On the nonlinear stability behaviour of distorted plane Couette flow

This paper discusses the nonlinear stability behaviour of distorted plane Couette flowto 2-dimensional disturbances,and compares it with that of distorted plane Poiseuille flow.The results show that plane Couette flow is more unstable than plane Poiseuille flow tofinite-amplitude disturbances.     

Plane stationary flows with ionizing shock waves in a magnetic field

The formulation of stationary, plane, and self-similar problems is considered when the flow parameters depend only on the polar angle, and the magnetic field lies in the flow plane. The case in which the magnetic field is perpendicular to the flow plane has been examined in [1]. The conditions are found under which the solution depends on an arbitrary parameter and the reasons for this nonuniqueness are explained. Self-similar solutions are constructed to describe the flow around an insulating wedge and a wall.     

Investigation of the Relative Motion of a Viscous Fluid and a Porous Medium Using the Brinkman Equations

Exact solutions are obtained for the following three problems in which the Brinkman filtration equations are used: laminar fluid flow between parallel plane walls, one of which is rigid while the other is a plane layer of saturated porous medium, motion of a plane porous layer between parallel layers of viscous fluid, and laminar fluid flow in a cylindrical channel bounded by an annular porous layer.     

Three-dimensional shock intersection

The flow field in the neighborhood of the three-dimensional intersection of two shocks of different families is investigated when in the plane perpendicular to the line of intersection the flow velocity component is subsonic behind at least one of the departing shocks. In the plane case these flows are not realized. The boundary of the domain of the key parameters for which these flows are possible is determined. The characteristics of the flow field are determined when: (1) behind the departing shocks the flow is homogeneous, and (2) the velocity vectors behind the departing and arriving shocks are parallel to a single plane which contains the intersection line. The flow in Mach-type shock intersection in the neighborhood of the intersection lines (triple points in the plane) is a particular case of the problem considered. It is shown that Mach-type shock intersection is not possible when the intensity of the arriving shocks is less than for their steady-state Mach intersection in the calculation plane. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 137–143, November–December, 1998.     

Two theorems on the steady-state conduction in anisotropic body

In this paper, steady-state plane conduction problems are considered. Two theorems are proven on the conductance of non-homogeneous and anisotropic plane conductors. Here, the conductor is a two-dimensional bounded plane domain with a conducting matter having two seperated boundary terminals and two seperated insulated boundary segments. Known theorems referring to homogeneous and isotropic conductors have been extended to non-homogeneous and anisotropic ones. The results of the paper can be directly used in following fields: heat flow, diffusion, irrotational hydraulic flow, flow of electrical current and anti-plane shear deformation. Application of formulae derived is illustrated in the examples of heat flow and anti-plane elastic shear deformation.     

Asymptotic representation of the solution in the hodograph plane in transonic flow problems around profiles

The flow around a slender profile by an ideal gas flow at a constant, almost sonic, velocity at infinity is considered. The behavior of the perturbed stream in the domain upstream of the compression shocks sufficiently remote from the streamlined body is studied. The question is investigated of what conditions the solution in the hodograph plane satisfies when it corresponds to a flow without singularities on the limit characteristic in the physical flow plane. It is known that cases are possible when a regular solution in the hodograph plane loses its regularity property upon being mapped into the physical plane [1]. A regular flow on the limit characteristic can be continued analytically downstream into the supersonic domain between the limit characteristic and the shock. The requirement of analyticity of the streamlined profile is essential for realizability of the flow under consideration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 84–88, January–February, 1976.In conclusion, the author is grateful to O. S. Ryzhov for discussing the research.     

Instability of plane Couette flow of viscoelastic liquids

A review of our work on the stability of plane Couette flow of a viscoelastic liquid is given. The first part of the review is based on the assumption of a “short memory” of the fluid. The Reynolds-Orr energy criterion intimates the possibility of instability at very low Reynolds numbers. A linear stability analysis for disturbances in the flow plane shows that beyond the stability limit given by the energy criterion there are always disturbances which grow with time. A critical assessment of the short memory theory shows the severe limitations of its applicability.In the second part of the paper, the assumption of short memory is dropped. The stability of plane Couette flow with respect to special disturbances perpendicular to the flow plane is investigated for a Maxwell fluid. The flow is unstable if the product of Reynolds number and Weissenberg number is higher than a certain limit, which has the value one for a simple Maxwell fluid. This result can also be interpreted as follows: The flow becomes unstable if the velocity at the boundary walls is higher than the shear wave velocity of the fluid.     

Experimental and numerical investigation on steady blowing flow control within a compact inlet duct

A combined experimental and numerical investigation of flow control actuation in a short, rectangular, diffusing S-shape inlet duct using a two-dimensional tangential control jet was conducted. Experimental and numerical techniques were used in conjunction as complementary techniques, which are utilized to better understand the complex flow field. The compact inlet had a length-to-hydraulic diameter ratio of 1.5 and was investigated at a free-stream Mach number of 0.44. In contrast to the baseline flow, where the flow field was fully separated, the two-dimensional control jet was able to eliminate flow separation at the mid-span portion of the duct and changed considerably the three-dimensional flow field, and ultimately, the inlet performance. A comparison between the baseline (no actuation) and forced flow fields showed that secondary flow structures dominated both flow fields, which is inevitably associated with total pressure loss. Contrary to the baseline case, the secondary flow structures in the forced case were established from the core flow stagnating on the lower surface of the duct close to the aerodynamic interface plane. High fidelity spectral analysis of the experimental results at the inlet’s exit plane showed that the baseline flow field was dominated by pressure fluctuations corresponding to a Strouhal number based on hydraulic diameter of 0.26. Not only did the two-dimensional tangential control jet improve the time-averaged pressure recovery at the inlet exit plane (13.3% at the lower half of the aerodynamic interface plane), it essentially eliminated the energy content of the distinct unsteady fluctuations which characterized the baseline flow field. This result has several implications for the design of a realistic engine inlet; furthermore, it depicts that a single non-intrusive static pressure measurement at the surface of the duct can detect flow separation.     

Study of shear-thinning/thickening effects on plane Couette-Poiseuille flow with uniform crossflow

The shear-thinning/thickening effects on the plane Couette-Poiseuille flow with a uniform crossflow are studied. The detailed solution procedures for both theo- retical and numerical purposes are given. In order to clarify the difference between the Newtonian flow and the power-law flow, all cases of the plane Couette-Poiseuille flows with uniform crossflows for different power indexes are assigned to the phase diagram in the parameter plane corresponding to the Couette number and the crossflow Reynolds number. The effects of shear-thinning/thickening on the phase diagram are discussed. An important feature of the shear-thinning circumstance distinguished from the shear- thickening circumstance is discovered.     

A kinetic–hydrodynamic simulation of microstructure of liquid crystal polymers in plane shear flow

We study the microstructure formation and defects dynamics arising in liquid crystalline polymers (LCPs) in plane shear flow by a kinetic–hydrodynamic coupled model. The kinetic model is an extension of the Doi theory with a non-local intermolecular potential, including translational diffusion and density variation. LCP molecules are ensured anchoring at the boundary by an additional boundary potential, meanwhile mass conservation of LCPs holds in the whole flow region. Plane Couette flow and Poiseuille flow are studied using the kinetic–hydrodynamic model and the molecular director is restricted in the shear plane. In plane Couette flow, the numerical results predict seven in-plane flow modes, including four in-plane modes reported by Rey and Tsuji [Macromol. Theory Simul. 7 (1998) 623–639] and three new complicated in-plane modes with inner defects. Furthermore, some significant scaling properties were verified, such as the thickness of the boundary layer is proportional to molecular length, the tumbling period is proportional to the inverse of shear rate. In plane Poiseuille flow, the micro-morph is quasi-periodic in time when flow viscosity and molecular elasticity are comparable. Different local states, such as flow-aligning, tumbling or wagging, arise in different flow region. The difference of the local states, or difference of the tumbling rates in near-by regions causes defects and form branch pattern in director spatial–temporal configuration figure.     

On the steady simple shear flows of the one-mode Giesekus fluid

Analytical solutions for the plane Couette flow and the plane Poiseuille flow of the one-mode Giesekus fluid without any retardation time have been obtained by considering the domain of definition for each of the two branch solutions which arise due to the presence of the quadratic stress terms in the constitutive equations. For each fixed value of the mobility parametera, the limiting value of the Weissenberg number for the upper branch solution, i.e., the physically realistic solution is determined in terms of the corresponding dimensionless shear stress for the plane Couette flow and in terms of the corresponding dimensionless pressure gradient for the plane Poiseuille flow. In the case of the plane Couette flow, it is shown that fora falling in the range 0a1/2 only the physically realistic solution exists while for 1/2<a 1 a nonphysical solution coexists with the realistic one. In the case of the plane Poiseuille flow, it is shown that the non-physical solution cannot even exist around the center plane of the channel, and the effects of the mobility parameter and the dimensionless pressure gradient on the flow variables are investigated. Possible extensions of the present approach to other steady simple shear flows with and without the introduction of the retardation time are also discussed.     

Mixing Intensification on the Edge of a Jet by Means of Longitudinal Vortices

The action of an artificially generated spanwise flow in the form of periodical longitudinal vortices on a plane turbulent mixing layer is investigated. It is shown that the disturbances result in a significant increase in the thickness of the mixing region. For two kinds of spanwise flow, namely, vortices whose centers lie in the plane separating the streams and vortices located above this plane, the dependence of the mixing layer thickness on the vortex amplitude and vertical dimension and on the longitudinal coordinate is found.     

Investigation of micro-jet active control of a precessing jet using PIV

A circular jet entering an open-ended concentric circular chamber can rotate or precess about the jet axis for certain flow conditions and chamber configurations. Active flow control of a precessing jet provides the ability to influence the flow field inside the chamber and the resulting flow after the chamber exit. Twelve micro-jets surrounding the jet at the chamber inlet are used as actuation. At the chamber exit, four pressure probes and three-component velocity measurement using stereo particle image velocimetry (stereo-PIV) is used to monitor the flow. A phase plane method using signals from the pressure sensors is developed to monitor the location of the jet high-velocity region in real-time. Phase-locked stereo-PIV, triggered by the micro-jet actuation signal, is used to investigate the flow field and validate the pressure phase plane results. The effectiveness of the micro-jet actuation and the validation of the pressure phase plane measurements demonstrate actuation and the sensing needed for future closed-loop control of the precessing jet.     

Stokes flow before plane boundary with mixed stick-slip boundary conditions

A general theorem for the Stokes flow over a plane boundary with mixed stick-slip boundary conditions is established. This is done by using a representation for the velocity and pressure fields in the three-dimensional Stokes flow in terms of a biharmonic function and a harmonic function. The earlier theorem for the Stokes flow due to fundamental singularities before a no-slip plane boundary is shown to be a special case of the present theorem. Furthermore, in terms of the Stokes stream function, a corollary of the theorem is also derived, providing a solution to the problem of the axisymmetric Stokes flow along a rigid plane with stick-slip boundary conditions. The formulae for the drag and torque exerted by the fluid on the boundary are established. An illustrative example is given.     

Rivulet flow of a viscoelastic liquid

A theoretical investigation is made of the rivulet flow of a viscoelastic liquid down an inclined plane. It is shown that elasticity causes a significant change in the shape of the rivulet, with height rise at the center. There is also a change in the relationship between the flow rate and the geometry of the rivulet. Elasticity is found to cause a flow in the cross-sectional plane in the form of counter-rotating vortices. In the Newtonian case the flow is purely axial.     

Weissenberg effect and its dependence upon the experimental geometry

The flow of a second-order fluid with a free surface between two coaxially mounted cylinders of finite length, the inner one of which is rotating, is being studied. In the case of slow flow and small shear rates the flow can be divided into a primary flow in the plane perpendicular to the axis of rotation and a secondary flow in the meridional plane. These flow components are numerically calculated and the results are compared with the analytical results for the semi-infinite cylinder approximation. The influence of the finiteness of the cylinders (end effect) upon the free surface deformation is analysed. The numerical results for the secondary flow are compared with results obtained by flow visualisation.     

Stability of a binary-mixture advective flow in a plane horizontal layer with perfectly heat conducting boundaries

The stability of a steady-state advective flow of a binary mixture in a plane horizontal layer with perfectly heat conducting solid boundaries in the presence of a uniform longitudinal temperature gradient is investigated. The problem is solved with account for thermodiffusion. The limits of the stability of the steady-state flow with respect to long-wave disturbances are found analytically. The stability of the main flow with respect to plane and spiral disturbances with finite wavelengths is studied numerically. The stability maps in the “Rayleigh number-Sorét parameter” plane are constructed for a number of typical liquid and gaseous mixtures.     

On two-dimensional laminar flows of a particulate suspension in the presence of gravity field

Three simple two-dimensional streaming motions of a mixture of solid particles with a continuous carrier fluid, or gas, in the presence of the gravity field are considered. These include flow of a mixture over an infinite stationary rigid plane perpendicular to the direction of the gravity field, flow near an oscillating rigid plane and flow in a mixture induced by a suddenly accelerated plane. The nature of the boundary conditions at the interface between a layer of sediment settling on the rigid boundary and the mixture above it suggests an introduction of the independent variables that enable simple analytical expressions for the solutions of the first two flows and a numerical solution by means of a Laplace transform in the last case.     

Natural Convection from a Vertical Plate in a Porous Medium Using Brinkman's Model

A regular perturbation analysis is presented for the following laminar natural convection flows of Newtonian fluids with temperature-dependent effective viscosity: a freely-rising plane plume, the flow above a horizontal line source on an adiabatic surface (a plane wall plume) and the flow adjacent to a vertical uniform flux surface for porous medium. The temperature-dependent effective viscosity introduces nonsimilarity into the governing equations. Numerical results are presented for the flow and heat transfer characteristics.     

On the stability of distorted laminar flow(III)—The nonlinear stability analysis of distorted parallel shear flow

Based on the hydrodynamic stability theory of distorted laminar flow and the kind of distortion profiles on the mean velocity in parallel shear flow given in paper [1], this paper investigates the nonlinear stability behaviour of parallel shear flow, carries on stability calculation taking account of the perturbations of background turbulence noise under certain assumption, and obtains some results in accordance qualitatively with those of experiment for plane Poiseuille flow and pipe Poiseuille flow.The author thanks Prof. Zhou Heng sincerely for his kind offer of his computer program of the artificial neutrality method on the stability in subcritical range of plane Poiseuille flow.