Philip A. Thompson

A Fundamental Derivative in Gasdynamics

The quantity which is here called the fundamental derivative has been defined as the nondimensional form Γ≡12ρ3c4(∂2Υ/∂P2)s. The relation of Γ to other thermodynamic variables is discussed. It is already known that the existence of conventional compression shocks requires Γ>0. It is shown that other dynamic behavior of compressible fluids is fixed by the sign of Γ. Particular emphasis is given to phenomena corresponding to negative Γ. These phenomena include the area variation of a transonic passage, the form of a Prandtl‐Meyer wave, the behavior of adiabatic flow with friction, and nonlinear wave propagation. Formulas and numerical values are given for Γ in various substances.

Negative shock waves

Negative or rarefaction shock waves may exist in single-phase fluids under certain conditions. It is necessary that a particular fluid thermodynamic quantity Γ ≡ −½δ In (δP/δν)s/δ In ν be negative: this condition appears to be met for sufficiently large specific heat, corresponding to a sufficient level of molecular complexity. The dynamic formation and evolution of a negative shock is treated, as well as its properties. Such shocks satisfy stability conditions and have a positive, though small, entropy jump. The viscous shock structure is found from an approximate continuum model. Possible experimental difficulties in the laboratory production of negative shocks are briefly discussed.

Existence of Real Fluids with a Negative Fundamental Derivative Γ

The gasdynamic behavior of fluids is inverted if a certain thermodynamic property Λ ≡ ρ3c4(∂2υ/∂P2)8/2 is negative. Several different methods of calculation, based on published thermodynamic data and standard equations of state, yield Γ<0 for relatively complex organic compounds.

Wave splitting in a fluid of large heat capacity

The splitting of a single pressure discontinuity into a propagating two-wave system is studied for the case of saturated-liquid expansion (liquid-evaporation wave splitting) and vapour compression (vapour-condensation wave splitting). Experimental results from the Max-Planck-Institut fur Stromungsforschung and from Rensselaer Polytechnic Institute show that splitting occurs in test fluids of large molar heat capacity, such as iso-octane ( C v 0 / R ≈ 37). Each of the two forms of splitting results in a single-phase forerunner wave carrying a pressure discontinuity followed by a phase-change wave, also with a pressure discontinuity. The thermodynamic state between the forerunner wave and the phase-change wave is metastable (supersaturated liquid or vapour). The waves are quantitatively described by systems of adiabats, e.g. shock adiabats. It appears that nucleation processes are predominantly homogeneous. In vapour-compression shock-wave splitting, a combined wave (liquefaction shock) splits into discrete forerunner and condensation waves at a triple point, the intersection of a liquefaction shockfront, forerunner shock and condensation discontinuity: such a point occurs just at critical supersaturation (i.e. the Wilson-line state), where condensation is spontaneous and immediate. For shock waves that produce a metastable state of subcritical supersaturation, condensation is delayed, that is, the condensation discontinuity propagates more slowly; for a split-shock system, the condensation discontinuity propagates subsonically. The pressure amplitude of a real split-shock system is much larger than that predicted by an equilibrium model. In liquid-evaporation wave splitting, the forerunner wave is an acoustic expansion wave and the second wave an evaporation wave with a propagation velocity approximately determined by the Chapman-Jouguet condition for deflagration. Such evaporation wavefronts are increasingly distinct as the temperature approaches the critical-point value. The evaporation rates across the wavefront are comparable to those found in vapour explosions.

Direct observation of shock splitting in a vapor–liquid system

The splitting of a single liquefaction shock wave into a vapor‐phase forerunner shock and a following condensation shock is observed photographically in shock‐tube experiments. The behavior of the two‐shock system is qualitatively described by equilibrium shock conditions, but the state downstream of the forerunner shock is found to be far from equilibrium. It is proposed that this state is limited by the critical supersaturation for homogeneous nucleation. The overall pressure jump for the two‐shock system is much greater than would be predicted by an equilibrium calculation.

Rarefaction and Liquefaction Shock Waves in Regular and Retrograde Fluids with Near-Critical End States

Near-critical states are obtained behind reflected liquefaction shock and rarefaction waves in shock tube experiments. Thermodynamic properties C ν , c, and Γ (nonlinearity coefficient) are determined using a near-critical, linear parametric equation of state, and the possibility of rarefaction shocks are investigated based on these results. The power law behavior of C ν as predicted by the parametric EOS is incorporated into a BWR type classical EOS using a switching function. Hence a single EOS, valid over an extended temperature range, can be used in shock calculations with reasonably accurate predictions in the critical region.

Estimate of shock thickness based on entropy production

The shock thickness is estimated for a stationary shock in an ideal gas by equating the internal entropy production to the entropy increase found from the Rankine–Hugoniot equation. For elementary assumed profiles, the thickness is expressed by a simple formula. For realistic dependence of viscosity on temperature, results are in qualitative agreement with experiment.

An experimental study of reflected liquefaction shock waves with near-critical downstream states in a test fluid of large molar heat capacity

Near-critical states have been achieved downstream of a liquefaction shock wave, which is a shock reflected from the endwall of a shock tube. Photographs of the shocked test fluid (iso-octane) reveal a rich variety of phase-change phenomena. In addition to the existence of two-phase toroidal rings which have been previously reported, two-phase structures with a striking resemblance to dandelions and orange slices have been frequently observed. A model coupling the flow and nucleation dynamics is introduced to study the two-wave system of shock-induced condensation and the liquefaction shock wave in fluids of large molar heat capacity

Shock‐wave series for real fluids

Power series based on the Rankine–Hugoniot equation relate the states upstream and downstream of a shock. The usual forms for such series are limited to very weak shocks, e.g., to a shock Mach number approximately 1.1. Improvement is obtained by restricting the shock series to a form which is exact, for the special case of a perfect gas, with the lowest‐order term only. Series in this form show reasonable accuracy for real fluids up to shock Mach numbers of order 1.5.

Flow Visualization of a Shock Wave by Simple Refraction of a Background Grid

Experiments with high-molecular-weight fluids yield large absolute density changes across gas-phase shock waves, producing correspondingly large changes in the index of refraction n. In the experiments described here, the change Δn across the shock is of order 10-2. Simple photography of the shock against a ruled background grid shows a visible shockfront by refraction of the grid lines. Comparison of the photographic image with an image generated by optical ray tracing yields a measurement of the density behind the shock.