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A vertical jet of water impinging on a horizontal surface produces a radial film flow followed by a circular hydraulic jump. We report a phenomenon where fairly large (1 ml) drops of liquid levitate just upstream of the jump on a thin air layer between the drop and the film flow. We explain the phenomenon using lubrication theory. Bearing action both in the air film and the water film seems to be necessary to support large drops. Horizontal support is given to the drop by the hydraulic jump. A variety of drop shapes is observed depending on the volume of the drop and liquid properties. We show that interaction of the forces due to gravity, surface tension, viscosity and inertia produces these various shapes.

World University root node

Department of Mechanical Engineering

Journal of Fluid Mechanics

Levitation of a drop over a film flow

Radial thin film flows are obtained by the impingement of circular free liquid jets on surfaces. The surfaces may be in the form of a circular plate, cone or that of a sphere. These flows are governed by the effects of inertia, viscosity, gravity and surface tension. Based on the film Reynolds number and Froude number, a circular hydraulic jump can be obtained in such flows. In this paper a new integral method is proposed for such axisymmetric laminar flows. The boundary layer approximation is used. The equations are solved using a cubic velocity profile, considering the radial hydrostatic pressure gradient in the film flow. In the new approach the coefficients of the cubic profile depend on the pressure gradient and body force terms and are allowed to vary with radial distance. Thus for example, separation can be predicted. The effect of the jet Reynolds number, Froude number and the surface dimension is considered. For flows with the circular hydraulic jump, the region upstream and downstream of the jump is solved separately using the boundary condition at the surface edge. Radial thin film flows are obtained by the impingement of circular free liquid jets on surfaces. The surfaces may be in the form of a circular plate, cone or that of a sphere. These flows are governed by the effects of inertia, viscosity, gravity and surface tension. Based on the film Reynolds number and Froude number, a circular hydraulic jump can be obtained in such flows. In this paper a new integral method is proposed for such axisymmetric laminar flows. The boundary layer approximation is used. The equations are solved using a cubic velocity profile, considering the radial hydrostatic pressure gradient in the film flow. In the new approach the coefficients of the cubic profile depend on the pressure gradient and body force terms and are allowed to vary with radial distance. Thus for example, separation can be predicted. The effect of the jet Reynolds number. Froude number and the surface dimension is considered. For flows with the circular hydraulic jump, the region upstream and downstream of the jump is solved separately using the boundary condition at the surface edge.

International Journal of Heat and Mass Transfer

Integral analysis applied to radial film flows

This paper deals with the stability and transition to turbulence of wall-bounded unsteady velocity profiles with reverse flow. Such flows occur, for example, during unsteady boundary layer separation and in oscillating pipe flow. The main focus is on results from experiments in time-developing flow in a long pipe, which is decelerated rapidly. The flow is generated by the controlled motion of a piston. We obtain analytical solutions for laminar flow in the pipe and in a two-dimensional channel for arbitrary piston motions. By changing the piston speed and the length of piston travel we cover a range of values of Reynolds number and boundary layer thickness. The velocity profiles during the decay of the flow are unsteady with reverse flow near the wall, and are highly unstable due to their inflectional nature. In the pipe, we observe from flow visualization that the flow becomes unstable with the formation of what appears to be a helical vortex. The wavelength of the instability ≃ 3δ + ̄ where δ + ̄ is the average boundary layer thickness, the average being taken over the time the flow is unstable. The time of formation of the vortices scales with the average convective time scale and is ≃ 33/(Δ + ̄u/δ + ̄), where Δu = (umax - umin) and umax, umin and δ are the maximum velocity, minimum velocity and boundary layer thickness respectively at each instant of time. The time to transition to turbulence is ≃ 39/(Δ + ̄u/δ + ̄). Quasi-steady linear stability analysis of the velocity profiles brings out two important results. First that the stability characteristics of velocity profiles with reverse flow near the wall collapse when scaled with the above variables. Second that the wavenumber corresponding to maximum growth does not change much during the instability even though the velocity profile does change substantially. Using the results from the experiments and the stability analysis, we are able to explain many aspects of transition in oscillating pipe flow. We postulate that unsteady boundary layer separation at high Reynolds numbers is probably related to instability of the reverse flow region.

Transition of unsteady velocity profiles with reverse flow

Experiments indicate that turbulent free convection over a horizontal surface (e.g. Rayleigh-Bénard convection) consists of essentially line plumes near the walls, at least for moderately high Rayleigh numbers. Based on this evidence, we propose here a two-dimensional model for near-wall dynamics in Rayleigh-Bénard convection and in general for convection over heated horizontal surfaces. The model proposes a periodic array of steady laminar two-dimensional plumes. A plume is fed on either side by boundary layers on the wall. The results from the model are obtained in two ways. One of the methods uses the similarity solution of Rotem & Classen (1969) for the boundary layer and the similarity solution of Fuji (1963) for the plume. We have derived expressions for mean temperature and temperature and velocity fluctuations near the wall. In the second approach, we compute the two-dimensional flow field in a two-dimensional rectangular open cavity. The number of plumes in the cavity depends on the length of the cavity. The plume spacing is determined from the critical length at which the number of plumes increases by one. The results for average plume spacing and the distribution of r.m.s. temperature and velocity fluctuations are shown to be in acceptable agreement with experimental results.

A model for near-wall dynamics in turbulent Rayleigh-Bénard convection

A new turbulence model has been proposed for the axisymmetric plume. In this model the equations for the turbulent kinetic energy (k), its dissipation (ε), temperature fluctuations (t′2), and intermittency (γ) have been solved. The equation for the dissipation (ε) accounts for the effect of entrainment, which in turn requires the solution of a transport equation for the intermittency. The effect of buoyancy on Reynolds stress and radial heat flux has been modelled by the algebraic stress model (ASM). For the buoyancy production of turbulent kinetic energy a simple model (which assumes it to be proportional to √kt′2) has been used. A detailed comparison of the predictions has been made with measurements. The present k - ε - t′2 - γ model is shown to simulate accurately the intermittency and the effects of buoyancy on mean and turbulent quantities. © 1997 by Elsevier Science Inc.

Applied Mathematical Modelling

A new turbulence model for the axisymmetric plume

The axially translating flexible beam with a prismatic joint can be modelled by using the Euler-Bernoulli beam equation together with the convective terms. In general, the method of separation of variables cannot be applied to solve this partial differential equation. In this paper, a non-dimensional form of the Euler Bernoulli beam equation is presented, obtained by using the concept of group velocity, and also the conditions under which separation of variables and assumed modes method can be used. The use of clamped-mass boundary conditions leads to a time-dependent frequency equation for the translating flexible beam. A novel method is presented for solving this time-dependent frequency equation by using a differential form of the frequency equation. The assumed model Lagrangian formulation of dynamics is employed to derive closed form equations of motion. It is shown by using Lyapunov's first method that the dynamic responses of flexural modal variables become unstable during retraction of the flexible beam, while the dynamic response during extension of the beam is stable. Numerical simulation results are presented for the uniform axial motion induced transverse vibration for a typical flexible beam. © 1996 Academic Press Limited.

Journal of Sound and Vibration

The modelling of axially translating flexible beams

AIAA Journal

Comparison of four turbulence models for wall-bounded flows affected by transverse curvature

This paper deals with the radially flowing film produced by a vertical liquid jet impinging on a horizontal surface. Typically, circular hydraulic jump is formed at some radius. A new integral method to calculate the film flow is proposed. Flow visualization pictures and film thickness measurements for a range of flow rates are presented. A newly discovered phenomenon of surface wave-induced transition is discussed.

Journal of the Indian Institute of Science

On radial film flow on a horizontal surface and the circular hydraulic jump

A new model is proposed to explain the Nielsen's equilibrium condition. This model takes into account the effects of both turbulent entrainment and diffusion on the growth/erosion of the gradient zone. The existing turbulent entrainment model is modified to make it applicable near the equilibrium condition. The new model predictions indicate the dependence of the equilibrium condition on the mixed layer depth, apart from the salinity and temperature gradients in the gradient zone. © 1995.

Solar Energy

Modeling the dynamics of the mixed layer in solar ponds

Experimental results are presented that show that the translational velocities of piston generated vortex rings often undergo oscillations, similar to those recently discovered for drop generated rings. An attempt has been made to minimize uncertainties by utilizing both dye and hydrogen bubbles for visualization and carefully repeating measurements on the same ring and on different realizations under the same nominal piston conditions. The results unambiguously show that under most conditions, both for laminar and turbulent rings and for rings generated from pipes and orifices, the oscillations are present. The present results, together with the earlier results on drop generated rings, give support to the view that translational velocity oscillations are probably an inherent feature of translating vortex ring fields. © 1995 American Institute of Physics.

Physics of Fluids

Translational velocity oscillations of piston generated vortex rings

We present, results from a two dimensional numerical simulation of a stably stratified layer heated from below. The aim of the simulation is to study the mechanism of erosion of a stably stratified medium by the convective velocity. We solve transport equations for momentum, heat and species using Patankar's SIMPLER method. Tracer particles are used to get flow pattern inside the mixed layer and in the gradient zone. We have observed randomly moving thermals issuing from the bottom heated surface and from the interface separating the mixed layer and the gradient zone. Stable fluid from the gradient zone gets entrained into the mixed layer with the falling thermal from the interface. This is the main mechanism of the turbulent entrainment. Results from the numerical simulation indicate the need to include the effects of the turbulent entrainment in modelling the equilibrium condition.

Lecture Notes in Physics

Study of the erosion of stably stratified medium heated from below

The planform structure of turbulent Rayleigh-Bénard convection is obtained from visualising a liquid crystal sheet stuck to the bottom hot surface. The bottom plate of the convection cell is Plexiglas and the top plate is glass. Water is the 3est liquid and the Rayleigh number is 4 × 107. The planform pattern reveals randomly moving hot streaks surrounded by cold regions suggesting that turbulent Rayleigh-Bénard convection is dominated by quasi-two-dimensional randomly moving plumes. Simultaneous temperature traces from two vertically separated thermocouples indicate that these plumes may be inclined forward in the direction of horizontal motion. The periodic eruption of thermals observed by Sparrow et al [3] and which forms the basis of Howard's model is not observed. © 1994.

International Communications in Heat and Mass Transfer

Planform structure of turbulent Rayleigh-Bénard convection

A new model has been developed for the integral analysis of high Schmidt number (or equivalently high Prandtl number) laminar buoyant jets discharged horizontally. This model assumes top-hat density profile across the inner core of jet and Gaussian velocity profile. Entrainment coefficient corresponding to pure laminar jet has been taken in the analysis. The prediction of the jet trajectory agree well with experimental data in the regions where the jet remains laminar. © 1992.

A note on high Schmidt number laminar bouyant jets discharged horizontally

Effect of pulsed slot suction on a turbulent boundary layer

Thick boundary layers at zero pressure gradient are produced by generators kept along the leading edge of a flat surface and sustained by roughness elements distributed over the surface. Size, shape and spacing of the generators, distribution of the roughness elements and the roughness fetch are varied and the mean velocity profile measured at various locations. The results obtained show that layer thicknesses of about 25 cm can be obtained in a relatively short fetch of 1 m and that mean velocity profiles corresponding to both rural and urban terrain conditions can be simulated.