Laminar boundary layer in flow of a nonequilibrium ionized gas

The motion of a hypersonic body is accompanied by an increase in the gas temperature in the boundary layer up to tens of thousands of degrees, which causes the gas to ionize. Under these conditions there are problems in calculating coefficients of viscosity, diffusion, and heat conduction. Investigations have shown that it is invalid to extrapolate the widely used approximations for transport coefficients in the high temperature region [1–3]. This paper considers the laminar boundary layer in the vicinity of the stagnation point of a blunt body in a stream of monatomic nonequilibrium ionized gas. The main thrust is a more accurate calculation of transport coefficients and an investigation of their effect on profiles of the gasdynamic parameters. A specific calculation is performed for argon by way of example.     

Breakdown of the near-electrode layer in a flow of ionized gas

An electric discharge in a flow of ionized gas is widely used in many physics and engineering problems. Among them are problems associated with current flow in various magnetohydrodynamic devices (generators, accelerators), arc shunting in a plasmatron, physical experiments in shock tubes, etc. It is known that with cold electrodes providing the contact between the plasma and the external circuit and relatively high pressures, two modes of current flow occur: at low current, the discharge is of a distributed nature; as the applied voltage increases, the discharge abruptly shifts into a discharge with a clearly developed cathode spot at some critical current density (we call this form of discharge an arc discharge). Existing experimental data [1–20] refers to varying experimental conditions. Furthermore, the critical voltage (or current) at which the transition of the discharge from a distributed discharge to an arc discharge occurs varies within very broad limits. From an analysis of the experimental data, a condition is formulated which the discharge parameters satisfy at the time of transition from a distributed discharge to an arc discharge.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 16–23, May–June, 1973.The author thanks Yu. A. Nikuev for invaluable help in the analysis of experimental data.     

Low speed MHD Couette flow of a slightly rarefied and electrically conducting gas

The effects of an applied magnetic field on the steady, laminar, low speed plane Couette flow of a slightly rarefied and electrically conducting gas are studied. Consideration is given to the slip-flow regime, wherein the gas rarefaction begins to play its important role. The generally accepted method of analysis for slip flows is utilized, i.e. the continuum magnetohydrodynamic equations of motion are used throughout the gas, together with the first and the second order slip velocity and temperature jump boundary conditions. Considerations are further given to (1) the case of zero electric field and (2) the case of a nonconducting channel in which the net current across the channel is zero.     

Asymptotic theory of neutral stability of the Couette flow of a vibrationally excited gas

An asymptotic theory of the neutral stability curve for a supersonic plane Couette flow of a vibrationally excited gas is developed. The initial mathematical model consists of equations of two-temperature viscous gas dynamics, which are used to derive a spectral problem for a linear system of eighth-order ordinary differential equations within the framework of the classical linear stability theory. Unified transformations of the system for all shear flows are performed in accordance with the classical Lin scheme. The problem is reduced to an algebraic secular equation with separation into the “inviscid” and “viscous” parts, which is solved numerically. It is shown that the thus-calculated neutral stability curves agree well with the previously obtained results of the direct numerical solution of the original spectral problem. In particular, the critical Reynolds number increases with excitation enhancement, and the neutral stability curve is shifted toward the domain of higher wave numbers. This is also confirmed by means of solving an asymptotic equation for the critical Reynolds number at the Mach number M ≤ 4.     

Numerical study of the generalized cylindrical Couette flow of rarefied gas

We study the cylindrical Couette flow of a rarefied gas between two cylinders in the generalized setup in which the inner of which not only rotates but also slides along its axis. The analysis is based on the numerical solution of the S-model kinetic equation. The influence of ratio of cylinder radiuses, velocities of the inner cylinder and Knudsen number on shear stresses, mass-flow rates as well as macroscopic parameters is investigated in the broad range of Knudsen numbers.     

Slip boundary conditions on a catalytic surface in a multicomponent gas flow

The boundary conditions for the velocity slip and temperature and concentration jumps on the surface of a body in a rarefied multicomponent gas flow are obtained. The mathematical treatment is given in detail because of the need to examine critically some previous results which disagree with each other in spite of the fact that the initial premises and the methods of solution were the same. The results of this study, which are given in a convenient form, represent the boundary conditions for both the simplified and the complete Navier-Stokes equations in problems of hypersonic rarefied gas flow past bodies with a catalytically active surface.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 159–168, January–February, 1996.     

Calculation of the chemically nonequilibrium flow of a multicomponent gas through a nozzle

We shall describe a method for calculating the flow of a complex gas mixture not in chemical equilibrium, based on selection of the time scale for the problem solution and sectioning of the reaction velocities in the near-equilibrium mode of their flow. The method permits derivation of calculation programs possessing wide applicability for systems with a large number of reactions. The application of the method is illustrated through calculations on a system, the components of which contain H,O,C,N, and Cl elements. Calculation of the parameters of the flow of a multicomponent gas through a nozzle under given conditions requires consideration of the kinetics of the chemical reactions which occur in the system. Mathematically, such a problem leads to the simultaneous numerical solution of the gas-dynamics equations for the flow parameters and the chemical kinetic equations for the concentration of the individual components. Several difficulties exist, the most basic of which are calculation of the region of near equilibrium flow and the transonic flow region with transition across the sonic point, and calculation of a large number of chemical reaction velocities of greatly differing magnitudes, in the case of a complex gas mixture. In order to obtain a stable solution in the near-equilibrium flow region, several methods have recently been proposed, which permit consolidation of the integration step. We note the use of a local linearization of the chemical kinetic equation system, as employed in [1]. This method in practice is useful for relatively slow change in component concentrations. In [2] at each integration step the kinetic equations are transformed into a system of L algebraic equations (where L is the number of reactions), and with an increase in the number of reactions (L20) the laboriousness of such a calculation increases sharply. The implicit differential schemes of integration presented in [3, 4] appear more acceptable, but in fact they too have been tested only for systems with a relatively small number of reactions. The difficulty of calculating the transonic flow region, as is well known, is connected with selection of the unique value of mass flow G, at which the transition to supersonic flow is realized. This may be avoided by defining over the length of the nozzle one of the gas-dynamic functions (for example, pressure distribution [4]), which are not highly sensitive to chemical nonequilibrium, the values being taken from supplementary calculations of nonequilibrium flow through the nozzle. Several investigators have limited their examination to the supersonic flow region (see, for example, [5]), but with this method the results may lack sufficient accuracy, since in some cases (for high gradients of the gas-dynamic magnitudes) the transonic region produces a comparable contribution to the general effect of nonequilibrium. We will describe below a method with which a practically universal system of calculating the nonequilibrium flow of a complex gas mixture through a nozzle can be realized. In practice, up to 60 of the most significant reactions may be considered, out of a practically unlimited number initially present. The method is based on sectioning of the reaction velocities in the near-equilibrium mode. This permits attainment of a stable solution in the near-equilibrium flow region with acceptable machine-time expenditures. The method describes the transition through the sonic point, since the systematic error introduced by its use (within the limits of calculation accuracy) improves the convergence of the iteration process used in finding G. In applying the method it is useful to select the more important of those reactions theoretically possible, and also to conduct calculations for equilibrium flow conditions of the individual chemical reactions; the latter permits evaluation of the maximum possible contribution of reactions for which the velocity constants are unknown.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 159–163, September–October, 1971.The authors are grateful to A. I. Vol'pert and L. N. Stesik for their interest in and evaluation of the study.     

Unsteady Couette flow in a rotating system

The unsteady Couette flow between two infinite horizontal plates induced by the non-torsional oscillations of one of the plates in a rotating system under the boundary layer approximations is investigated. An exact solution of the governing equations has been obtained by using Laplace transform technique. It is shown that when the oscillating plate situated at an infinite distance from stationary plate then the problem reduces to the unsteady boundary layer problem in a rotating system with non-torsional oscillations of the free-stream velocity.     

Unsteady Couette flow in a rotating system

We have studied the unsteady Couette flow of a viscous incompressible fluid confined between parallel plates, rotating with an uniform angular velocity about an axis normal to the plates. The flow is induced by the motion of the upper plate and the fluid and plates rotate in unison with the same constant angular velocity. An exact solution of the governing equations have been obtained for small and large time $\tau$ by applying Laplace transform technique. It is found that the primary velocity decreases with increase in rotation parameter for small as well as large time. It is interesting to note that a back flow occurs in the region $0.0⩽\eta ⩽0.7$ for large time with increase in K when $K=4$ and 5. The secondary velocity increases in magnitude for small time with increase in rotation parameter. It is observed that the secondary velocity increases in magnitude for small values of rotation parameter. On the other hand, for large values of rotation parameter $K^2$, it decreases near the stationary plate and increases near the moving plate. The shear stress due to primary flow decreases with increase in rotation parameter $K^2$. On the other hand, it increases due to secondary flow with increase in rotation parameter for small time. It is noticed that for large time there exists separation in the primary and secondary flows due to high rotation.     

Linear stability of the Couette flow of a vibrationally excited gas. 2. viscous problem

Based on the linear theory, stability of viscous disturbances in a supersonic plane Couette flow of a vibrationally excited gas described by a system of linearized equations of two-temperature gas dynamics including shear and bulk viscosity is studied. It is demonstrated that two sets are identified in the spectrum of the problem of stability of plane waves, similar to the case of a perfect gas. One set consists of viscous acoustic modes, which asymptotically converge to even and odd inviscid acoustic modes at high Reynolds numbers. The eigenvalues from the other set have no asymptotic relationship with the inviscid problem and are characterized by large damping decrements. Two most unstable viscous acoustic modes (I and II) are identified; the limits of these modes were considered previously in the inviscid approximation. It is shown that there are domains in the space of parameters for both modes, where the presence of viscosity induces appreciable destabilization of the flow. Moreover, the growth rates of disturbances are appreciably greater than the corresponding values for the inviscid flow, while thermal excitation in the entire considered range of parameters increases the stability of the viscous flow. For a vibrationally excited gas, the critical Reynolds number as a function of the thermal nonequilibrium degree is found to be greater by 12% than for a perfect gas.     

Nonuniform convective Couette flow

An exact solution describing the convective flow of a vortical viscous incompressible fluid is derived. The solution of the Oberbeck–Boussinesq equation possesses a characteristic feature in describing a fluid in motion, namely, it holds true when not only viscous but also inertia forces are taken into account. Taking the inertia forces into account leads to the appearance of stagnation points in a fluid layer and counterflows, as well as the existence of layer thicknesses at which the tangent stresses vanish on the lower boundary. It is shown that the vortices in the fluid are generated due to the nonlinear effects leading to the occurrence of counterflows and flow velocity amplification, compared with those given by the boundary conditions. The solution of the spectral problem for the polynomials describing the tangent stress distribution makes it possible to explain the absence of the skin friction on the solid surface and in an arbitrary section of an infinite layer.     

Linear stability of the Couette flow of a vibrationally excited gas. 1. Inviscid problem

Stability of the Couette flow of a vibrationally excited diatomic gas with a parabolic profile of static temperature is studied within the framework of the linear theory. A set of explicit asymptotic estimates are obtained for inviscid disturbances described by a system of linearized equations of two-temperature gas dynamics. It is shown that the first Rayleigh condition (theorem) is satisfied for unstable modes, and the classification of inviscid modes into even and odd modes is valid. A generalized condition of the presence of an inflection point on the velocity profile, which is necessary for disturbances to evolve, is obtained. The sufficient condition in Howard’s semi-circle theorem is refined. Complex phase velocities of two-dimensional even and odd inviscid modes are numerically calculated as functions of the Mach number, degree of excitation of vibrational levels of energy, and characteristic relaxation time. In the Couette flow problem, in contrast to the case of a free shear layer, the growth rate of the most unstable second mode increases with increasing Mach number and tends to a certain limit for which an asymptotic expression in the form of an ordinary differential equation is obtained. The calculated results show that the effect of reduction of the growth rate on the background of the relaxation process is clearly expressed in the range of flow parameters considered.     

Couette flow of granular materials

The flow of granular materials between rotating cylinders is studied using a continuum model proposed by Rajagopal and Massoudi (A method for measuring material moduli for granular materials: flow in an orthogonal rheometer, DOE/PETC/TR90/3, 1990). For a steady, fully developed condition, the governing equations are reduced to a system of coupled non-linear ordinary differential equations. The resulting boundary value problem is non-dimensionalized and is then solved numerically. The effect of material parameters, i.e., dimensionless numbers on the volume fraction and the velocity fields are studied.     

3D-PTV measurements in a plane Couette flow

Genuine plane Couette flow is hard to realize experimentally, and no applications of modern spatially resolving measurement techniques have been reported for this flow so far. In order to resolve this shortcoming, we designed and built a new experimental facility and present our first results here. Our setup enables us to access the flow via 3D particle tracking velocimetry and therefore to obtain truly three-dimensional flow fields for the first time experimentally in plane Couette flow. Results are analyzed in terms of basic flow properties, and a clear distinction of flow regimes (laminar for Re < 320, transitional for 320 < Re < 400, and turbulent when Re > 400) could be made. Comparison with DNS data shows good agreement in the turbulent regime and builds trust in our data. Furthermore, vortical coherent structures are studied in detail with the additional help of kalliroscope imaging, and the typical vortex spacing is determined to be roughly one gap width. As a noteworthy result, we find that the onset of the turbulent regime coincides with the range of Reynolds numbers at which a distance of 100 wall units is comparable to the gap width.