Stability of two-layer fluid flow

The problem of two-layer convective flow of viscous incompressible fluids in a horizontal channel with solid walls in the presence of evaporation is considered in the Oberbeck–Boussinesq approximation assuming that the interface is an undeformable thermocapillary surface and taking into account the Dufour effect in the upper layer which is a mixture of gas and liquid vapor. The effects of longitudinal temperature gradients at the boundaries of the channel and the thicknesses of the layer on the flow pattern and the evaporation rate are studied under conditions of specified gas flow and the absence of vapor flow on the upper boundary of the channel. It is shown that the long-wavelength asymptotics for the decrement is determined from the flow characteristics, the longwavelength perturbations occurring in the system decay monotonically, and the thermal instability mechanism is not potentially the most dangerous.     

The turbulent boundary layer of dissociated gas in the initial section of a tube

Many theoretical and experimental papers [1–4] have been devoted to investigating the turbulent boundary layer in the initial section of a channel. For the most part, however, the flow of an incompressible fluid with constant parameters is considered. There are many practical cases in which it is of interest to treat the development of the turbulent boundary layer of gas in the initial section of a pipe when conditions are strongly nonisothermal. A solution of a problem of this type, based on the theory of limit laws, is given in paper [1]. The present article extends this solution to the case of the flow of a high-enthalpy gas when the effect of gas dissociation on the turbulent boundary layer characteristics must be taken into account. We shall consider the flow of a mixture of i gases which is in a frozen state inside the boundary layer, and in an equilibrium state on its boundaries. Formulas are derived for the laws of friction and heat exchange, and a solution is given for the turbulent boundary layer equations in the initial section of the pipe when the wall temperature is constant and the gas flows at a subsonic velocity.Finally the authors are grateful to S. S. Kutateladze for discussing the paper.     

Numerical solution of the system of one-dimensional unsteady Navier-Stokes equations for a compressible gas

In many problems encountered in modern gasdynamics, the boundary layer approximations are inadequate to account for the dissipative factors-viscosity and thermal conductivity of the gas-and the solution of the complete system of Navier-Stokes equations is required. This includes, for example, flows with large longitudinal pressure gradients, which in order of magnitude are comparable with or exceed the transverse gradients (temperature jumps, sharp flow rotations, compression shocks, etc.). In many cases, for example in flows with low density, the scale of action of the longitudinal gradients becomes significant, which leads to the need for considering the flow structure in the vicinity of the large gradients. The formulation of certain problems of this type leads to a system of one-dimensional Navier-Stokes equations.We present a difference scheme for the solution of the system of one-dimensional stationary and nonstationary Navier-Stokes equations and give examples of the calculation of the structure of the stationary shock wave front, unsteady gas flow under the influence of sudden heating of one of the boundaries, and unsteady gas flow in the vicinity of the decay of an initial discontinuity. The solution of the stationary problems is accomplished as a result of stabilization as t .The author wishes to thank V. Ya. Likhushin and V. S. Avduevskii for interest in the study and for their valuable counsel during the investigation.     

On the stability of certain nonparallel flows of a viscous incompressible liquid in a channel

The stability of very simple nonparallel flows of a viscous incompressible liquid in an infinite plane channel described by the exact solutions of the Navier-Stokes equations is studied. Such solutions are realized between two parallel porous plates when the liquid (or gas) is forced in at one wall and drawn out at the same velocity at the other, with a steady flow of liquid along the channel. In this case the transverse velocity component is constant, and the profile of the longitudinal velocity component is independent of the longitudinal con-ordinate x, being an asymmetric function of the transverse coordinate y. A study of the hydrodynamic stability then reduces to the solution of an equation differing from the Orr-Sommerfeld equation by virtue of the presence of additional terms containing the transverse velocity component of the main flow. By numerically solving both this equation and the ordinary Orr-Sommerfeld equation and comparing the corresponding results for various inflowing Reynolds numbers R₀=v₀h/ (v₀ is the inflow velocity, h is the width of the channel), the effect of the nonparallel and asymmetrical nature of such flows on their stability is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 125–129, July–August, 1970.     

Effect of a magnetic field on the flow in the vicinity of the stagnation point of a blunt body with ablation of a protective layer

During hypersonic flow around a blunt-nosed body, the gas which passes through the bow shock is heated to high temperatures, where dissociation, ionization, and inverse phenomena (recombination) take place in the gas. If an ionized gas moves in a magnetic field, the ponderomotive force which is set up changes the nature of its motion close to the stagnation point, decreasing the frictional stress and heat transfer at the wall (at the contact surface of the gas and the body about which the gas flows). In this case, the intense heat fluxes from the strongly heated gas to the body about which the gas flows cause phase changes in the surface of the body (melting, sublimation, etc.). These processes, in turn, affect the flow in the vicinity of the stagnation point due to realization of the heat of phase transition, the conduction of heat from the entrained mass, and the diffusion of evaporating material into the boundary layer. References [1, 2] are devoted to a study of the joint influence of the magneto-gasdynamic and ablation effects. The magnetogasdynamic layers and the wall profile of the external velocity (flow around wedges) are discussed in [1], and special cases of such boundary layers-flow close to the stagnation line (the two-dimensional case) and close to the stagnation point (the axisymmetric case) of a blunt body are considered in [2]. Melting and evaporation are taken into account by setting the longitudinal and the transverse velocity components at the wall not equal to zero-the first taking into account the flow of the molten material and the second pyrolysis of the vapor of the surface material into the gaseous boundary layer. However, the values of these components, also the enthalpy on the wall h_w(in[1, 2] h_W 0), are not known beforehand and must be determined from the boundary conditions at the wall which express the mass and heat balances. The general formulation of the problem given in the gasdynamics case by G. A. Triskii in [3, 4], and elsewhere includes a consideration of the boundary-layer equations in the gas, the boundary-layer equations in the melted zone, and the heat conductivity equations in the solid with boundary conditions at the outer edge of the boundary layer, on the gas-molten zone interface, on the molten zone-solid interface, and inside the solid. This approach to the problem can also be utilized in the magnetogasdynamic case, as it is in this article with certain simplifying assumptions as compared with [3, 4]. In this sense, the present article is an extension of the results of [3, 4] to the field of magnetogasdynamics.In conclusion, the author thanks K. A. Lur'e for proposing the subject and useful discussions.     

Some self-similar viscous gas flows in a channel

We consider the class of self-similar axisymmetric and two-dimensional laminar flows of a viscous gas in a long channel with smooth contour, in which the longitudinal component of the velocity and the gas temperature are functions of a single dimensionless transverse coordinate. Such flows correspond to exponential (axisymmetric flow) or linear (two-dimensional flow) increase of the radius or height of the channel and corresponding exponential or hyperbolic decrease of the static pressure along the channel.     

Numerical study of boundary layer transition in flowing film evaporation on horizontal elliptical cylinder

A numerical study of the onset of longitudinal transition between turbulent and laminar regimes during the evaporation of a water film is presented. These water film streams along a horizontal elliptical tube under the simultaneous effects of gravity, pressure gradients, caused by the vapor flow and curvature, and viscous forces. At the interface of water vapor, the shear stress is supposed to be negligible. Outside the boundary layer, the vapor phase velocity is obtained from potential flow. In the analysis Von Karmans turbulence model is used and the inertia and convection terms are retained. Transfers equations are discretised by using the implicit Keller method. The effects of an initial liquid flow rate per unit of length, Froude number, temperature difference between the wall and the liquid–vapor interface and ellipticity on the transition position have been evaluated. The transition criterion has been given in term of the critical film Reynolds number (Re)_C.     

On the steady motion of two immiscible fluids in an undulating porous reservoir

We describe steady two-dimensional flows of two immiscible fluids through an undulating porous medium of constant thickness, with impermeable or slightly permeable boundaries. Flows in the same or opposite directions are called, respectively, direct or counter flows. Three special classes of flow are determined:

1. The pressure dominated case occurs for high direct flows and has the interface approximately a constant vertical distance from the impermeable boundaries.
2. The gravity dominated case occurs for low direct flows and has the interface very close to the lower (upper) boundary for downward (upward) sloping boundaries except at crossovers.
3. Counter flows require the interface to decrease in the direction of flow of the lower fluid.

Numerical examples illustrate the three classifications above. For incompressible flows the interface and pressure equations uncouple. A stability analysis shows that the direction of integration of the differential equation for the interface must be opposite to the flow direction for direct flows; for counter flows the direction of integration depends on whether the interface is above or below a critical height. Direct flows through cyclic geometries are asymptotically cyclic upstream. If the reservoir is ‘leaky’, asymptotically self-similar flows result when the (small) permeability ratio is scaled to the dynamical flow parameters.     

Asymptotic Solutions of Problems of One-Dimensional Time-Dependent Combustible Gas Flows in the Presence of Thermal Action

In a development of studies [1, 2], asymptotic solutions of the Navier-Stokes equations are found for one-dimensional combustible gas flows in the presence of various forms of thermal action on a moving surface (x=x _w(t)). In the problems considered, the temperature or the heat flux q _w(t) is specified on the surface or the surface is the interface between a combustible gas and a moving heated piston or another gas (for example, in a shock tube). Use is made of the fact that, as t , in many cases the values of v _w=(dx/dt)_w and q _w are bounded. This leads to a steady-state flow in the flame zone in the coordinate system moving with its front and homogeneous uniform flow ahead of and behind it. Solutions of all these problems are given for the burnt-gas boundary layer region adjacent to the surface. The numerical calculations performed confirm the results obtained. A velocity law leading to time invariability of the flow pattern obtained with allowance for the interaction between the boundary layer and the burnt-gas homogeneous flow is found, including in the problem of the breakdown of an arbitrary discontinuity. The results are generalized to include the case of motion at an angle of incidence with an additional velocity component aligned with the surface.     

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.     

Calculations of subsonic inviscid flows in axisymmetric channels with division of the flow

The flow of gas in a channel with division of the flow by a longitudinal baffle is characterized by the ratio m of the flow rates in the exterior (G_e) and interior (G_i) parts of the channel: m = G_e/G_i. For example, flow in a channel with a wall screen forming a narrow annular slit with the channel wall corresponds to m 1. In the case of a gas extraction pipe in the channel m 1. In two-profile gas turbine engines m is of the order of unity or several units [1]. Flow in a channel with a baffle is characterized by the presence of a liquid boundary between the flows up to the start of the baffle. For given shape of the channel and baffle, the position of the separating stream surface depends on the conditions of the problem. In the present paper, the influence of the flow rate ratio m on the flow pattern in such a channel is investigated numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 154–156, July–August, 1981.     

Investigation of isothermal flow of a mixture of gas and particles in a channel of variable section

Isothermal flow of a gas with particles is investigated analytically, which makes it possible to analyze all possible flow regimes in channels of different shapes. It is shown that in a channel of constant section there are two possibilities: either an equilibrium regime is established with constant flow parameters, or the gas reaches the velocity of sound, and then further flow in the channel is impossible (blocking of the channel). In a contracting nozzle, blocking also occurs if the channel is sufficiently long. In an expanding nozzle when there are particles in the gas with a velocity lower than the gas velocity, it is possible to have flow regimes with transition through the velocity of sound: a subsonic flow goes over into a supersonic flow and, conversely, it is also possible to have a flow in which there is blocking of the channel, which is quite different from the flow of a pure gas in an expanding nozzle and is due to the influence of interphase friction on the flow. The variation of the pressure along the flow can be nonmonotonic with points of local maximum or minimum which do not coincide with the singular point at which the gas velocity reaches the velocity of sound. In the case of nonequilibrium gas flows with particles in a Laval nozzle, the velocity of the gas may become equal to the isothermal velocity of sound not only in the exit section of the nozzle or in its expanding part, as noted in [4–6], but also at the minimal section, since it is possible to have flows for which the velocities of the phases are equalized at this section.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 61–68, October–December, 1981.     

Rarefied gas flow in a channel with specular‐diffuse boundary conditions for arbitrary knudsen numbers. Onsager relations

The problem of a rarefied gas flow in a channel for arbitrary Knudsen numbers has been solved analytically for the first time in the case where the scattering of gas molecules on the channel walls can be described by speculardiffuse boundary conditions. The mean free path of gas molecules is assumed to be constant, i.e., the collision frequency is proportional to molecular velocity. The gas moves under the action of a streamwise temperature gradient. Exact relations for heat and mass fluxes and for meanmass velocity are obtained. It is shown that the Onsager relations are valid within the entire range of Knudsen numbers in the problem of heat and mass transfer in a channel. The dependence of heat and mass fluxes on the Knudsen number (channel thickness) is analyzed. A comparison with available results is performed.     

Gas-jet synthesis of diamond-like films from an H2 + CH4 gas mixture glow

Synthesis of diamond-like coatings from a high-velocity flow of gas mixtures in flow regimes from free-molecular to continuum with flow velocities from hundreds to thousands meters per second at different specific flow rates and temperatures in the case of activation of gases on hot surfaces is studied experimentally. Deposition of carbon films at low (less than 0.15 Pa) and high (2600 Pa) pressures from a mixture of hydrogen and methane is considered. The hydrogen flow is computed by the Direct Simulation Monte Carlo (DSMC) method in accordance with test conditions with given surface temperatures and chemical transformations on the surfaces. It is found that coatings obtained at the high pressure contain particles typical for diamonds and unusual inclusions shaped as prisms with a hexagonal cross section.     

Unified gas kinetic scheme combined with Cartesian grid method for intermediate Mach numbers

We develop a method to seamlessly simulate flows over a wide range of Knudsen numbers past arbitrarily shaped immersed boundaries. To achieve seamless computation, ie, not use any zone division to distinguish between continuum and non‐continuum regions, we use the unified gas kinetic scheme (UGKS), which is based on the Bhatnagar‐Groos‐Krook (BGK) approximation of the Boltzmann equation. We combine UGKS with an appropriately designed Cartesian grid method (CGM) to allow us to compute flows past arbitrary boundaries. The CGM we use here satisfies boundary conditions at the wall by using a constrained least square interpolation procedure. However, it differs from the usual, continuum CGMs in 2 ways. Firstly, to allow us capture non‐continuum effects at the boundaries, the CGM used herein interpolates the microscopic velocity distribution function in addition to the macroscopic variables. Secondly, even for the macroscopic variables, we use a gas kinetic method–based density interpolation procedure at the boundaries that allows the CGM to interface well with the UGKS method. We demonstrate the robustness and efficacy of the method by testing it on stationary immersed boundaries at various Knudsen numbers ranging from continuum to transition regimes.     

Stokes flow over a cavity on a superhydrophobic surface containing a gas bubble

Within the approximation of Stokes hydrodynamics, several problems of a steady-state flow over a two-dimensional cavity containing a gas bubble are solved using the method of boundary integral equations. In contrast to previous publications, the method developed makes it possible to study the situation in which the cavity is only partially filled with gas, and the edges of a curved phase interface do not coincide with the cavity corners. Using periodic boundary conditions for the velocity, the flows with pure-shear and parabolic velocity profiles, and also the flow over a group of cavities were considered. The aim of the study was to calculate the effective (average) slip velocity over a microcavity, as applied to flows near textured superhydrophobic surfaces. A parametric numerical study of the effective velocity slip as a function of the radius of curvature of the interface and the position of the interface relative to the cavity boundaries was performed. The accuracy of the method is validated by the calculations of a number of limiting flows over a cavity, for which a quantitative agreement with the results known in the literature is demonstrated.     

Interaction of a turbulent jet and a cocurrent confined incompressible flow

The effects induced in a coaxial circular channel flow by an axisymmetric turbulent jet are investigated for various values of the velocity and radius ratios 0.16m<1 and 2.5f30.9. The problem is solved by means of an e-L model of turbulence [1, 2]. The calculation scheme differs from the usual one for boundary layers, jets and wakes in that the pressure p is assumed to be unknown and is determined by assigning the boundary conditions for the radial velocity component and the transverse gradient of the longitudinal velocity component on both boundaries. On the basis of the calculations and the experimental data of [3, 4] generalized relations are obtained. These make it possible to estimate the turbulence characteristics of an axisymmetric jet in a confined cocurrent flow when the pressure is variable along the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 14–19, September–October, 1986.In conclusion, the author wishes to thank G. S. Glushko for constructive discussion of the results and useful advice.     

Diffuse interface model for high speed cavitating underwater systems

High speed underwater systems involve many modelling and simulation difficulties related to shocks, expansion waves and evaporation fronts. Modern propulsion systems like underwater missiles also involve extra difficulties related to non-condensable high speed gas flows. Such flows involve many continuous and discontinuous waves or fronts and the difficulty is to model and compute correctly jump conditions across them, particularly in unsteady regime and in multi-dimensions. To this end a new theory has been built that considers the various transformation fronts as ‘diffuse interfaces’. Inside these diffuse interfaces relaxation effects are solved in order to reproduce the correct jump conditions. For example, an interface separating a compressible non-condensable gas and compressible water is solved as a multiphase mixture where stiff mechanical relaxation effects are solved in order to match the jump conditions of equal pressure and equal normal velocities. When an interface separates a metastable liquid and its vapor, the situation becomes more complex as jump conditions involve pressure, velocity, temperature and entropy jumps. However, the same type of multiphase mixture can be considered in the diffuse interface and stiff velocity, pressure, temperature and Gibbs free energy relaxation are used to reproduce the dynamics of such fronts and corresponding jump conditions. A general model, based on multiphase flow theory is thus built. It involves mixture energy and mixture momentum equations together with mass and volume fraction equations for each phase or constituent. For example, in high velocity flows around underwater missiles, three phases (or constituents) have to be considered: liquid, vapor and propulsion gas products. It results in a flow model with 8 partial differential equations. The model is strictly hyperbolic and involves waves speeds that vary under the degree of metastability. When none of the phase is metastable, the non-monotonic sound speed is recovered. When phase transition occurs, the sound speed decreases and phase transition fronts become expansion waves of the equilibrium system. The model is built on the basis of asymptotic analysis of a hyperbolic total non-equilibrium multiphase flow model, in the limit of stiff mechanical relaxation. Closure relations regarding heat and mass transfer are built under the examination of entropy production. The mixture equation of state (EOS) is based on energy conservation and mechanical equilibrium of the mixture. Pure phases EOS are used in the mixture EOS instead of cubic one in order to prevent loss of hyperbolicity in the spinodal zone of the phase diagram. The corresponding model is able to deal with metastable states without using Van der Waals representation.     

Numerical solution of an inverse problem of nozzle theory for a two-phase gas-particle mixture

In the present paper gas flows with monodisperse and polydisperse particles in plane and axisymmetric nozzles are calculated by the inverse method [1, 2]. The gas velocity distribution is specified on the axis of symmetry of the nozzle, while the gas and particle parameters are specified in the entrance section. As a result of the numerical integration of a system of equations describing a flow of gas with condensate particles in it we determine the gas and particle parameters, the gas streamlines, and the particle trajectories with allowance for the mutual influence of the gas and particles. One of the gas streamlines is taken as the nozzle contour and the limiting trajectories and pure gas zone are found. A difference method is described which makes it possible to calculate the subsonic, transonic, and supersonic flow regions using a single algorithm, its features are noted, and the results of the calculation for monodisperse mixtures with particle diameters 1 and 5 m and fractions by weight 0.3 are given. A comparison is made with the results of calculations by other methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 106–114, July–August, 1986.The authors express their gratitude to N. B. Ponomarev and G. E. Dumnov for their useful discussions and help in carrying out the calculations.     

Single-pixel resolution ensemble correlation for micro-PIV applications

A new correlation method for particle image velocimetry (PIV) is proposed that yields velocity data at single-pixel spatial resolution. This method is an extension of the ensemble correlation method for PIV. This single-pixel ensemble correlation method is particularly suited for (quasi-) stationary and periodic flows, which are typically encountered in many micro-PIV applications, such as microfluidics and micro-scale biological flows. The method can yield data at the same level of precision and reliability as conventional PIV data. The main advantage of the new method is that it can resolve steep velocity gradients and obtain unbiased measurements of the velocity in the vicinity of flow boundaries (viz. walls). The performance as a function of the ensemble size is investigated by means of synthetic PIV images. Both ensemble correlation and single-pixel correlation are applied to micro-channel flow. With single-pixel ensemble correlation we obtained a spatial resolution of 300 nm. The results demonstrate that ensemble correlation over-estimates the measured channel width, whereas single-pixel correlation yields a result that is in agreement with the actual channel dimensions.Parts of this paper were previously presented at the 11th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, 8–11 July 2002, and the 5th International Symposium on Particle Image Velocimetry, Busan, 22–24 September 2003.

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