We report homogeneous ice nucleation rates between 202 K and 215 K, thereby reducing the measurement gap that previously existed between 203 K and 228 K. These temperatures are significantly below the homogenous freezing limit, TH ≈ 235 K for bulk water, and well within no-man's land. The ice nucleation rates are determined by characterizing nanodroplets with radii between 3.2 and 5.8 nm produced in a supersonic nozzle using three techniques: (1) pressure trace measurements to determine the properties of the flow as well as the temperature and velocity of the droplets, (2) small angle X-ray scattering (SAXS) to measure the size and number density of the droplets, and (3) Fourier Transform Infrared (FTIR) spectroscopy to follow the liquid to solid phase transition. Assuming that nucleation occurs throughout the droplet volume, the measured ice nucleation rates Jice,V are on the order of 1023 cm−3 s−1, and agree well with published values near 203 K.

Freezing water in no-man's land

Steady two-dimensional flow of vapor/carrier gas mixtures (moist air) with nonequilibrium condensation is investigated in theory and experiment. Aside from Laval nozzles, transonic flow over airfoils at M<x ^ 1 is considered. The numerical calculation is based on the Euler equation linked with the classical theory of homogeneous nucleation and droplet growth using a new diabatic, time-dependent, explicit, finite volume method. By means of this model, the effects of energy supply in local supersonic flow over airfoils are investigated including a detailed analysis of the two-dimensional structure in the diabatic case. The variation of the pressure drag coefficient due to the heating is the sum of the following connected processes: the reduction of the wave drag in the local supersonic area producing a higher Mach number behind the shock, the shock shifting, and the pressure increase in the rear section of an airfoil due to evaporation. Both airfoil series investigated, circular arc and NACA-0012, show the same tendency concerning the shifting of the normal shock. But the pressure drag coefficients vary in opposite directions with different supply conditions.

Transonic flow around airfoils with relaxation and energy supply by homogeneous condensation

A special shock tube process combining a reflected expansion wave with a weak shock wave is analyzed and calibrated. The process is employed to transfer water vapor carried in argon into a known supersaturated state for a short period of time (0.5 ms). During that period steady state homogeneous nucleation takes place followed by condensational growth. Nucleation and growth rates are measured by a 90° Mie-light scattering technique in the temperature range 200–260 K. The results are compared with existing theoretical models.

Nucleation and growth rates of homogeneously condensing water vapor in argon from shock tube experiments

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

A detailed analysis of supercritical transonic nozzle flows with stationary normal shock waves is presented. A classification scheme based on the normal shock location is obtained using asymptotic methods, and four distinct supercritical flow regimes are distinguished. A simple shock fitting technique that determines the shock location within any desired precision is introduced. Consequently, an algorithm that exhibits the asymptotic solution in each supercritical flow regime is developed for the expansion of moist air in nozzles with atmospheric supply conditions by utilizing the classical nucleation theory and the Hertz–Knudsen droplet growth law. Good agreement with the recent static pressure measurements and visualized shock locations is achieved in relatively slender nozzles when the condensed phase is assumed to consist purely of water drops.

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Asymptotic solution of transonic nozzle flows with homogeneous condensation. I. Subcritical flows

A generalized form of the similarity law for the condensation onset Mach number of water vapor in air in the transonic and supersonic range for water vapor flow in moist air is derived from well known basic approaches for supersonic nozzles. These statements are confirmed by extensive experimental investigations in Laval nozzles, as well as by results of other authors and computations on the basis of the Euler equation linked with the classical theory of nucleation and droplet growth. In this experimental research priority is given to the qualitative description of the two-dimensional condensation processes, and their effects in transonic flows in nozzles of different geometrical configuration (e. g. slightly or well curved). A quantitative discussion of 2-D structures in condensation regions requires the introduction of a characteristic angle along streamlines. It is then directly possible to describe the different types of compression disturbances in supersonic flows with heat addition.

G. Schnerr

Papers

2-D transonic flow with energy supply by homogeneous condensation: onset condition and 2-D structure of steady Laval nozzle flow