Unsteady Near-Field Plumes

Free Flows

Underlying Flow Regime 1-07

Description

Introduction

Free vertical buoyant plumes and free-jets are related phenomena, both having a core region of higher momentum flow surrounded by shear layers bounding regions of quiescent fluid. However, whereas for jets the driving force for the fluid motion is a pressure drop through an orifice, for plumes the driving force is buoyancy due to gradients in fluid density. Plumes can develop due to density gradients caused by temperature differences, for example in fires, or can be generated by fluids of different density mixing, such as hydrogen releases in air. There are many flows of both engineering and environmental importance that feature buoyant plumes, ranging from flows in cooling towers and heat exchangers to large geothermal events such as volcanic eruptions. There has been considerable attention paid to the mean flow behaviour of plumes in the far field, e.g. Chen & Rodi [5] or List [6] [7], which are examined in a companion UFR. However, there has been less study of the near-field unsteady dynamics of plumes.

In the present work, only non-reacting plumes are considered. This choice has been made in order to avoid the additional complexities associated with combustion, soot production and radiation in fire plumes. For helium plumes, the difference in density between helium and air is a factor of seven which is similar to that in fire plumes [8]. The principal difference between fire and helium plumes arises from the fact that heat is released locally from the flame in fire plumes whereas in helium plumes the buoyancy is produced only near the source where there are large concentration gradients.

The near-field of buoyant plumes features two key instabilities. The first is the Rayleigh-Taylor instability related to the presence of dense fluid above less-dense fluid. The two layers of different-density fluid are in equilibrium if they remain completely plane-parallel but the slightest disturbance causes the heavier fluid to move downwards under gravity through the lighter fluid. At the interface between the two fluids, irregularities are magnified to form fingers or spikes of dense fluid separated by bubbles of lighter fluid. The size of these irregularities grows exponentially with time and the smaller the density difference, the larger the wavelength of the instability. There has been considerable research into the dynamics of Rayleigh-Taylor instability (e.g. [9][10] [11][12]) as a consequence of its importance in nuclear weapons, atmospheric flows and astrophysics. Figure 2 shows the classic spike and bubble flow structures characteristic of R-T instability produced by two fluids of different density mixing, taken from Cook et al. [13].

Figure 2 Rayleigh-Taylor instability, from Cook et al. [13]. The heavy fluid is in black

The second instability in buoyant plumes is the Kelvin-Helmholtz instability related to the shear-layer interface between the rising plume and the ambient fluid. This forms axisymmetric roll-up vortex sheets on the boundary between the two layers of fluid travelling at different velocities, and is a feature in practically all turbulent shear flows including jets and wakes.

There is some uncertainty over the relative significance of the R-T and K-H instabilities in buoyant plumes. Buckmaster & Peters [14], Ghoniem et al. [15], Coats [16], and Albers & Agrawal [17] have suggested that the K-H instability plays the dominant role in plumes whilst others, including DesJardin et al. [1] , Tieszen et al. [2] and Cetegen & Kasper [18], suggest that the R-T instability is more important. For more details of the instability mechanisms and the transition to turbulence in buoyant flows, see also Gebhart et al. [19].

The Puffing Cycle

Medium to large scale plumes are characterised by the repetitive shedding of coherent vortical structures at a well-defined frequency, a phenomenon known as “puffing”. DesJardin et al. [1] present a detailed analysis of the plume puffing cycle, which they decompose into a number of stages. In the first stage, the less-dense plume fluid is rising close to the plume axis. Near the base of the plume, there is a layer of dense air overlying the less-dense plume fluid. There are two instabilities near the edge of the plume: one related to the misalignment of the vertical pressure-gradient and radial density gradient (the baroclinic torque) and another due to the misalignment of the vertical gravity and the radial density gradient (the gravitational torque). These produce a rotational moment on the fluid, increasing its vorticity and pulling air into the plume. The fluid motion coalesces to produce a large toroidal vortex which is self-propagated vertically upwards. As the vortex shifts vertically, fluid is pumped through to the core of the plume resulting in higher velocities on the plume axis. Radial velocities are induced near the base of the plume and air is drawn in producing an unstable stratification of denser fluid above less-dense fluid, ready for the cycle to begin again.

Using Direct Numerical Simulation (DNS), Jiang & Luo [20] [21] found that the gravitational torque is responsible for much of the initial production of vorticity in plumes. The term is highest towards the edge of the plume where the density gradient vector is pointing radially outwards at right-angles to the gravitational vector. The baroclinic torque was found to dominate the vorticity transport once the puffing structure has been established.

The toroidal vortex structure produced in small puffing plumes of helium in air, with a source diameter of under 10 cm, is relatively coherent. As the size of the plume is increased, the strength of secondary azimuthal instabilities increase which destabilize the toroidal vortex, producing finger-like instabilities. These are shown clearly near the base of the plume in the LES of DesJardin et al. [1] (see Figure 3). The secondary instabilities generate streamwise vorticity that enhances the mixing process. DesJardin et al. suggest that capturing these instabilities may be important in numerical simulations of pool fires where combustion is predominantly mixing-controlled.

Figure 3 An instantaneous snapshot of the puff cycle from DesJardin et al. [1] showing the finger-like azimuthal instabilities near the base of the plume. The isocontour of streamwise vorticity is shown at ±10% of the peak value.

Characteristic Dimensionless Parameters

There are a number of dimensionless parameters which are used to characterise buoyant plumes. For plumes produced by a release of buoyant gas, the inlet Reynolds number, Re, is given by:

Re={\frac {\rho _{0}V_{0}D}{\mu }}

\left(1\right)

where $\rho _{0}$ is the plume fluid density, $V_{0}$ is the inlet velocity, D is the characteristic inlet length scale or inlet diameter and $\mu$ is the dynamic viscosity. The Reynolds number represents the ratio of inertial forces to viscous forces. At high Reynolds numbers, the destabilizing inertial forces dominate the viscous forces and the flow is turbulent. For isothermal pipe flows, this occurs for Re > 3000. Between 2000 < Re < 3000 the flow is transitional, for Re < 2000 the flow is usually laminar.

A useful parameter for describing buoyant flows is the densimetric Froude number, Fr, which represents the ratio of inertial forces to buoyancy forces. It is defined here as:

Fr={\frac {V_{0}}{\sqrt {gD(\rho _{\infty }-\rho _{0})/\rho _{\infty }}}}

\left(2\right)

where g is the gravitational acceleration and $\rho _{\infty }$ is the ambient fluid density. The densimetric Froude number varies from near zero for pure plumes to infinity for pure jets. Some texts choose to define Fr using the square of the definition given above (e.g. Chen & Rodi [5]).

The Richardson number, Ri, is simply the inverse of the square of the Froude number:

Ri={\frac {(\rho _{\infty }-\rho _{0})gD}{(\rho _{\infty }V_{0}^{2})}}

\left(3\right)

In some texts, the density difference in the Froude and Richardson numbers is made dimensionless using the plume source density, $\rho _{0}$ , instead of the ambient density, $\rho _{\infty }$ .

Subbarao & Cantwell [22] note that the Richardson number can be interpreted as the ratio of two timescales: the time for a fluid element to move one jet diameter due to inertia, $\tau _{1}=D/V_{0}$ , and the time for a fluid element to move the same distance under the action of buoyancy, $\tau _{2}=\left[\rho _{\infty }D/g\left(\rho _{\infty }-\rho _{0}\right)\right]^{1/2}$ , where:

Ri=\left({\frac {\tau _{1}}{\tau _{2}}}\right)^{2}

\left(4\right)

In addition to Reynolds{}-number effects, the transition from laminar to turbulent flow is affected by the strength of buoyancy. In a buoyant plume that is initially laminar but transitions to turbulent flow at some distance further downstream, the point at which transition occurs moves closer to the source as either the Reynolds number or the Richardson number is increased [22].

Frequency of Pulsatile Plume Motion

The dimensionless Strouhal number, St, is used to describe the oscillation frequency of unsteady plumes. It is defined as follows:

St={\frac {fD}{V_{0}}}

\left(5\right)

where $f$ is the frequency of the oscillation.

A number of empirical correlations for the puffing frequency of plumes have been developed based on the Richardson number. Cetegen & Kaspar [18] found that for axisymmetric helium-air plumes with $Ri<100$ , the Strouhal number was related to the Richardson number by:

St=0.8\left.Ri\right.^{0.38}

\left(6\right)

The graph of St versus Ri taken from their paper showing this relationship is reproduced in Figure 4. Between $100<Ri<500$ there is a transitional region as the plume becomes more turbulent and mixing is enhanced. For $Ri>500$ the Strouhal number was found to scale according to:

St\propto \left.Ri\right.^{0.28}

\left(7\right)

Figure 4 Correlation of puffing frequency in terms of Strouhal number and modified Richardson number, from Cetegan & Kaspar [18].

For planar helium plumes (produced by rectangular nozzles) with Richardson number in the range $1<Ri<100$ , Cetegen et al. [23] found that the Strouhal number varied according to:

St=0.55\left.Ri\right.^{0.45}

\left(8\right)

A similar relationship for planar plumes was obtained in the more recent DNS of planar plumes by Soteriou et al. [24], who obtained the correlation:

St=0.536\left.Ri\right.^{0.457}

\left(9\right)

The difference between the puffing frequency in planar and axisymmetric plumes has been attributed to the difference in mixing rates and the strength of the buoyancy flux in the two cases. If the planar and axisymmetric Strouhal number correlations given by Equations (6) and (8) are extrapolated to higher Richardson numbers, they suggest that planar plumes exhibit higher frequency pulsations for $Ri>211$ (where the two correlations cross over).

Figure 5 Puffing frequency of pool fires as a function of the burner diameter D, from Cetegan & Ahmed [25].

For axisymmetric fire plumes, Cetegan & Ahmed [25] found the following relationship between the puffing frequency, $f$ , and the diameter of the burner or source, $D$ :

f=1.5\left.D\right.^{-1/2}

\left(10\right)

Their correlation is compared to the experimental data in Figure 5. It is remarkably consistent, considering that the fire plumes used in their study involved solid, liquid and gas fuel sources. The dependence of the puffing frequency on the source diameter is slightly stronger in helium plumes, where $f\propto D^{-0.62}$ [18]. For planar helium plumes, Soteriou et al. [24] showed that the frequency varied according to $f\approx 0.5{\sqrt {g/D}}$ .

Observations from plume experiments [18][22][26] and CFD simulations [24] have shown that the pulsation frequency in plumes does not strongly depend on the Reynolds number. The relative unimportance of the Reynolds number suggests that the instability mechanism controlling the pulsatile behaviour is essentially inviscid [24]. Once the conditions are met for the plume to become oscillatory, viscosity no longer appears to play a significant role in the puffing frequency. The helium plume experiments and simulations reported by Soteriou et al. [24] showed that the puffing frequency is unaffected by having the nozzle orifice flush to a solid surface or having the pipe from which the buoyant fluid escapes mounted free from the surrounding walls.

Onset of Pulsatile Flow Behaviour

The onset of unsteady flow behaviour in plumes is controlled by the balance of inertial, viscous and buoyancy forces. When viscous forces dominate, the plume remains steady.

Cetegen et al. [23] and Soteriou et al. [24] investigated in depth the transition from steady to unsteady flow behaviour in planar non-reacting plumes using both experiments and direct numerical simulation. Figure 6a shows some of their results, where plumes are characterised as either stable or unstable. The graph axes are the source Reynolds number and the inverse density ratio, $1/S=\rho _{0}/\rho _{\infty }$ . Clearly, as either the Reynolds number is increased or the inverse density ratio decreases, the plume becomes less stable.

Experiments with both axisymmetric and planar plumes have found that pulsations are not produced when the density ratio exceeds $\rho _{0}/\rho _{\infty }\approx 0.6$ [18][23][27][28]. Simulations by Soteriou et al.[24] showed that pulsations could in fact be produced at density ratios closer to one, but that the Froude and Reynolds numbers at which these pulsations were obtained would not be easily achieved experimentally.

Using their simulations, Soteriou et al. [24] were able to examine separately the effects of the Reynolds number, the density ratio and the Froude number on the onset of transition. They obtained a transition relationship between Reynolds and Richardson numbers of $Re=183\left.Ri\right.^{-0.627}$ (see Figure 7). The plume was unsteady for Reynolds or Richardson numbers above the line shown in the graph (i.e. for $Re>183\left.Ri\right.^{-0.627}$ or ${\mathit {Ri}}>\left(Re/183\right)^{0.627}$ ).

Cetegen et al. [23] showed experimentally that when the nozzle orifice is mounted flush to a wall, the transition from a stable to an oscillatory plume occurs at a lower threshold velocity. The presence of a flat plate surrounding the nozzle prevents any coflow which results in higher induced cross-stream velocities. These cause the plume immediately downstream of the nozzle to contract more and produce a thinner column of buoyant fluid that is more susceptible to perturbations.

In terms of the onset of unsteady flow behaviour, axisymmetric plumes are significantly more stable than planar plumes. This is shown clearly in the results of Cetegen et al.[23] (Figure 6b), where the conditions for stability of axisymmetric plumes are shown in addition to the planar plume behaviour with and without a flat plate.