Baroclinic Vorticity Generation Research Articles

Measurements and prediction of air entrainment rates of pool fires

Motivated by the various applications of entrainment rate correlations in fire research and the large uncertainty in the efficacy of existing correlations and experimental data, the first particle imaging velocimetry (PIV)-based measurements of fire-induced flow field around pool fires burning methanol, heptane, and toluene were obtained. Air entrainment rates for 15-cm and 30-cm pool fires burning the three different fuels were calculated based on the mean velocity field. The entrainment data for the six fires could be correlated well using the fire Froude number as the nondimensional parameter. An existing kinematic approach to the prediction of the fire-induced flow field was extended to the present fires. The driving processes for the entrainment flow, namely, the volumetric heat release and the baroclinic vorticity generation, were evaluated based on correlations of buoyant diffusion flame structure in the literature. The predicted entrainment velocities were substantially higher than the measurements but were in qualitative agreement with the data. On this basis the heat release rate and vorticity correlations used in the analysis were corrected by using a smaller radius for the 1/c point in the velocity profile. The modified predictions were in better agreement with the experimental data. Therefore, further evaluation of the kinematic approach with proper heat release rate and vorticity distributions is warranted.

Transport element method for axisymmetric variable-density flow and its application to the spread and dispersion of a dense cloud

The Lagrangian transport element method for the solution of axisymmetric constant-density flow is extended to variable-density flow. In this method, thin vortex ring-elements of the Rankine type are used to discretize the vorticity field as well as the density gradient field. At the axi s, since the ring element concept does not apply, several Hill's spherical vortices are used. A simple algebraic expression for determining the density gradient is derived. Using this expression, the rate of baroclinic vorticity generation in the vorticity equation can be easily obtained. The transport element method, free of using Boussinesq approximation and turbulent closure modeling, is used to simulate variable-density flows such as the spread and dispersion of an axisymmetric dense gas cloud following its sudden release. The results show that multiple large scale concentric rings are generated at the cloud surface. Most of the cloud material is associated with the spreading front where the largest and strongest ring is located. The generation of these rings is explained using the baroclinic-vorticity generation mechanism. The entrainment of surrounding air into the dense cloud is shown to occur mainly at the cloud top through the large-scale engulfment by the vortex rings. The results on the cloud dispersion pattern and the rate of the horizontal spreading of the cloud compare well with experimental observations.

Compressible vortex reconnection

Reconnection of two antiparallel vortex tubes is studied as a prototypical coherent structure interaction to quantify compressibility effects in vorticity dynamics. Direct numerical simulations of the Navier-Stokes equations for a perfect gas are carried out with initially polytropically related pressure and density fields. For an initial Reynolds number (Re = Γ /v, circulation divided by the kinematic viscosity) of 1000, the pointwise initial maximum Mach number (M) is varied from 0.5 to 1.45. At M=0.5, not surprisingly, the dynamics are essentially incompressible. As M increases, the transfer of Γ starts earlier. For the highest M, we find that shocklet formation between the two vortex tubes enhances early Γ transfer due to viscous cross-diffusion as well as baroclinic vorticity generation. The reconnection at later times occurs primarily due to viscous cross-diffusion for all M. However, with increasing M, the higher early Γ transfer reduces the vortices’ curvature growth and hence the Γ transfer rate; i.e. for the Re case studied, the reconnection timescale increases with M. With increasing M, reduced vortex stretching by weaker ‘bridges’ decreases the peak vorticity at late times. Compressibility effects are significant in countering the stretching of the bridges even at late times. Our observations suggest significantly altered coherent structure dynamics in turbulent flows, when compressible.

Effects of the free-stream density ratio on free and forced spatially developing shear layers

The effects of the free-stream density ratio on the evolution of the incompressible, high Reynolds and Froude number, confined mixing layer are investigated numerically. Two-dimensional simulations of the spatially developing flow with and without external forcing are obtained using the Lagrangian transport element method. Results indicate that a nonunity density ratio alters the flow characteristics significantly. In the unforced flow, it increases the layer growth as the slow stream becomes denser, biases the speed of both the linear instability waves and the rollup eddies toward that of the denser stream, and modifies entrainment in favor of the dense fluid. These results, which are in agreement with experimental and analytical evidence, are analyzed in terms of the evolution of the vorticity field and, in particular, of the action of the mechanism of baroclinic vorticity generation. It is found that this mechanism creates vorticity of opposite signs across each eddy, which, through simple kinematical arguments, is linked to the alteration of the eddy speed and the modification of the local entrainment patterns. High-amplitude external forcing modifies the growth behavior of the layer while leaving its entrainment characteristics and the eddy speeds unaffected. In this case the layer growth is no longer monotonically varying with the free-stream density ratio. Instead, it is a strong function of the momentum ratio, reaching a minimum at a momentum ratio of unity and increasing more significantly for higher values of this parameter. Enhancement of the layer growth via forcing occurs only when the momentum ratio is substantially different from unity. It is found that the forced layer growth characteristics are related to the layer orientation, which is also a function of the momentum ratio. Using this fact and basic principles, a simple analytical model is derived to explain the numerical results. It is suggested that the unforced flow behaves differently due to its initial instability characteristics that are bypassed when forcing is present.

A computational model for the rise and dispersion of wind-blown, buoyancy-driven plumes—III. Penetration of atmospheric inversion

Abstract The computational model of Zhang and Ghoniem (1993, Atmospheric Environment 27, 2295–2311; 1994. Atmospheric Environment, 28, 3005–3018) is applied to simulate the dispersion of a wind-blown, buoyancy-driven plume in a stratified atmosphere characterized by a sharp density drop across a thin horizontal inversion layer. Results show that the plume may completely penetrate, partially penetrate or get fully trapped below the inversion layer depending on the plume buoyancy, the height, strength, and thickness of the inversion layer. Plume penetration is favored by the lower height, weaker strength and larger thickness of the inversion layer. The presence of inversion accelerates the plume bifurcation into two diverging, downwind drifting material lumps, supported by the formation of two counter-rotating streamwise eddies, below the inversion. As a plume impinges on an inversion, internal gravity waves are generated along the layer, absorbing some of the plume energy and reducing its penetration potential. Concomitant with the wave activities, re-entrainment plays an important role in determining the final equilibrium height and the generation of weak oscillations in the plume trajectory, ambient circulation and trapping fraction. The mechanism of baroclinic-vorticity generation is used to interpret various buoyancy-related phenomena in this problem. The plume trapping fraction, defined as the percentage of plume material held below the initial inversion height, is calculated under different conditions. Comparison with laboratory experiments shows that the predicted terminal trapping fraction agrees well with the measurements.

The vorticity dynamics of an exothermic, spatially developing, forced, reacting shear layer

The effects of combustion exothermicity on the vorticity dynamics of a low Mach number, forced,spatially developing, high Reynolds number, reacting shear layer are investigated using the results of a two-dimensional numerical simulation. The chemical reaction is modeled by single-step, irreversible, Arrhenius kinetics with a high Damkohler number, a moderate Karlovitz number, and significant heat release. The numerical solution is obtained using the Lagrangian transport-element method. Results indicate that a fast exothermic reaction, with the concomitant density variation, modifies the shape, size, speed, and orientation of the large-scale vortical structures and their downstream interactions, leading to an overall reduction of the cross-stream growth of the mixing region. These changes are traced to the two primary mechanisms by which density variation due to heat release modifies vorticity: volumetric expansion and baroclinic generation. It is shown that the weakening of the vorticity due to volumetric expansion diminishes the cross-stream mixing zone by aligning the eddy major axis with the flow direction. Baroclinic vorticity generation, on the other hand, is responsible for the formation of a band of positive vorticity on the outer perimeter of the large eddies, whose vorticity is predominantly negative, which inhibits entrainment and alters the eddy interaction mechanism from pairing of adjacent addies to tearing of smaller eddies by their larger neighbors. Both mechanisms contribute to the acceleration of the eddies in the streamwise direction.

A comparison of Arctic and Antarctic mesoscale vortices

The mesoscale (less than 100km) vortices occurring in the two polar regions are considered in terms of their geographical and seasonal distribution, satellite cloud signatures and forcing mechanisms. Environmental conditions important in the development of the vortices are considered, including sensible heat flux, stability throughout the troposphere and synoptic factors, such as baroclinicity and upper air cold pools. A scheme to classify the observed vortices in the two polar regions in terms of the physical mechanisms behind their formation and development is proposed. The processes considered important are baroclinic instability, convection and vorticity generation through cyclonic vorticity advection and topographic forcing. The major difference between the systems observed in the two polar regions is the lack of deep convection in the southern hemisphere, which precludes the development of many of the vigorous types of system found in the north and the major role that topography plays in the Antarctic coastal region. The most common type of vortex found in the Antarctic occurs over the ice‐free ocean to the west of synoptic scale disturbances and is similar to the type of northern system know as a comma cloud.

Influence of initial conditions on compressible vorticity dynamics

Prompted by the lack of a unique choice of pressure (P) and density (ρ) fields for a compressible free vortex and by the observed dependence of turbulence dynamics on initial P and ρ in compressible simulations, we address the effects of initial conditions on the evolution of a single vortex, on the prototypical phenomenon of vortex reconnection, and on two-dimensional turbulence. Two previous choices of initial conditions used for numerical simulations of compressible turbulence have been: (i) both P and ρ uniform (constant initial conditions, CIC), and (ii) uniform ρ with P determined from the Poisson equation (constant density initial conditions, CDIC). We find these initial conditions to be inappropriate for compressible vorticity dynamics studies. Specifically, in compressible reconnection, the effects of baroclinic vorticity generation and shocklet formation cancel each other during early evolution for CDIC, thus leading to almost incompressible behavior. Although CIC captures compressibility effects, it incorrectly changes the initial vorticity distribution by introducing strong acoustic transients, thereby significantly altering the evolving dynamics. Here, a new initial condition, called polytropic initial condition (PIC), is proposed, for which the Poisson equation is solved for initially polytropically related P and ρ fields. PIC provides P and ρ distributions within vortices which are consistent with those observed in shock-wedge interaction experiment and also leads to compressible solutions with no acoustic transients. At low Mach number (M), we show that the effects of all these three initial conditions can be predicted by low-M asymptotic theories of the Navier-Stokes equations. At high M, it is shown here that inappropriate initial conditions may alter the evolutionary dynamics and, hence, lead to wrong conclusions regarding compressibility effects. We argue that PIC is a more appropriate choice.

The three-dimensional structure of periodic vorticity layers under non-symmetric conditions

Numerical simulations of a three-dimensional temporally growing shear layer are obtained at high Reynolds number and zero Froude number using a vortex scheme modified for a variable-density flow. Attention is focused on the effect of initial vorticity and density distributions on the interaction between instability modes which lead to the generation and intensification of streamwise vorticity. Results show that the three-dimensional instabilities evolve following the formation of concentrated span wise vorticity cores. The deformation of each core along its span resembles the amplification of the translative instability. The generation of vortex rods, which wrap around individual cores while stretching between neighbouring cores, suggest a mode similar to the Corcos instability. The instability modes leading to the formation of both structures, energized by the extensional strain generated by the cores, grow simultaneously. A similar series of events occurs in variable-density shear layers and in shear layers which start with an asymmetric vorticity distribution. Baroclinic vorticity generation in the variable-density layer leads to the formation of asymmetric cores whose volumetric composition is biased towards the lighter fluid. The structures are propelled, by their asymmetric vorticity distribution, in the direction of the heavier stream while their eccentric spinning forces an uneven stretching of the vortex rods. The origin of the asymmetry is established by comparing these with the results of a shear layer with an initially asymmetric vorticity distribution in a uniform-density flow. The strong late-stage asymmetry exhibited by the former is not observed in the latter. Thus, baroclinic vorticity generation is responsible for the observed symmetry. We also find that initially asymmetric vorticity distribution does not, as suggested before, lead to asymmetric spacing between the streamwise rods, it is concluded that the experimentally observed asymmetric spacing must arise after pairing.

Vorticity generation and evolution in shock-accelerated density-stratified interfaces

The results of direct numerical simulations of inviscid planar shock-accelerated density-stratified interfaces in two dimensions are presented and compared with shock tube experiments of Haas [(private communication, 1988)] and Sturtevant [in Shock Tubes and Waves, edited by H. Gronig (VCH, Berlin, 1987), p. 89] . Heavy-to-light (‘‘slow/fast or s/f) and light-to-heavy (‘‘fast/slow,’’ or f/s) gas interfaces are examined and early-time impulsive vorticity deposition and the evolution of coherent vortex structures are emphasized and quantified. The present second-order Godunov scheme yields excellent agreement with shock-polar analyses at early time. A more physical vortex interpretation explains the commonly used (i.e., linear paradigm) designations of ‘‘unstable’’ and ‘‘stable’’ for the f/s and s/f interfaces, respectively. The later time events are Rayleigh–Taylor like and can be described in terms of the evolution of a vortex layer (large-scale translation and rotation): asymmetric tip vortex ‘‘roll-up’’ and ‘‘binding;’’ layer ‘‘instability;’’ convective mixing; and baroclinic vorticity generation from secondary shock–interface interactions.

A Note on Spin up Effects in a Rotating Mixture

The centrifugal batch separation of an initially homogeneous mixture of particles and fluid is considered in containers of finite length for which end-wall effects can be important. The “full” nonlinear theory can be reduced to the determination of an appropriate analytic function. Vortex-line stretching due to suction into the Ekman layers counteracts the baroclinic generation of vorticity caused by the pressure and density fields.

Dynamics of vortex interaction with a density interface

We present results from an experimental and numerical investigation into the dynamics of the interaction between a planar vortex pair or axisymmetric vortex ring with lengthscale a and circulation Γ encountering a planar interface of thickness δ across which the fluid density increases from ρ1 to ρ2. Similarity considerations indicate that baroclinic generation of vorticity and its subsequent interaction with the original vortex is governed by two dimensionless parameters, namely (a/δ) A and R, where A ≡ (ρρ2 − ρ1)/(ρ2 + ρ1) and R ≡ (a3g/Γ2). For thin interfaces (δ [Lt ] a), the interaction is governed only by the parameters A and R. Furthermore, in the Boussinesq limit (A → 0), the dynamics are governed solely by the product AR and the interaction is entirely invertible with respect to the initial locations and direction of propagation of the vortices. We document details of the interaction dynamics in the Boussinesq limit over a range of the parameter AR. Results show that, for relatively small values of AR, rather than the vortex simply rebounding at the interface, its outermost layers are instead successively ‘peeled’ away by baroclinically generated vorticity and form a topologically complex backflow in which the ring fluid, the light fluid and the heavy fluid are intertwined. For larger values of AR the vortex barely penetrates the interface, and our results suggest that in the limit AR → ∞ the interaction with a density interface becomes similar to the interaction at a solid wall. We also present results for thick interfaces in the Boussinesq limit, as well as larger density jumps for which the density parameter A enters as a second similarity quantity. Comparison of the experimental and numerical results demonstrate that many of the features of such interactions can be understood within the context of inviscid fluids, and that inviscid vortex methods can be used to accurately simulate the dynamics of such interactions.

Flame Generation of Vorticity: Vortex Dipoles from Monopoles

Abstract Numerical simulation of premixed flame propagation through a two-dimensional vorticity distribution exhibits the coupling between combustion heat release and fluid dynamics. Effects of both thermal expansion and vorticity generation via baroclinic torques are considered. The rotational part of the velocity field is described by discrete vorticity. The irrotational velocity. given by a velocity potential. is determined by a Poisson equation where the chemical reaction determines the spatial distribution of volume expansion. The transport of a single reacting scalar is computed on an Eulerian mesh with a small-Mach-number flow assumption so that the density gradient is nonzero only within the reaction zone. The vector cross product of density gradient and pressure gradient defines the baroclinic generation of vorticity. With a single initial vortical region-the monopole interacting with a premixed flame-the baroclinic effect produces opposite circulation, and thus. creates a dipole configuration. We use the dipole description in a loose sense. because in some solutions the two signs of vorticity are spatially intermingled. However. in comparison to the monopole. the "dipole" solution has stronger local velocity fluctuations but weaker long-range velocities. Thus, the turbulence in the burnt gas is more intense with smaller length scales than the configuration before the flame-vortex interaction.

Origin and manifestation of flow-combustion interactions in a premixed shear layer

The interactions between the flow field and the combustion process in a premixed shear layer are investigated using the results of numerical simulation. The reaction is governed by a finite-rate Arrhenius kinetics, the flow is compressible and at high Reynolds number, heat release is moderate and molecular heat and mass diffusivities are finite. The thickness of the reaction zone and that of the vorticity layer are approximately the same. Lagrangian simulations are obtained using the vortex and transport element methods. Results indicate that at the early stages, a reacting shear layer behaves like a laminar flame. During the growth of the roll-up eddy, the rate of burning is strongly enhanced by the entrainment fluxes that lead to the swelling of the reaction zone, and the total rate of product formation can be approximated by the unstrained laminar burning velocity times the flame length measured along the line of maximum reaction rate. Following the burning of the eddy core, the strain field along the eddy boundaries causes a noticeable thinning of the reaction zone and reduces the rate of burning. Baroclinic vorticity generation due to the acceleration of fluid elements in the density gradient is the most important mechanism by which combustion affects the flow field. It augments the overall volumetric entrainment into the eddy core, and causes an entrainment asymmetry with a bias towards the products. The generated vorticity extends the growth period of the eddy and imparts on it an extra mean convective motion.

On the generation and evolution of internal gravity waves

The tidal generation and evolution of internal gravity waves is investigated experimentally and theoretically using a two-dimensional two-layer model. Time-dependent flow is created by moving a profile of maximum submerged depth 7.7 cm through a total stroke of 29 cm in water above a freon-kerosene mixture in an 8.6-m-long 30-cm-deep 20-cm-wide transparent channel, and the deformation of the fluid interface is recorded photographically. A theoretical model of the interface as a set of discrete vortices is constructed numerically; the rigid structures are represented by a source distribution; governing equations in Lagrangian form are obtained; and two integrodifferential equations relating baroclinic vorticity generation and source-density generation are derived. The experimental and computed results are shown in photographs and graphs, respectively, and found to be in good agreement at small Froude numbers. The reasons for small discrepancies in the position of the maximum interface displacement at large Froude numbers are examined.

Coriolis-force attenuation of blocking in a stratified flow

Even very small Coriolis forces are shown to alter significantly the nature of the upstream wake of an object in slow (small Froude number) translation through a non-diffusive stratified fluid. If the Ekman number is of order one, the far upstream extent of the wake is reduced. If the fluid rotation is sufficient to make the Ekman number small, the contraction of the wake is much greater. We study a particular case in detail; the Ekman number is small enough to make horizontal boundary layers Ekman layers. In this case, the wake is confined to the vicinity of the object, the upstream flow arising from a combination of Ekman pumping and baroclinic vorticity generation. The upstream flow is described by an eigenfunction whose amplitude is dependent on object geometry. If the object is a semi-infinite rectangular parallelepiped, that amplitude is determined by detailed examination of the shear layer at the face of the parallelepiped and its interaction with the Ekman layer on the top surface of the object