock wave speed is continuous.

Normal shock-wave structure in two-phase vapour-droplet flows

This article introduces an algorithm that determines the thermodynamic conditions behind incident and reflected shocks in aerosol-laden flows. Importantly, the algorithm accounts for the effects of droplet evaporation on post-shock properties. Additionally, this article describes an algorithm for resolving the effects of multiple-component-fuel droplets. This article presents the solution methodology and compares the results to those of another similar shock calculator. It also provides examples to show the impact of droplets on post-shock properties and the impact that multi-component fuel droplets have on shock experimental parameters. Finally, this paper presents a detailed uncertainty analysis of this algorithm’s calculations given typical experimental uncertainties.

AEROFROSH: a shock condition calculator for multi-component fuel aerosol-laden flows

The structure of the relaxation zone behind a shock wave of moderate strength in a mixture of gas, vapour and droplets is analysed. A model is presented for shock induced evaporation, which is based on wet-bulb equilibrium and on the absence of relative motion between droplets and gas. Experimental and numerical data on heterogeneous condensation induced by an unsteady rarefaction wave and on re-evaporation due to shock wave passage are reported for a mixture of water vapour, nitrogen gas and condensation nuclei. Pressure, temperature, saturation ratio and droplet size are experimentally obtained and are very well predicted by a numerical simulation based on the non-linear quasisteady wet-bulb model for phase transition, as well for the expansion wave as for the shock wave. During expansion, droplet number density decays much faster than predicted, which is not yet satisfactorily explained. Shock induced droplet evaporation is studied for post-shock saturation ratios ranging from 5×10−3 to 0.2, corresponding to shock Mach numbers of 1.2 to 1.9. The evaporation times are well predicted by the theoretical model. No evidence is found for droplet break-up for Weber numbers up to 13, and droplet radii of the order of 1μm.

Shock wave structure in a mixture of gas, vapour and droplets

A unified theory for aerodynamic and condensation shock waves in vapor‐droplet flows in the presence of an inert carrier gas is presented. Same conservation equations apply across discontinuous models for both types of wave. Exact (as well as approximate), explicit analytical jump conditions across such discontinuities are derived subject to several boundary conditions. Collectively they may be called the generalized Rankine–Hugoniot equations for vapor‐droplet mixtures. All the equations derived are general and can be applied in the case of a pure vapor‐droplet flow by letting the mass fraction of the carrier gas go to zero. Much physical insight may be obtained from this integral analysis. It is shown that four types of aerodynamic shock waves (viz., equilibrium partly dispersed, equilibrium fully dispersed, partly dispersed with complete evaporation, and fully dispersed with complete evaporation) may occur. Conditions for each type of these waves to occur are specified and the appropriate jump conditio...

A unified theory of aerodynamic and condensation shock waves in vapor‐droplet flows with or without a carrier gas

A model is presented for the droplet evaporation process induced by a shock wave propagating in a fog. The model is based on the existence of a quasi-steady wet bulb state of the droplets during evaporation. It is shown that for moderate shock strength, Ma = < 2, and droplet radii in the range of 1–5 the, the major part of the evaporation process is governed by a balance between heat conduction and vapour diffusion. The formation of a fog by means of an unsteady adiabatic expansion of humid nitrogen is described. Experimental results of shock induced evaporation are shown for shock Mach numbers from 1.2 to 2.1, droplet mass fraction of 5 · 10-3, and initial droplet radii of 1–1.4 μm. The expected linear relation between droplet radius squared and time during evaporation is observed. Characteristic evaporation times appear to be strongly dependent on shock strength. A variation of about two decades, predicted by theory, was experimentally observed for the Mach number range studied.

Shock wave induced evaporation of water droplets in a gas-droplet mixture

Heterogeneous condensation and evaporation induced by an unsteady rarefaction wave and by the passage of a shock wave in a water‐nitrogen mixture is studied. Time resolved measurements of modal droplet radius, droplets radius, droplet number density, pressure, and gas density are presented. Condensation on Cr2O3‐particles of about 10 nm starts at a saturation ratio of 3.4. Typical droplet radii and droplet number densities are 1.5 μm and 1011 m−3. A shock wave with Mach number 1.36 causes full evaporation in 4 ms.

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HJ Huib Smolders

Papers

Shock wave induced evaporation of water droplets in a gas-droplet mixture

Wave induced growth and evaporation of droplets in a vapour-gas mixture