The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, BerlinPrinted on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) PrefaceThe monograph is devoted to modern mathematical models and numerical methods for solving gas-and fluid-dynamic problems based on them. Two interconnected mathematical models generalizing the Navier-Stokes system are presented; they differ from the Navier-Stokes system by additional dissipative terms with a small parameter as a coefficient. The new models are called the quasi-gas-dynamic and quasi-hydrodynamic equations. Based on these equations, effective finite-difference algorithms for calculating viscous nonstationary flows are constructed and examples of numerical computations are presented. The universality, the efficiency, and the exactness of the algorithms constructed are ensured by the fulfillment of integral conservation laws and the theorem on entropy balance for them.The book is a course of lectures and is intended for scientists and engineers who deal with constructing numerical algorithms and performing practical calculations of gas and fluid flows and also for students and postgraduate students who specialize in numerical gas and fluid dynamics. Moscow Tatiana Elizarova December 2008 v Contents

A comprehensive study of a steady axisymmetric supersonic jet of CO 2 , including experiment, theory, and numerical calculation, is presented. The experimental part, based on high-sensitivity Raman spectroscopy mapping, provides absolute density and rotational temperature maps covering the significant regions of the jet: the zone of silence, barrel shock, Mach disk, and subsonic region beyond the Mach disk. The interpretation is based on the quasi-gasdynamic (QGD) system of equations, and its generalization (QGDR) considering the translational-rotational breakdown of thermal equilibrium. QGD and QGDR systems of equations are solved numerically in terms of a finite-difference algorithm with the steady state attained as the limit of a time-evolving process. Numerical results show a good global agreement with experiment, and provide information on those quantities not measured in the experiment, like velocity field, Mach numbers, and pressures. According to the calculation the subsonic part of the jet, downstream of the Mach disk, encloses a low-velocity recirculation vortex ring.

A comprehensive numerical and experimental study of normal shock waves in hypersonic axisymmetric jets of N 2 is presented. The numerical interpretation is based on the quasi-gasdynamic (QGD) approach, and its generalization (QGDR) for the breakdown of rotational-translational equilibrium. The experimental part, based on diagnostics by high-sensitivity Raman spectroscopy, provides absolute density and rotational temperatures along the expansion axis, including the wake beyond the shock. These quantities are used as a reference for the numerical work. The limits of applicability of the QGD approach in terms of the local Knudsen number, the influence of the computational grid on the numerical solution, the breakdown of rotation-translation equilibrium, and the possible formation of a recirculation vortex immediately downstream from the normal shock wave are the main topics considered.

The one-dimensional problem of the structure of a stationary shock wave in nitrogen is solved in the frame of the Navier-Stokes ͑NS͒ equations. Proper interpretation of the bulk viscosity coefficient included in the shear stress tensor leads to a numerical solution close to the experiment, showing that the NS equations provide more accurate solutions to the problem than supposed previously.The width and the density profile of one-dimensional stationary shock waves have often been employed as a test problem for numerical models of rarefied gas flows. Previous studies on argon, helium, and nitrogen shock waves have shown the difficulties and limitations of the Navier-Stokes ͑NS͒ equations. 1-3 Experimental data for these studies are compiled in Ref. 4. Earlier calculations 5,6 have shown substantial differences with the experiments for Mach numbers MaϾ 2. This motivated several attempts towards improving NS modeling for rarefied flows, as shown in Ref. 7 and the literature in it.In a recent work, 8 it has been shown that argon shock waves up to Ma= 10 can be calculated in the NS approach with far better precision for the density profile than thought before, on the order of 30%. Argon is, however, a monatomic species with no internal degrees of freedom. Diatomic molecules pose different problems due to the internal degrees of freedom.In this Brief Communication, the one-dimensional shock wave problem is solved for molecular nitrogen, the results being compared with Alsmeyers compilation. 4 Molecular nitrogen has one vibrational and two rotational degrees of freedom. The vibrational collision number being much larger than the rotational one 9 suggests that vibration does not need to be taken into account here, to a good approximation. On the contrary, its comparatively low rotational collision number Z ϳ 5-16 ͑Ref. 1͒ in the thermal range 300ഛ T ഛ 6000 K of shock waves up to Ma= 10 suggests a more efficient transfer of energy between rotational and translational degrees of freedom, which needs to be considered in the present problem. This contribution of the internal degrees of freedom to the shock wave problem is taken here into account by means of the bulk viscosity and its effect on the NS system.A traditional shooting method to a matching point is employed for solving the steady-state NS system. This method, with specific features for the boundary conditions, converges very fast. A relaxation technique previously employed for solving the NS system 8 imposes two direct boundary conditions and delivers the same results, but converges about thousand times slower.The NS equations for stationary one-dimensional plane flow readwhere stands for the gas density, u for the velocity, and p = RT for the pressure at the temperature T, with R is the gas constant; H = ͑E + p͒ / is the total enthalpy per unit volume, E = u 2 /2+ p / ͑␥ −1͒ is the total energy per unit volume, and ␥ the specific heat ratio. The shear-stress tensor ͑⌸͒ and the heat flux vector ͑q͒ in Eqs. ͑2͒ and ͑3͒ are defined aswhere is the viscosity coefficie...

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