M. S. Cramer

On the propagation of waves exhibiting both positive and negative nonlinearity

One-dimensional small-amplitude waves in which the local value of the fundamental derivative changes sign are examined. The undisturbed medium is taken to be a Navier–Stokes fluid which is at rest and uniform with a pressure and density such that the fundamental derivative is small. A weak shock theory is developed to treat inviscid motions, and the method of multiple scales is used to derive the nonlinear parabolic equation governing the evolution of weakly dissipative waves. The latter is used to compute the viscous shock structure. New phenomena of interest include shock waves having an entropy jump of the fourth order in the shock strength, shock waves having sonic conditions either upstream or downstream of the shock, and collisions between expansion and compression shocks. When the fundamental derivative of the undisturbed media is identically zero it is shown that the ultimate decay of a one-signed pulse is proportional to the negative 1/3-power of the propagation time.

Negative nonlinearity in selected fluorocarbons

The Martin–Hou equation has been employed to compute the fundamental derivative of gasdynamics for seven commercially available fluorocarbons. Each fluid was found to have a region of negative nonlinearity large enough to include the critical isotherm. Negative nonlinearity twice as strong as that found in previous investigations is reported. In addition, estimates for the fundamental derivative at saturation are given.

Nonclassical Dynamics of Classical Gases

In the present article we examine the dynamics of single-phase, equilibrium, i.e., classical, fluids in the dense gas regime. The behavior of fluids of moderately large molecular weight is seen to differ significantly from that of air and water under normal conditions. New phenomena include the formation and propagation of expansion shocks, sonic shocks, double sonic shocks, and shock-splitting. The more complicated existence conditions for shock waves are described and related to the dissipative structure. We also give a brief description of transonic flows and show that the critical Mach number for conventional blade shapes can be increased by a factor of 30–50% for these fluids.

Steady, isentropic flows of dense gases

Steady isentropic flows of fluids in their dense gas regime are examined. It is shown that the Mach number may increase, rather than decrease, with density or pressure if the specific heats of the fluid are sufficiently large. Conditions are also reported under which isentropic expansions through converging–diverging nozzles are not possible, regardless of the imposed exit pressure. In such cases, the nozzle must be replaced with one having multiple throats. Applications to external transonic flows are briefly considered.

Exact solutions for sonic shocks in van der Waals gases

Exact closed‐form solutions for finite amplitude sonic shocks are presented for the case of a van der Waals gas having a constant specific heat. Solutions are provided for both single and double sonic shocks. Sample calculations are presented that include sonic shocks embedded in smooth inviscid flows.

Shock splitting in single-phase gases

On determine les conditions precises du dedoublement de choc et on decrit la formation de la configuration de choc dedouble a partir de conditions initiales lisses. Exemples numeriques

Shock Formation in Fluids Having Embedded Regions of Negative Nonlinearity

The steepening of one‐dimensional finite‐amplitude waves in inviscid Van der Waals gases is described. The undisturbed medium is taken to be unbounded, at rest and uniform. The specific heat is taken to be large enough to generate an embedded region of negative nonlinearity in the general neighborhood of the saturated vapor line and thermodynamic critical point. Under these conditions the shock formation process may differ significantly from that predicted by the perfect gas theory. The present study illustrates these differences for both isolated pulses and periodic wave trains. It is further shown that as many as three shocks, two compression and one expansion, may be formed in a single pulse or, in the case of wave trains, repeated element. It is also shown that the convected sound speed may become identical to the thermodynamic sound speed of the undisturbed medium at three distinct values of the density; the first of these corresponds to the density of the undisturbed medium while the other two are r...

THE DISSIPATIVE STRUCTURE OF SHOCK WAVES IN DENSE GASES

The present study provides a detailed description of the dissipative structure of shock waves propagating in dense gases which have relatively large specific heats. The flows of interest are governed by the usual Navier–Stokes equations supplemented by realistic equations of state and realistic models for the density dependence of the viscosity and thermal conductivity. New results include the first computation of the structure of finite-amplitude expansion shocks and examples of shock waves in which the thickness increases, rather than decreases, with strength. A new phenomenon, referred to as impending shock splitting, is also reported.

Numerical estimates for the bulk viscosity of ideal gases

We estimate the bulk viscosity of a selection of well known ideal gases. A relatively simple formula is combined with published values of rotational and vibrational relaxation times. It is shown that the bulk viscosity can take on a wide variety of numerical values and variations with temperature. Several fluids, including common diatomic gases, are seen to have bulk viscosities which are hundreds or thousands of times larger than their shear viscosities. We have also provided new estimates for the bulk viscosity of water vapor in the range 380–1000 K. We conjecture that the variation of bulk viscosity with temperature will have a local maximum for most fluids. The Lambert-Salter correlation is used to argue that the vibrational contribution to the bulk viscosities of a sequence of fluids having a similar number of hydrogen atoms at a fixed temperature will increase with the characteristic temperature of the lowest vibrational mode.

Nozzle flows of dense gases

Numerical solutions for steady inviscid flows in conventional converging–diverging nozzles are obtained. The fluids considered are Bethe–Zel’dovich–Thompson fluids, i.e., those having specific heats so large that the fundamental derivative of gas dynamics is negative over a finite range of pressures and temperatures. Three general classes of flow are delineated which include two nonclassical types in addition to the usual classical flows; the latter are qualitatively similar to those of perfect gases. The nonclassical flows are characterized by isentropes containing as many as three sonic points. Numerical solutions depicting finite‐strength expansion shocks, steady flows with shock waves standing upstream of the nozzle throat, and steady flows containing as many as three shock waves are presented. Flows having arbitrarily large‐exit Mach numbers are found to be possible only if a sonic expansion shock is formed in the nozzle. This observation contrasts with prediction based on the perfect gas theory whic...