Source Momentum Flux Research Articles

Optimization of Gravity Wave Source Parameters for Improved Seasonal Prediction of the Quasi-Biennial Oscillation

AbstractA methodology is presented for objectively optimizing nonorographic gravity wave source parameters to minimize forecast error for target regions and forecast lead times. In this study, we employ a high-altitude version of the Navy Global Environmental Model (NAVGEM-HA) to ascertain the forcing needed to minimize hindcast errors in the equatorial lower stratospheric zonal-mean zonal winds in order to improve forecasts of the quasi-biennial oscillation (QBO) over seasonal time scales. Because subgrid-scale wave effects play a large role in driving the QBO, this method leverages the nonorographic gravity wave drag (GWD) parameterization scheme to provide the necessary forcing. To better constrain the GWD source parameters, we utilize ensembles of NAVGEM-HA hindcasts over the 2014–16 period with perturbed source parameters and develop a cost function to minimize errors in the equatorial lower stratosphere compared to analysis. Thus, we may determine the set of GWD source parameters that yields a forecast state that most closely agrees with observed QBO winds over each optimization time interval. Results show that the source momentum flux and phase speed spectrum width are the most important parameters. The seasonal evolution of optimal parameter value, specifically a robust semiannual periodicity in the source strength, is also revealed. Changes in optimal source parameters with increasing forecast lead time are seen, as the GWD parameterization takes on a more active role as QBO driver at longer forecast lengths. Implementation of a semiannually varying source function at the equator provides RMS error improvement in QBO winds over the default constant value.

Highly buoyant bent-over plumes in a boundary layer

Highly buoyant plumes, such as wildfire plumes, in low to moderate wind speeds have initial trajectories that are steeper than many industrial waste plumes. They will rise further into the atmosphere before bending significantly. In such cases the plume's trajectory will be influenced by the vertical variation in horizontal velocity of the atmospheric boundary layer. This paper examined the behavior of a plume in an unstratified environment with a power-law ambient velocity profile. Examination of previously published experimental measurements of plume trajectory show that inclusion of the boundary layer velocity profile in the plume model often provides better predictions of the plume trajectory compared to algebraic expressions developed for uniform flow plumes. However, there are many cases in which uniform velocity profile algebraic expressions are as good as boundary layer models. It is shown that it is only important to model the role of the atmospheric boundary layer velocity profile in cases where either the momentum length (square root of source momentum flux divided by the reference wind speed) or buoyancy length (buoyancy flux divided by the reference wind speed cubed) is significantly greater than the plume release height within the boundary layer. This criteria is rarely met with industrial waste plumes, but it is important in modeling wildfire plumes.

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

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.

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

The flow structure of jets from transient sources and implications for modeling short‐duration explosive volcanic eruptions

AbstractWe used laboratory experiments to examine the rise process in neutrally buoyant jets that resulted from an unsteady supply of momentum, a condition that defines plumes from discrete Vulcanian and Strombolian‐style eruptions. We simultaneously measured the analog‐jet discharge rate (the supply rate of momentum) and the analog‐jet internal velocity distribution (a consequence of momentum transport and dilution). Then, we examined the changes in the analog‐jet velocity distribution over time to assess the impact of the supply‐rate variations on the momentum‐driven rise dynamics. We found that the analog‐jet velocity distribution changes significantly and quickly as the supply rate varied, such that the whole‐field distribution at any instant differed considerably from the time average. We also found that entrainment varied in space and over time with instantaneous entrainment coefficient values ranging from 0 to 0.93 in an individual unsteady jet. Consequently, we conclude that supply‐rate variations exert first‐order control over jet dynamics, and therefore cannot be neglected in models without compromising their capability to predict large‐scale eruption behavior. These findings emphasize the fundamental differences between unsteady and steady jet dynamics, and show clearly that: (i) variations in source momentum flux directly control the dynamics of the resulting flow; (ii) impulsive flows driven by sources of varying flux cannot reasonably be approximated by quasi‐steady flow models. New modeling approaches capable of describing the time‐dependent properties of transient volcanic eruption plumes are needed before their trajectory, dilution, and stability can be reliably computed for hazards management.

The merging of two unequal axisymmetric parallel turbulent jets

Results of an experimental study of the merging of unequal parallel round turbulent jets are presented. Experiments were conducted for a jet axial separation to nozzle diameter ratio of 3.0 and the Reynolds numbers ranging from 8 000 to 15 000. The distance to the point where the jets are merged was measured for a range of jet source momentum flux ratios. Three different merger criteria were used based on the mean velocity profile, mean passive tracer concentration profile, and Reynolds stress profile. The results show that the concentration profile merges closest to the jet sources followed by the velocity profile with the Reynolds stress profile merging furthest from the nozzles. For all three profiles the merge distance is relatively insensitive to the momentum flux ratio, consistent with previous findings for slot jets and buoyant round jets. The measured merge distances are consistent with previously published results for equal round jets, though the poor spatial resolution of data in the literature means that limited comparison is possible. There are no studies of unequal jet merger currently in the literature that could be used for comparison.

Physical Model Study of an Alternating Diffuser for Thermal Discharge

Physical model tests were conducted of the mixing of heated water from a proposed thermal diffuser. Dilutions were measured and flows imaged by a three-dimensional laser-induced fluorescence system that provides vastly more data than conventional thermocouple techniques. The flows were quite three-dimensional. For zero current speed, the effluent mixed over the water depth, but only in a limited region, and a two-layer stratified flow developed toward the edges of the effluent field and farther downstream. For zero and slow currents, the lowest surface dilution occurred where the jet centerline intersected the water surface and could be reasonably predicted by simple free-jet calculations. The prediction of an entrainment model, UM3, was close to the observed results because the flow is three-dimensional and the entraining water can be supplied from the ends. CORMIX considerably underestimated the observed dilution because it uses a two-dimensional analysis and neglects the effect of source momentum flux....

Transition behaviour of weak turbulent fountains

Numerical simulations of fully turbulent weak fountain flow are used to provide direct evidence for the scaling behaviour of fountain flow over the Froude number range Fr = 0.1–2.1 and Reynolds number range Re = 20–3494. For very weak flow at Fr < 0.4, the flow mean penetration height, Zm, scales with Zm/R0 = A1Fr2/3 + A2Fr2/3 where R0 is the source radius. A1 and A2 are constants which quantify the separate effects of the radial acceleration of fountain fluid from the source (A1) and the backpressure from the surrounding intrusion, if present, on the upflow (A2). The evidence presented in this work suggests that the mechanisms for the two parts in the scaling of Zm scale with Fr2/3. The intrusion behaviour varies with the Reynolds number (Re) but there is no Re affect on the fountain penetration height. For Re < 250 the radial intrusion flow is subcritical and has different behaviour. For Fr between 0.4 and 2.1 the effect of source momentum flux increases and the flow structure changes to one where there is a coherent upflow and a cap region where the flow stagnates and then reverses. The two regions have separate scaling behaviour such that the overall height, through this transition range of Froude numbers, can be described by Zm/R=C1Fr2/3 + C2Fr2, where C1 and C2 are constants. Over this transition range the effect of source velocity profile is more significant than the Reynolds number effects and the effect of inlet turbulence is minor.

Characterising line fountains

We present analytical solutions for the initial rise heightzmof two-dimensional turbulent fountains issuing from a horizontal linear source of width 2b0into a quiescent environment of uniform density. Using the initial rise height prediction as a measure we classify line fountains into three types depending on their source conditions. For source Froude numbersFr0≫ 1, the near-source flow of the ‘forced’ fountain is dominated by source momentum flux and behaves like a jet; the asymptotic solution to the fountain equations yields, in agreement with previous studies,zm/b0~Fr04/3. ForFr0=O(1) the fountain is ‘weak’ and fluid is projected vertically to a height that is consistent with an energy-conserving flow – the sensitivity of the rise height withFr0increases aszm/b0~Fr02. ForFr0≪ 1, the fountain is ‘very weak’ and we find thatzm/b0~Fr02/3. As the local value of the Froude number decreases with height, all three forms of fountain behaviour identified are expected above a highly forced source and we provide scalings for the three lengths that contribute to the total rise height. Comparisons between our predicted rise heights and the previous experimental results show good agreement across a wide range ofFr0. The collated data highlights that experiments have focused in the majority on fountains above sources with intermediateFr0. Notably there is a lack of measurements on very weak line fountains and of independent experimental confirmation of the initial rise heights across the range ofFr0.

Warm discharges in cold fresh water. Part 1. Line plumes in a uniform ambient

Turbulent buoyant plumes in cold fresh water are analysed, assuming a quadratic dependence of density on temperature. The model is based on the assumption that entrainment velocity is proportional to vertical velocity in the plume. Numerical and asymptotic solutions are obtained for both rising and descending plumes from virtual sources with all possible combinations of buoyancy, volume and momentum fluxes. Physical sources can be identified as points on trajectories of plumes from virtual sources.The zero-buoyancy condition, at which the plume and the ambient have equal densities but their temperatures are on opposite sides of the temperature of maximum density, is of particular importance. If an upwardly buoyant plume rising through a body of water reaches the surface before passing through its zero-buoyancy level, it will form a surface gravity current; otherwise, the plume water will return to the source as a fountain. The height at which zero buoyancy is attained generally decreases as the source momentum flux increases: greater plume velocity produces greater entrainment and hence more rapid temperature change. Descending plumes, if ejected downwards against upward buoyancy, may be classified as strongly or weakly forced according to whether they reach the zero-buoyancy condition before being brought to rest. If they do, they continue to descend with favourable buoyancy; otherwise, they may form an inverted fountain. Once a descending plume has attained downward buoyancy, it can continue to descend indefinitely, ultimately behaving like a plume in a fluid with a linear equation of state. In contrast, a rising plume will eventually come to rest, however large its initial upward buoyancy and momentum fluxes are.

Time-dependent plumes and jets with decreasing source strengths

The classical bulk model for isolated jets and plumes due to Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234, 1956, p. 1) is generalized to allow for time-dependence in the various fluxes driving the flow. This new system models the spatio-temporal evolution of jets in a homogeneous ambient fluid and Boussinesq and non-Boussinesq plumes in stratified and unstratified ambient fluids.Separable time-dependent similarity solutions for plumes and jets are found in an unstratified ambient fluid, and proved to be linearly stable to perturbations propagating at the velocity of the ascending plume fluid. These similarity solutions are characterized by having time-independent plume or jet radii, with appreciably smaller spreading angles ( is the conventional entrainment coefficient. These new similarity solutions are closely related to the similarity solutions identified by Batchelor (Q. J. R. Met. Soc., vol. 80, 1954, p. 339) in a statically unstable ambient, in particular those associated with a linear increase in ambient density with height.If the source buoyancy flux (for a rising plume) or source momentum flux (for a rising jet) is decreased generically from an initial to a final value, numerical solutions of the governing equations exhibit three qualitatively different regions of behaviour. The upper region, furthest from the source, remains largely unaffected by the change in buoyancy flux or momentum flux at the source. The lower region, closest to the source, is an effectively steady plume or jet based on the final (lower) buoyancy flux or momentum flux. The transitional region, in which the plume or jet adjusts between the states in the lower and upper regions, appears to converge very closely to the newly identified stable similarity solutions. Significantly, the predicted narrowing of the plume or jet is observed. The size of the narrowing region can be determined from the source conditions of the plume or jet. Minimum narrowing widths are considered with a view to predicting pinch-off into rising thermals or puffs.

Boussinesq plumes and jets with decreasing source strengths in stratified environments

Solutions to the equations of Morton et al. (Proc. R. Soc. Lond. A, vol. 234, 1956, p. 1) describing turbulent plumes and jets rising in uniformly stratified environments are identified for the first time. The evolution of plumes and jets with sources whose driving flux decreases with time is considered in a stratified environment. Numerical calculations indicate that as the source buoyancy flux, for a Boussinesq plume (or source momentum flux, for a Boussinesq jet), is decreased, a transitional narrowing region with characteristic spreading angle tan -1 (2α/3) is formed, where α is the well-known entrainment coefficient. The plume or jet dynamics are modelled well by a separable solution to the governing equations which predicts stalling in the plume at a critical stall time t s = π/N and stalling in the jet at a critical stall time t s = π/(2N), where N is the buoyancy frequency of the ambient background stratification. This stall time is independent of the driving source conditions, a prediction which is verified by numerical solution of the underlying evolution equations.

The mixing in a room by a localized finite-mass-flux source of buoyancy

The mixing produced by a turbulent buoyant plume with finite mass flux in a room is examined analytically and numerically. The entrainment of ambient fluid into the ascending buoyant plume leads to a return flow in the room which carries fluid downwards from the top of the room. The cycling of ambient fluid through the buoyant plume and the return flow causes the density to become uniform and gradually evolve towards that of the source fluid. As a result the buoyancy flux associated with the input fluid decreases and the plume motion becomes dominated by the source momentum flux. We develop an asymptotic model of the mixing using buoyant plume theory for a momentum-dominated flow. This provides an analytical description of the evolution of the density in the room which is in excellent accord with a full numerical simulation, and provides an improved description of the experimental filling-box data originally presented by Baines & Turner (1969).

Thermal stratification produced by plumes and jets in enclosed spaces

The airflow and thermal stratification produced by a localised heat source located at floor level in a closed room is of considerable practical interest and is commonly referred to as a ‘filling box’. In rooms with low aspect ratios H/ R≲1 (room height H to characteristic horizontal dimension R) the thermal plume spreads laterally on reaching the ceiling and a descending horizontal ‘front’ forms separating a stably stratified, warm upper region from cooler air below. The stratification is well predicted for H/ R≲1 by the original filling box model of Baines and Turner (J. Fluid. Mech. 37 (1968) 51). This model represents a somewhat idealised situation of a plume rising from a point source of buoyancy alone—in particular the momentum flux at the source is zero. In practical situations, real sources of heating and cooling in a ventilation system often include initial fluxes of both buoyancy and momentum, e.g. where a heating system vents warm air into a space. This paper describes laboratory experiments to determine the dependence of the ‘front’ formation and stratification on the source momentum and buoyancy fluxes of a single source, and on the location and relative strengths of two sources from which momentum and buoyancy fluxes were supplied separately. For a single source with a non-zero input of momentum, the rate of descent of the front is more rapid than for the case of zero source momentum flux and increases with increasing momentum input. Increasing the source momentum flux effectively increases the height of the enclosure, and leads to enhanced overturning motions and finally to complete mixing for highly momentum-driven flows. Stratified flows may be maintained by reducing the aspect ratio of the enclosure. At these low aspect ratios different long-time behaviour is observed depending on the nature of the heat input. A constant heat flux always produces a stratified interior at large times. On the other hand, a constant temperature supply ultimately produces a well-mixed space at the supply temperature. For separate sources of momentum and buoyancy, the developing stratification is shown to be strongly dependent on the separation of the sources and their relative strengths. Even at small separation distances the stratification initially exhibits horizontal inhomogeneity with localised regions of warm fluid (from the buoyancy source) and cool fluid. This inhomogeneity is less pronounced as the strength of one source is increased relative to the other. Regardless of the strengths of the sources, a constant buoyancy flux source dominates after sufficiently large times, although the strength of the momentum source determines whether the enclosure is initially well mixed (strong momentum source) or stably stratified (weak momentum source).

Ocean Outfalls. I: Submerged Wastefield Formation

Experiments to study the dilution and formation of wastefields resulting from discharge into a linearly density‐stratified steady current of arbitrary speed and direction are reported in three papers. The experiments were conducted using a model multi‐port diffuser for a range of parameters typical of ocean wastewater outfalls. In this paper, results are given from which the established wastefield characteristics for the “line plume” source conditions can be predicted. In general, the wastefield rise height and thickness decrease as the current speed increases. Dilution increases with current speed for all current directions, with diffusers perpendicular to the current resulting in higher dilutions than when parallel. Concentration profiles for perpendicular currents show little horizontal structure as the individual plumes rapidly merge. The results for dilution show no dependency on port spacing or source momentum flux over the parameter range tested, and the dominant source parameter is the buoyancy fl...

Horizontal Round Buoyant Jet in Shallow Water

A common outfall configuration in practice involves a buoyant discharge through a horizontal circular pipe into a shallow body of water. A detailed experimental study of the near‐field flow and mixing characteristics is presented. The proximity of the bed and the free surface has a significant influence on the flow pattern. Experimental results are adequately represented by length and buoyancy scales defined by source momentum and source buoyancy flux. Jet path, centerline dilution and horizontal and vertical half widths are compared with an integral model in an unconfined environment.

ATMOSPHERIC DISPERSION OF LARGE SCALE SPILLS

Large scale spills of volatile cryogenic or pressurized ambient temperature liquids often produce denser-than-air clouds which disperse in the atmosphere in a manner that is different than trace gases. These differences are due to density or gravity-induced effects such as turbulence damping from the stable density stratification, alteration of the ambient velocity field due to gravity flow, and the source momentum flux. Also important are thermodynamic effects such as aerosol formation and flash vaporization upon release, evaporative cooling, or heat transfer from the ground modifying cloud buoyancy and turbulence. In addition, chemical reaction effects such as hydrolysis with atmospheric water vapor, polymerization, or decomposition which also affects cloud density and thermodynamics can be important. These effects are very important close to the release point but also can have dramatic effects on the cloud as it disperses downwind. Large scale tests involving releases of heavy gases have been conducted...

Internal Hydraulics of Diffusers with Uniform Lateral Momentum Distribution

This study deals with the internal hydraulics of a multiport diffuser that ensures uniform momentum distribution along the span of main pipe. Multiport diffuser systems consisting of lateral outlets are generally found suitable for mixing chemicals and for the disposal of large quantities of industrial effluents. For effluents whose mixing is dominated by the source momentum flux, it is important to maintain a uniform lateral momentum distribution along the length of the main pipe. Experiments to verify the proposed model were also conducted. Results in the form of design charts for specific cases of hydraulically smooth as well as rough pipes are presented.

Total emission of soot and thermal radiation by free turbulent diffusion flames

The emission of soot and thermal radiation by free turbulent diffusion flames of CO, H 2, CH 4, C 2H 6, C 3H 8, C 2H 4, and C 2H 2 was studied over a range spanning the transition from forced convection to natural convection (burner diameters 0.7–10.9 mm, buoyancy Richardson ratios Ri L ≡ π g ϱ ∞L 3 4 G ̇ 0 of 2–10,000, where L is flame length, G ̇ 0 is source momentum flux, ϱ ∞ is ambient air density, and g is acceleration due to gravity. Both soot emission, expressed as the ratio χ S between soot output and carbon input, and radiant cooling, expressed as the ratio χ R between radiant heat loss and chemical energy input (heat of combustion), rise with Ri L and tend to plateaus as natural convection becomes dominant Ri L > 2000). However, it also appears that they fundamentally depend on parameters other than Ri L, χ S being strongly correlated with the second Damkohler ratio or chemical loading factor and χ R with the analogous thermal loading factors. The behavior observed is in reasonable accord with theoretical expectations. The plateau values of χ R at high Ri L are 0.18–0.4 for the hydrocarbon fuels, and for χ S they are 0.0015–0.0045 for the hydrocarbons other than acetylene. Acetylene flames emit up to 100 times as much soot as the others. Buoyancy is unimportant below Ri L = 200 since χ S and χ R are then independent of burner orientation (vertical or horizontal). Also, the soot emission for hydrocarbon fuels other than acetylene seems to stablize, indicating χ S ⋍ 0.0003 at Ri L < 200.

Effective source flux parameters for use in analytical plume rise models

The derivation of analytical or closed form plume rise and growth formulas is reviewed in order to clarify (a) the relationship between the various forms of the formulas which appear in the literature and (b) the relationship between the different source momentum and buoyancy flux definitions which appear in the literature. In particular, the effective source radius, source momentum flux parameter, and source buoyancy flux parameter, consistent with the assumptions inherent in analytical models, are discussed. The importance of using corrected source flux terms is illustrated by comparing predictions with field observations of a plume from a large industrial source.