Mathematical modeling of jet interaction with a high-enthalpy flow in an expanding channel

Results of modeling the interaction of a plane supersonic jet with a supersonic turbulent high-enthalpy flow in a channel are reported. The problem is solved in a two-dimensional formulation at external flow Mach numbers M_∞ = 2.6 and 2.8 and at high values of the total temperature of the flow T ₀ = 1800–2000 K. The mathematical model includes full averaged Navier-Stokes equations supplemented with a two-equation turbulence model and an equation that describes the transportation of the injected substance. The computations are performed by using the ANSYS Fluent 12.1 software package. Verification of the computational technique is performed against available experimental results on transverse injection of nitrogen and helium jets. The computed and experimental results are demonstrated to agree well. For the examined problems, in addition to surface distributions of characteristics, fields of flow parameters are obtained, which allow one to reproduce specific features that can be hardly captured in experiments. Parametric studies show that an increase in the angle of inclination and the mass flow rate of the jet leads to an increase in the depth of jet penetration into the flow, but more intense separated flows and shock waves are observed in this case.     

Self-sustained oscillations of a confined impinging jet

The present paper investigates the dynamics of a laminar plane jet impinging on a flat plate in a channel. An experimental parametric study is carried out to determine the flow regimes at different levels of confinement and Reynolds numbers. For very confined jets, the flow is steady whatever the Reynolds number. The overall structure of the flow is symmetric with respect to the jet axis and is characterized by the presence of recirculation zones at the channel walls. The dynamics is radically different for less confined jets. Above a critical Reynolds number, the flow bifurcates in the form of an oscillating flapping mode of the impinging jet. Analyses of the experimental results provide with a quantitative characterization of this regime in terms of amplitude, wavelength and frequency. This self-oscillating bifurcated flow induces strong sweepings of the target plate by the jet and intense vortex dipole ejections from the impacted wall. Such a regime is expected to be particularly useful in the enhancement of the local heat transfer at relatively low cost in terms of flow rate.     

Absolute, convective, and global instability of a magnetic liquid jet

An infinite or semi-infinite jet of non-conductive magnetic liquid in a uniform longitudinal magnetic field can be absolutely or convectively unstable for different values of the flow parameters. Though the higher field inhibits the absolute instability, this inhibition is maximum at some field intensity. A critical value of the surface tension exists, above which the instability is absolute for any intensity of the field. If the jet has a large but finite length and proper boundary conditions are held at its beginning and end, it is always globally unstable. The unstable global mode is based on a pair of waves that propagate in opposite directions and reflect from one into the other at the flow boundaries.     

Interaction of a heavy fluid flow in a channel with a transverse jet barrier

An experimental and numerical analysis of the interaction between a plane horizontal water flow in a rectangular channel (free water current) and a plane thin water jet (water jet curtain) is presented; the jet flows out vertically from either a slot nozzle in the bottom of the channel or the crest of a rigid spillway at a velocity appreciably (several times) greater than the water velocity in the channel. Numerical calculations were carried out using the STAR-CD software package preliminarily tested against the experimental data obtained. The dependence of the water level in the channel at a certain distance ahead of the jet barrier on the main jet parameters and the water flow rate in the horizontal channel is studied. It is found that in the region of the interface between the flows both steady and unsteady (self-oscillatory) flow patterns can be realized. Steady stream/jet interaction patterns of the “ejection” and “ejection-spillway” types are distinguished and a criterion separating these regimes is obtained. The notion of a rigid spillway equivalent to a jet curtain is introduced and an approximate dependence of its height on the relevant parameters of the problem is derived. The possibility of effectively controlling the water level ahead of a rigid spillway with a sharp edge by means of a plane water jet flowing from its crest is investigated. The boundary of transition to self-oscillation interaction patterns in the region of the flow interface is determined. The structure of these flows and a possible mechanism of their generation are described. Within the framework of the inviscid incompressible fluid model in the approximate formulation for a “thin” jet, an analytical dependence of the greatest possible depth of a reservoir filled with a heavy fluid at rest and screened by a vertical jet barrier on the jet parameters is obtained.     

Filling of channels with a non-Newtonian fluid in the gravity field

The flow of a pseudoplastic fluid with a free surface realized during the filling of a channel at a constant flow rate in the entry section is modeled. The fluid flow direction coincides with that of gravity. The mathematical problem is formulated on the basis of the complete equations of motion. A numerical technique for solving the problem in the two-dimensional formulation is developed on the basis of the SIMPLE algorithm and the method of invariants. A parametric investigation of the free surface evolution is carried out for different values of the governing criteria and rheological parameters. Three filling regimes are found to exist, namely, a complete filling regime, a transitional regime characterized by air cavity formation, and a jet regime. Criterial dependences separating these regimes are presented.     

Particle removal from a surface by a bounded vortex flow

A bounded vortex flow consists of an axisymmetric vortex that is confined top and bottom between two plates (the “confinement plate” and “impingement plate”, respectively) and surrounded laterally by a swirling annular slot jet. The bottom of the vortex terminates on the boundary layer along the impingement plate and the top of the vortex is drawn into a suction port positioned at the center of the confinement plate. The circumferential flow within the annular jet is important for supplying circulation to the central wall-normal vortex. This flow field is proposed as a method for mitigation of dust build-up on a surface, where the vortex–jet combination supplements the more traditional vacuum port by enhancing the surface shear stress and related particle transport rate. The paper reports on a computational study of the velocity field and particle transport by a bounded vortex flow. Fluid flow computations are performed using a finite-volume approach for an incompressible fluid and particle transport is simulated using a discrete-element method. Computations are performed for different values of two dimensionless parameters – the ratio of the plate separation distance and the average radial location of the jet inlet (the dimensionless confinement height) and the ratio of flow rate withdrawn at the suction outlet and that injected by the jet (the flow rate ratio). For small values of the flow rate ratio, the impinging jet streamlines pass down to the boundary layer along the bottom surface and then travel up the vortex core. By contrast, for large values of flow rate ratio, the annular jet is quickly entrained into the suction outlet and no wall-normal vortex is formed. Particles are observed to roll along the impingement surface in a direction determined by the fluid shear stress lines. Particles roll outward when they lie beyond a separatrix curve of the surface shear stress lines, where particles within this separatrix curve roll inward, piling up at the center of the flow field. A toroidal vortex ring forms for the small confinement height case with flow rate ratio equal to unity, which yields double separatrix curves in the shear stress lines. The inward rolling particles intermittently lift up due to collision forces and burst away from the impingement surface, eventually to become entrained into the flow out the suction port or resettling back onto the impingement surface.     

Propagation of acoustic perturbations along a supersonic jet flowing out into the atmosphere

The propagation of acoustic perturbations (specified in the outlet cross section of a particular channel) along a supersonic jet flowing out of the channel is considered; also considered is acoustic emission from the surface of the jet into the atmosphere. The solution of these problems is obtained by a numerical method on the linear approximation. The laws governing the propagation of the perturbations as a function of the perturbation frequency and other determining parameters are investigated; these parameters include the velocity and temperature of the jet, the velocity of the subsonic accompanying flow in the external medium, and the character of the perturbation in the initial cross section of the jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 92–99, March–April, 1977.     

Viscous-fluid outflow from a vessel with account for jet formation

The results of a numerical simulation of slow free-boundary viscous-fluid outflow from a vessel are presented with account for jet formation. The problem is formulated in the creeping motion approximation. For solving the problem, a numerical algorithm for plane geometry, based on an indirect variant of the boundary-element method, is used. As a result of parametric studies, the evolution of the free surface inside the vessel and the jet shape is determined for different values of the governing parameters. Flow regimes with rapid funneling and film formation on the vessel walls are detected. The existence of an asymptotic flow regime is demonstrated using dimensional analysis and confirmed by calculation.     

Experimental investigation of flow over a backward-facing step in proximity to a flexible wall

Turbulent flow between a flexible wall and a solid surface containing a backward-facing step (BFS) was investigated using digital particle image velocimetry and high-speed photography. Stationary sheet of paper under tension was positioned above the solid surface in proximity to the BFS. The incoming air flow emerged from a planar nozzle that was located in the solid wall upstream of the BFS. Flows corresponding to two values of the Reynolds number (3,000 and 3,600) based on the step height and the maximum flow velocity at the step location were characterized in terms of patterns of time-averaged velocity, out-of-plane vorticity, streamline topology, and turbulence statistics. In addition, paper sheet oscillation was characterized using high-speed photography. For the control case of a solid upper wall with the geometry that represented the time-averaged paper profile, hydrodynamic frequencies were characterized using unsteady pressure measurements. Frequencies of the natural vibration modes of the paper sheet were well separated from the hydrodynamic frequencies corresponding to the oscillations of the shear layer downstream of the BFS. As the inflow velocity increased, the paper sheet was pulled closer to the solid surface, which resulted in increased confinement of the incoming jet. The flow reattachment length calculated on the basis of time-averaged flow patterns increased with the increasing Reynolds number.     

Single cavitation bubble generation and observation of the bubble collapse flow induced by a pressure wave

This study utilizes a U-shape platform device to generate a single cavitation bubble for a detailed analysis of the flow field characteristics and the cause of the counter jet during the process of bubble collapse caused by sending a pressure wave. A high speed camera is used to record the flow field of the bubble collapse at different distances from a solid boundary. It is found that a Kelvin–Helmholtz vortex is formed when a liquid jet penetrates the bubble surface after the bubble is compressed and deformed. If the bubble center to the solid boundary is within one to three times the bubble’s radius, a stagnation ring will form on the boundary when impinged by the liquid jet. The fluid inside the stagnation ring will be squeezed toward the center of the ring to form a counter jet after the bubble collapses. At the critical position, where the bubble center from the solid boundary is about three times the bubble’s radius, the bubble collapse flow will vary. Depending on the strengths of the pressure waves applied, the collapse can produce a Kelvin–Helmholtz vortex, the Richtmyer–Meshkov instability, or the generation of a counter jet flow. If the bubble surface is in contact with the solid boundary, the liquid jet can only move inside-out without producing the stagnation ring and the counter jet; thus, the bubble collapses along the radial direction. The complex phenomenon of cavitation bubble collapse flows is clearly manifested in this study.     

Self-oscillation regimes of the penetration of free conical thin-walled turbulent jets through a fluid surface

Certain results of an experimental investigation of hydrodynamic effects occurring, when free thin-walled turbulent jets issuing from a conical slot nozzle with a vertical axis penetrate through the surface of a fluid in a rectangular reservoir, are presented. The ranges of the jet flow rate and the spacing between the nozzle and the free surface, on which stable regular transverse self-oscillations of the boundaries of dome-shaped jets are observable, are determined. For fixed values of the conicity angle α = 60° and the nozzle slot width δ = 0.1 cm the characteristic form of the dependence of the self-oscillation period on the jet flow rate and the spacing between the slot nozzle and the free surface (dome height) is presented. The self-oscillation regime generation mechanism, together with the possible reasons for the revealed bifurcation changeover of the oscillation mode at certain values of the governing parameters and the hysteresis effects, are discussed. The salient features of the flows occurring on the surface and within the fluid beneath the dome are described.     

Stability Analysis of a Vertical Curved Channel Flow with a Radial Temperature Gradient

The linear stability analysis of a Newtonian incompressible fluid in a vertical curved channel formed by two coaxial cylindrical surfaces with a radial temperature gradient and an azimuthal pressure gradient shows that critical modes are oscillatory and non-axisymmetric. We have derived a generalized Rayleigh discriminant which includes both the curvature and buoyancy effects. Centrifugal buoyancy induces weak asymmetry of the dependence of the control parameter critical values on the sign of the temperature gradient. The critical parameters depend on the temperature gradient, the radius ratio and the nature of the fluid. For a wide curvature channel flow, there are two critical modes: oscillatory Dean modes for small temperature gradients and oscillatory centrifugal-thermal modes for relatively large temperature gradients. Received 14 November 2001 and accepted 29 March 2002 Published online: 2 October 2002 Communicated by H.J.S. Fernando     

Self-oscillatory regimes of the penetration of vertical free turbulent jets through a liquid surface

The results of an experimental investigation of the penetration of vertical plane and round free turbulent jets through the surface of a liquid contained in a relatively narrow channel are presented. It is established that there exist the ranges of jet thicknesses, their velocities, and free region lengths, on which regular self-oscillatory regimes of the displacement of submerged jet regions and two-phase flow regions are observable. The mechanism of the generation of these regimes and the special features of the observable flows are discussed. The dependences of the self-oscillation periods on the main control parameters of the problem are established.     

Hollow droplets impacting onto a solid surface

The impact process of spherical hollow droplets impinging onto a solid surface has been experimentally studied. Formation of a counter-jet in a wide range of Reynolds and Weber numbers was revealed, this jet being similar to a Worthington jet. For characterizing the regime of liquid flow in the hollow droplet, we propose using the Euler number. Theoretically, the problem was treated using a simple model of axisymmetric liquid flow. The obtained results proved to be consistent with experimental data.     

Numerical simulation of confined laminar jet flows

Two‐dimensional incompressible jet development inside a duct has been studied in the laminar flow regime, for cases with and without entrainment of ambient fluid. Results have been obtained for the flow structure and critical Reynolds number values for steady asymmetric jet development and for the onset of temporal oscillations, at various values of the duct‐to‐jet width ratio (aspect ratio). It is found that at low aspect ratios and Reynolds numbers, jet development inside the duct is symmetric. For larger aspect ratios and Reynolds numbers, the jet flow at steady state becomes asymmetric with respect to the midplane, and for still higher values, it becomes oscillatory with respect to time. When entrainment is present, the instabilities of asymmetric development and temporal oscillations occur at a much higher critical Reynolds number for a given aspect ratio, indicating that the stability of the jet flow is higher with entrainment. Copyright © 2000 John Wiley & Sons, Ltd.     

Near-wake flow structure of elliptic cylinders close to a free surface: effect of cylinder aspect ratio

The flow fields behind elliptic cylinders adjacent to a free surface were investigated experimentally in a circulating water channel. A range of cylinder aspect ratios (AR=2, 3, 4) were considered, while the cross-sectional area of the elliptical cylinder was kept constant. The main objective of this study was to investigate the effect of cylinder aspect ratio and a free surface on the flow structure in the near-wake behind elliptic cylinders. For each elliptic cylinder, the flow structure was analyzed for various values of the submergence depth of the cylinder beneath the free surface. The flow fields were measured using a single-frame double-exposure PIV (Particle Image Velocimetry) system. For each experimental condition, 350 instantaneous velocity fields were obtained and ensemble-averaged to obtain the mean velocity field and spatial distribution of the mean vorticity statistics. The results show that near-wake can be classified into three typical flow patterns: formation of a Coanda flow, generation of substantial jet-like flow, and attachment of this jet flow to the free surface. The general flow structure observed behind the elliptic cylinders resembles the structure previously reported for a circular cylinder submerged near a free surface. However, the wake width and the angle of downward deflection of the shear layer developed from the lower surface of the elliptic cylinder differ from those observed for a circular cylinder. These trends are enhanced as cylinder aspect ratio is increased. In addition, the free surface distortion is also discussed in the paper.     

Bifurcation of a Main Steady-State Viscous Fluid Flow in a Plane Divergent Channel

The evolution of steady-state viscous incompressible fluid flows in a plane divergent channel is investigated. For the classical formulation of the Jeffery-Hamel problem a complete solution is given as a function of the determining parameters. For a fixed angle of divergence the behavior of the main unimodal flow is determined as a function of the Reynolds number. Critical values at which the flow pattern bifurcates and the steady-state unimodal flow ceases to exist are found. The mechanism of bifurcation is established and its diagram is constructed. This mechanism and the diagram were not previously known in the scientific literature in connection with the investigation of the Jeffery-Hamel problem. The critical Reynolds number at which bifurcation occurs is given as a function of the channel divergence angle. The results may be of interest for hydromechanical, technological, and geophysical applications.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 25–36.Original Russian Text Copyright © 2005 by Akulenko and Kumakshev.     

Oscillations of the fluid flow and the free surface in a cavity with a submerged bifurcated nozzle

The free surface dynamics and sub-surface flow behavior in a thin (height and width much larger than thickness), liquid filled, rectangular cavity with a submerged bifurcated nozzle were investigated using free surface visualization and particle image velocimetry (PIV). Three regimes in the free surface behavior were identified, depending on nozzle depth and inlet velocity. For small nozzle depths, an irregular free surface is observed without clear periodicities. For intermediate nozzle depths and sufficiently high inlet velocities, natural mode oscillations consistent with gravity waves are present, while at large nozzle depths long term self-sustained asymmetric oscillations occur.For the latter case, time-resolved PIV measurements of the flow below the free surface indicated a strong oscillation of the direction with which each of the two jets issue from the nozzle. The frequency of the jet oscillation is identical to the free surface oscillation frequency. The two jets oscillate in anti-phase, causing the asymmetric free surface oscillation. The jets interact through a cross-flow in the gaps between the inlet channel and the front and back walls of the cavity.     

Flow of viscous gas out of a cylindrical channel into vacuum

A study is made of the exhausting of a jet of viscous gas from a cylindrical channel into vacuum in the presence of a flat bounding surface outside the channel in the plane of its exit section. The problem is solved numerically using the complete system of Navier—Stokes equations. The developed flow model makes it possible to take into account the influence of an external medium into which the jet exhausts on the structure of the flow in the exit section of the channel, and also the influence of the subsonic part of the boundary layer in the channel on the flow field of the jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–128, January–February, 1981.     

Stability of a melt/solid interface with reference to a plume channel

The structure of free-convection flow in a plume channel formed as a result of melting above a local heat source placed on the basement of a solid mass is experimentally investigated. The channel shape and the flow pattern in it are functions of the relative power Ka = N/N ₁, where N is the plume source power and N ₁ is the heat removed to the surrounding mass. When the heat is withdrawn from the plume channel by heat conduction, the channel represents a system of convective cells on whose boundaries there are channel constrictions. The temperature fields and the cell flow patterns are investigated. For mantle plumes, such as the Hawaiian, Iceland, and Bouvet plumes and extended igneous provinces, the basement diameter and the values of the criterion Ka are determined.