scholarly journals Energy dispersion in turbulent jets. Part 2. A robust model for unsteady jets

2014 ◽  
Vol 763 ◽  
pp. 538-566 ◽  
Author(s):  
John Craske ◽  
Maarten van Reeuwijk
Keyword(s):  
Momentum Flux ◽  
Longitudinal Velocity ◽  
Sudden Change ◽  
Turbulent Jets ◽  
Local Deformation ◽  
Integral Model ◽  
Type I ◽  
Gaussian Form ◽  
Gaussian Profile ◽  
Unsteady Jets

AbstractIn this paper we develop an integral model for an unsteady turbulent jet that incorporates longitudinal dispersion of two distinct types. The model accounts for the difference in the rate at which momentum and energy are advected (type I dispersion) and for the local deformation of velocity profiles that occurs in the vicinity of a sudden change in the momentum flux (type II dispersion). We adapt the description of dispersion in pipe flow by Taylor (Proc. R. Soc. Lond. A, vol. 219, 1953, pp. 186–203) to develop a dispersion closure for the longitudinal transportation of energy in unsteady jets. We compare our model’s predictions to results from direct numerical simulation and find a good agreement. The model described in this paper is robust and can be solved numerically using a simple central differencing scheme. Using the assumption that the longitudinal velocity profile in a jet has an approximately Gaussian form, we show that unsteady jets remain approximately straight-sided when their source area is fixed. Straight-sidedness provides an algebraic means of reducing the order of the governing equations and leads to a simple advection–dispersion relation. The physical process responsible for straight-sidedness is type I dispersion, which, in addition to determining the local response of the area of the jet, determines the growth rate of source perturbations. In this regard the Gaussian profile has the special feature of ensuring straight-sidedness and being insensitive to source perturbations. Profiles that are more peaked than the Gaussian profile attenuate perturbations and, following an increase (decrease) in the source momentum flux, lead to a local decrease (increase) in the area of the jet. Conversely, profiles that are flatter than the Gaussian amplify perturbations and lead to a local increase (decrease) in the area of the jet.

scholarly journals Energy dispersion in turbulent jets. Part 1. Direct simulation of steady and unsteady jets

2014 ◽  
Vol 763 ◽  
pp. 500-537 ◽  
Author(s):  
John Craske ◽  
Maarten van Reeuwijk
Keyword(s):  
Steady State ◽  
Momentum Flux ◽  
Propagation Speed ◽  
Turbulent Jets ◽  
Type I ◽  
Type Ii ◽  
Governing Equations ◽  
Unsteady Jets ◽  
Longitudinal Mixing ◽  
Self Similar

AbstractWe study the physics of unsteady turbulent jets using direct numerical simulation (DNS) by introducing an instantaneous step change (both up and down) in the source momentum flux. Our focus is on the propagation speed and rate of spread of the resulting front. We show that accurate prediction of the propagation speed requires information about the energy flux in addition to the momentum flux in the jet. Our observations suggest that the evolution of a front in a jet is a self-similar process that accords with the classical dispersive scaling$z\sim \sqrt{t}$. In the analysis of the problem we demonstrate that the use of a momentum–energy framework of the kind used by Priestley & Ball (Q. J. R. Meteorol. Soc., vol. 81, 1955, pp. 144–157) has several advantages over the classical mass–momentum formulation. In this regard we generalise the approach of Kaminskiet al. (J. Fluid Mech., vol. 526, 2005, pp. 361–376) to unsteady problems, neglecting only viscous effects and relatively small boundary terms in the governing equations. Our results show that dispersion originating from the radial dependence of longitudinal velocity plays a fundamental role in longitudinal transport. Indeed, one is able to find dispersion in the steady state, although it has received little attention because its effects can then be absorbed into the entrainment coefficient. Specifically, we identify two types of dispersion. Type I dispersion exists in a steady state and determines the rate at which energy is transported relative to the rate at which momentum is transported. In unsteady jets type I dispersion is responsible for the separation of characteristic curves and thus the hyperbolic, rather than parabolic, nature of the governing equations, in the absence of longitudinal mixing. Type II dispersion is equivalent to Taylor dispersion and results in the longitudinal mixing of the front. This mixing is achieved by a deformation of the self-similar profiles that one finds in steady jets. Using a comparison with the local eddy viscosity, and by examining dimensionless fluxes in the vicinity of the front, we show that type II dispersion provides a dominant source of longitudinal mixing.

Second-order integral model for a round turbulent buoyant jet

2002 ◽  
Vol 459 ◽  
pp. 397-428 ◽  
Author(s):  
HONGWEI WANG ◽  
ADRIAN WING-KEUNG LAW
Keyword(s):  
Mass Flux ◽  
Richardson Number ◽  
Momentum Flux ◽  
Second Order ◽  
Integral Model ◽  
Width Ratio ◽  
Buoyant Jet ◽  
Turbulent Characteristics ◽  
Momentum Fluxes ◽  
Local Mean

The development of a second-order integral model for a round turbulent buoyant jet is reported based on new experimental data on turbulent mass and momentum transport. The mean and turbulent characteristics of a round vertical buoyant jet covering the full range from jets to plumes were investigated using a recently developed combined digital particle image velocimetry (DPIV) and planar laser-induced fluorescence (PLIF) system. The system couples the two well-known techniques to enable synchronized planar measurements of flow velocities and concentrations in a study area. The experimental results conserved the mass and momentum fluxes introduced at the source accurately with closure errors of less than 5%. The momentum flux contributed by turbulence and streamwise pressure gradient was determined to be about 10% of the local mean momentum flux in both jets and plumes. The turbulent mass flux, on the other hand, was measured to be about 7.6% and 15% of the mean mass flux for jets and plumes respectively. While the velocity spread rate was shown to be independent of the flow regime, the concentration-to-velocity width ratio λ varied from 1.23 to 1.04 during the transition from jet to plume. Based on the experimental results, a refined second-order integral model for buoyant jets that achieves the conservation of total mass and momentum fluxes is proposed. The model employs the widely used entrainment assumption with the entrainment coefficient taken to be a function of the local Richardson number. Improved prediction is achieved by taking into account the variation of turbulent mass and momentum fluxes. The variation of turbulent mass flux is modelled as a function of the local Richardson number. The turbulent momentum flux, on the other hand, is treated as a fixed percentage of the local mean momentum flux. In addition, unlike most existing integral models that assume a constant concentration-to-velocity width ratio, the present model adopts a more accurate approach with the ratio expressed as a function of the local Richardson number. As a result, smooth transition of all relevant mean and turbulent characteristics from jet to plume is predicted, which is in line with the underlying physical processes.

On a momentum-mass flux diagram for turbulent jets, plumes and wakes

1961 ◽  
Vol 10 (1) ◽  
pp. 101-112 ◽  
Author(s):  
B. R. Morton
Keyword(s):  
Differential Equation ◽  
Mass Flux ◽  
Momentum Flux ◽  
Turbulent Jets ◽  
Approximate Solutions ◽  
Main Stream ◽  
Mean Flow ◽  
Actual Distribution ◽  
The Relationship ◽  
Moving Fluid

Many salient features of fully developed turbulent jets, plumes and wakes with steady mean flow are shown clearly by the relationship between the momentum flux and the mass flux in the column of moving fluid. Using a simple model for the flows, this relationship can be found from the solution of a single ordinary differential equation and the character of many related flows can be represented immediately (except for actual distribution in space) on a single momentummass flux diagram.In this note some approximate solutions based on dimensional arguments are outlined briefly for cases of buoyant and non-buoyant wakes and jets directed along the axis of a uniform main stream, and momentum-mass flux curves are presented.

The dissipation and shear dispersion of entropy waves in combustor thermoacoustics

2013 ◽  
Vol 733 ◽  
Author(s):  
Aimee S. Morgans ◽  
Chee Su Goh ◽  
Jeremy A. Dahan
Keyword(s):  
Transfer Function ◽  
Gas Turbine ◽  
Acoustic Waves ◽  
Flow Simulation ◽  
Turbulent Channel Flow ◽  
Combustion Noise ◽  
Mean Flow ◽  
Gaussian Form ◽  
Gaussian Profile ◽  
Shear Dispersion

AbstractThis paper considers the effect of flow advection on entropy waves. The interest is in whether entropy waves persist in gas turbine combustors, between the flame, where they are generated, and the combustor exit, where their acceleration generates acoustic waves (known as ‘entropy noise’ or ‘indirect combustion noise’). Entropy wave advection within a simplified fully developed turbulent channel-flow simulation is investigated. Entropy wave dissipation is found to be negligible, with loss of entropy wave strength caused predominantly by mean flow shear dispersion. The impulse response of entropy perturbations downstream of where they are generated is shown to be well modelled by a Gaussian profile in time. This yields a (different) Gaussian form for the inlet–outlet transfer function of entropy perturbations. For representative gas turbine flows, the magnitude of this transfer function is such that significant entropy wave strength will remain at the combustor exit, confirming that entropy-generated acoustic waves are likely to be important.

Study of Turbulent Variable Density Jets With Different Asymmetric Geometries

Volume 1 ◽  
2004 ◽  
Author(s):  
Bachir Imine ◽  
Miloud Abidat ◽  
Omar Imine ◽  
Hichem Gazzah ◽  
Iskender Go¨kalp
Keyword(s):  
Kinetic Energy ◽  
Aspect Ratio ◽  
Turbulent Kinetic Energy ◽  
Comparative Studies ◽  
Longitudinal Velocity ◽  
Turbulent Jets ◽  
Reynolds Stress Model ◽  
Variable Density ◽  
Stress Model ◽  
Density Ratios

In the present study, the effects of inlet jet geometry on the process of mixture with variable density have been investigated numerically. Three density ratios were considered, namely 1.0, 1.8 and 0.66 for Air-air, CH4-Air and CO2-Air mixtures respectively. The jets are produced through rectangular, elliptic and triangular tubes with aspect ratio 1.33. A second-order Reynolds stress model (RSM) is used to investigate variable density effects in asymmetric turbulent jets. Comparative studies are presented in the case of the calculations of the average variables such as the longitudinal velocity, species and the turbulent kinetic energy. The results obtained show that the asymmetric geometry noticeably enhances mixture in comparison with the axisymmetric case. Typical phenomenon of 3D jets are observed.

Entrainment waves in decelerating transient turbulent jets

2009 ◽  
Vol 638 ◽  
pp. 117-140 ◽  
Author(s):  
MARK P. B. MUSCULUS
Keyword(s):  
Momentum Flux ◽  
Leading Edge ◽  
Turbulent Jets ◽  
Local Concentration ◽  
Entrainment Rate ◽  
Engine Fuel ◽  
Mixing Rate ◽  
Deceleration Phase ◽  
Fluid Concentration ◽  
Long Time

A simplified one-dimensional partial differential equation for the integral axial momentum flux during the deceleration phase of single-pulsed transient incompressible jets is derived and solved analytically. The wave speed of the derived first-order nonlinear wave equation shows that the momentum flux transient from the deceleration phase propagates downstream at twice the initial jet penetration rate. Transient-jet velocity data from the existing literature is shown to be consistent with this derivation, and an algebraic analytical solution matches the measured timing and decay of axial velocity after the deceleration transient. The solution also shows that a wave of increased entrainment accompanies the deceleration transient as it travels downstream through the jet. In the long-time limit, the peak entrainment rate at the leading edge of this ‘entrainment wave’ approaches an asymptotic value of three times that of the initial steady jet. The rate of approach to the asymptotic behaviour is controlled by the deceleration rate, which suggests that rate-shaping may be tailored to achieve a desired mixing state at a given time after the end of a single-pulsed jet. In the wake of the entrainment wave, the absolute entrainment rate eventually decays to zero. The local injected fluid concentration also decays, however, so that entrainment rate relative to the local concentration of injected fluid remains higher than in the initial steady jet. An analysis of diesel engine fuel-jets is provided as one example of a transient-jet application in which the considerable increase in the mixing rate after the deceleration phase has important implications.

scholarly journals Studies on the Organizer Problem in Pelmatohydra Oligactis

1945 ◽  
Vol 21 (3-4) ◽  
pp. 155-160 ◽  
Author(s):  
H. P. LI ◽  
T. YAO
Keyword(s):  
Sudden Change ◽  
The Body ◽  
Stage I ◽  
Type I ◽  
Type Ii ◽  
Simultaneous Induction ◽  
Stage 4 ◽  
Stage 1

1. Determination of the hypostome occurs very early in the development of a bud (our stage I) in Pelmatohydra oligactis. 2. The mode of induction by a developing hypostome (type II) is different from that by an adult hypostome (type I). In the former case, an implant always induces a long tube-like outgrowth before any tentacle rudiment becomes visible. In the latter case, simultaneous induction of tentacles and body takes place. Frequently, tentacle rudiments appear before visible differentiation of the body. 3. Type II induction is characteristic of developing hypostomes from some phase of stage 1 up to a stage at which the bud possesses short, but non-motile tentacles (our stage 4). 4. As soon as the detachment of a young polyp from the parent occurs, there is a sudden change in the mode of induction, from type II to type I. 5. Possible explanations of the course of differentiation in the hypostome and of the transition between the two types of induction are suggested.

scholarly journals Numerical Study on Dynamical Structures and the Destratification of Vertical Turbulent Jets in Stratified Environment

Water ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 2085
Author(s):  
Xuan Huang ◽  
Ling-ling Wang ◽  
Jin Xu
Keyword(s):  
Internal Wave ◽  
Horizontal Plane ◽  
Numerical Study ◽  
Turbulent Jets ◽  
Threshold Value ◽  
Pollutant Emission ◽  
Dimensionless Time ◽  
Gaussian Profile ◽  
Dimensionless Analysis ◽  
The Law

The law of pollutant emission and diffusion in stratified waters is a common issue. In this paper, numerical study on the interaction between vertical turbulent jets and the pycnocline is carried out to study the problems of jet’s emission through the large eddy simulation (LES). A trigonometric function disturbance (TFD) method is developed to ensure the velocity distribution of the jet in the horizontal plane yield to Gaussian profile. Numerical simulations are carried out in the range of 1.11 < Frp < 4.77, corresponding to 1393 < Rep < 5979, where the Froude number Frp and the Reynolds number Rep are defined at the entrance of pycnocline. The coherent structure and internal waves are observed at the pycnocline during the process of jets impinging. After the impingement, the destratification effects can be found. It can be found that Frp = 3 is a threshold value for the interaction between jets and the pycnocline. When Frp > 3, the interaction becomes intensely. Furthermore, the fitting formula of the radial momentum flux dissipation rate that is used to describe the decay of energy contained by the jets during the impinging process, is established through the dimensionless analysis. As a result, the influence range of the jet on the horizontal plane can be evaluated by Rep. It is also found that the destratification of jets is mainly affected by the velocity of the internal wave induced by jets. In addition, by employing the dimensionless time T related to that velocity, the law of destratification varies with dimensionless time is obtained, which can be summarized as follows: Due to the influence of the first internal wave, the thickness of the pycnocline increases rapidly and reaches a critical value at T = 1.4, after that, the increase of the thickness of the pycnocline becomes linear.

Decay of momentum flux in submerged jets

1985 ◽  
Vol 154 ◽  
pp. 91-110 ◽  
Author(s):  
Wilhelm Schneider
Keyword(s):  
Flow Field ◽  
Momentum Flux ◽  
Turbulent Jets ◽  
Critical Distance ◽  
Axisymmetric Jet ◽  
Axisymmetric Jets ◽  
Submerged Jets ◽  
Classical Boundary ◽  
Turbulent Plane ◽  
Plane Jet

Slender laminar and turbulent, plane and axisymmetric jets emerging from orifices in plane or conical walls are studied at large distances from the orifices. The entrainment of momentum coupled with the entrainment of volume into a jet is determined, and its effect on the flow field is analysed by combining inner and outer expansions with a multiple scaling approach.In turbulent (plane or axisymmetric) jets, the axial velocity decreases more rapidly than predicted by classical boundary-layer solutions, and the momentum flux vanishes as the distance from the orifice tends to infinity. The analysis unveils a source of discrepancies in previous experimental data on turbulent jets.In a laminar plane jet, the momentum flux changes but little. In a laminar axisymmetric jet, the momentum flux changes slowly, yet considerably. When a critical distance from the orifice is approached, the momentum flux in the jet becomes very small, the jet diameter very large, and a toroidal viscous eddy is predicted. The structure of the flow field is briefly discussed.

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