Experimental investigation and integral simulation of a thermal annular plume

Abstract

Thermal plumes issued with mean exit velocities of 0.527 and 0.702 m/s from internally closed annular nozzles are investigated experimentally, theoretically, and numerically. The experimental setup and the measurement and analysis methodologies for flow and mixing are described in detail. The experimental setup consists of an air heating box with an annular slot of 0.12-m mean diameter and 0.01-m width, accompanied by equipment for setup operation and measurement of mean velocities and temperatures. The trajectory of maximum velocities and the isovelocity contours indicate reattachment due to plume self-merging at $z_{\textrm{P0}}= 0.161$ m and around 0.20 m in correspondence to exit velocities. It was found that $z_{\textrm{P0}}$ depends mainly upon initial Froude number and annulus equivalent diameter. Experiments detected a vortex ring right after the exit followed by a stagnation point at $z_{\textrm{P1}}\cong 0.055$ m on the plume centreline. The annular plume core obtains almost uniform temperature and on average 15 times higher dilutions than bulk dilutions of the equivalnt round plume. To predict the self-merging trajectory, a novel integral model was developed named self-merging approach (SMA). SMA is based on the solution of the ordinary differential equations of momentum and buoyancy applied to an infinitesimal sector of the annular plume, which is equivalent to two identical slot plumes. These plumes are reattached due to outer entrainment, inner entrainment starvation, and a series of sinks along the annular plume centreline with strengths calculated by an original relationship. SMA predictions agree very well with the observations. The present findings contribute to a better understanding of the phenomenon and offer new ideas for designing buoyancy-operating disposal systems with low momentum, using annular- or rosette-type diffusers mainly in shallow water bodies.

Key points

• Novel prediction of annular plume reattachment assuming a pair of plane plumes with inner entrainment starvation

• The reattachment distance increases proportionally with initial Froude number and equivalent diameter of the annulus

• Excellent integral model predictions for centreline velocities and concentrations beyond the core of annular plume

1 Introduction

Non-isothermal fluids may initiate flows due to the temperature differences, which induce buoyant forces, as the density depends on temperature. Thus, heat sources in fluid environments will form thermal plumes due to natural convection [1]. Oftentimes, a thermal plume is a combined result of strong buoyancy under weak inertia, due to considerable different temperatures from the surrounding fluid and initial low momentum [2]. Therefore, plumes as flows where buoyancy dominates present similar flow characteristics. Thermal plumes convect thermal energy and, in many cases, carry, diffuse, and disperse various pollutants either emitted from the heat sources themselves or produced by chemical reactions in fluid environments. Thermal plumes are found both in nature and in human-originated releases. Well-known cases include: (a) Releases of gases in the atmosphere during volcanic eruptions or from earth faults [3,4,5]; (b) emissions of gaseous thermal pollutants in the troposphere from chimneys or cooling towers [6, 7]; (c) pollutant discharges from diffusers during the disposal of wastewater, brine discharges from desalination plants or thermal effluent discharges in the ocean [8,9,10].

One of the main reasons for studying the behaviour of thermal plumes is to mitigate their environmental impact. Engineering examples in wastewater disposal have been given by Fischer et al. [8], Roberts et al. [11], and elsewhere. In the troposphere and in aquatic environments, the temperature increase with height causes stratification, which may contribute to the entrapment of pollutants conveyed by plumes, resulting in adverse effects on humans and ecosystems. Some researchers consider thermal plumes to test fire models [12, 13] by simulating the spreading of fire smoke and toxic gases, aimed at helping designers and security engineers develop fire safety precautions and systems. In addition, the study of thermal plumes may help in the improvement of the ventilation systems in buildings [14,15,16,17]. Generally, in practical applications, the mean flow and mixing characteristics are among the most useful for better understanding and/or decision making.

The annular plume is a limiting case of the annular buoyant jet. Figure 1 depicts a general configuration of flow and mixing. Although the flow and mixing coming from annular jets have been extensively studied in the last 50 years by several researchers [18,19,20,21,22,23,24,25,26], specific studies focusing on annular plumes are very limited. The special case of ventilated annular jets, which had the inner space of the annular nozzle open in the atmosphere, has been experimentally investigated by Warda et al. [22] and Padhani et al. [26]. Wang et al. [27] have numerically simulated thermal annular plumes of four different ratios (outer diameter over inner diameter of the annulus: 5/1, 5/2, 5/3, 5/4) using FLUENT CFD combined with the realizable k-$\varepsilon$ turbulence model and standard wall function. Their study has provided useful implications regarding the mean flow characteristics.

Fig. 1

Configuration of thermal plume flow and mixing coming out from a horizontal annular source (or ring exit) of mean diameter $D_0$ and width $d_0$, internally closed, in a calm ambient environment; the vortex ring, the equivalent annular source of diameter $d_{eq}$ and the main Cartesian and cylindrical coordinate systems are depicted

It is important to underline that a plume, being the limiting behaviour of a buoyant jet with strong buoyancy and weak momentum, becomes vertical right after the exit, irrespective of the nozzle inclination. Therefore, an annular plume may be considered as the limiting case of a series of plumes forming a rosette-type diffuser, which consists of the maximum possible number of nozzles. Rosette-type diffusers with four or more ports have been investigated by several researchers [28,29,30,31,32,33]. The configuration details of rosette-type diffusers constitute an interesting engineering topic, because they may affect initial dilutions in shallow water bodies [31, 32].

Alternatively, the annular plume could be considered as the limiting case of a coaxial plume with a very small flow rate from the central nozzle compared to the flow rate from the outer annular nozzle, which discharges fluid with dominant buoyancy over momentum. Favre-Marinet and Schettini [34] have experimentally investigated the density field of coaxial jets with large density differences; they have concluded that buoyancy effects are negligible on mixing in the near field region of coaxial jets. Regarding annular jets, they have declared that no further experiments have been carried out to explore the near-field and the striking behaviour of the gravity-aided annular jet remains so far unexplained. A similar finding is indirectly indicated from the results provided by Li et al. [35] and Li and Flynn [36], who have examined coaxial plumes.

For a thermal annular buoyant jet, the dimensionless relative temperature at (x, y, z) of the fluid is defined as $c=\Delta T/\Delta T_0 \cong \Delta \rho /\Delta \rho _0$ when $\rho _a/\rho _0 \cong 1$. Thus, c is also about equal to the relative concentration based on fluid densities owned to fluid temperatures and it will be hereinafter also called relative concentration or briefly concentration; $\Delta T=T-T_a$; $\Delta \rho =\rho _a-\rho$; $\Delta T_0=T_0-T_a$; $\Delta \rho _0=\rho _a-\rho _0$; T, $\rho$ mean temperature and mean density at (x, y, z); $T_0$, $\rho _0$ and $T_a$, $\rho _a$ are the fluid temperatures (K or °C) and densities (kg m$^{-3}$) at the plume exit and ambient environment, correspondingly. Application of the dimensional analysis (DA) helps in a concise presentation of the results in diagrams. Many researchers have used DA for deriving empirical laws of relevant flow and mixing problems, to give but some [8, 37, 38].

Experiments have shown that the flow of a thermal annular plume is determined uniquely by the quantities $B_0$, $m_0$, $\nu$, $a_T$, $D_{out}$, $D_{in}$, $P_{atm}/\rho _a$ and z, which form a set of eight independent variables, where $B_0 = g_0^{\prime } A_0 w_0 c_0$ is the initial kinematic buoyancy flux (m$^{4}$ s$^{-3}$), $g_0^{\prime } = g\beta \Delta T_0$ or, alternatively, $g_0^{\prime } = g \Delta \rho _0/\rho _0$ is the effective gravitational acceleration (m s$^{-2}$), $\beta = T_a^{-1}$ is the volumetric temperature expansion coefficient (K$^{-1}$); $A_0=\pi d_{eq}^2 /4$ is the cross-sectional area (m$^2$) of the source, $w_0$ and $c_0$ are the mean exit velocity (m/s) and relative temperature or concentration of the fluid at the exit; $c_0=1$; $m_0 = A_0 w_0^2$ is the initial kinematic momentum flux (m$^4$ s$^{-2}$); $\nu$ is the fluid kinematic viscosity (m$^2$/s); $a_T$ is the fluid thermal diffusivity (m$^2$/s) defined as the ratio of thermal conductivity k (W m$^{-1}$ K$^{-1}$) over the product of the fluid density $\rho$ (kg m$^{-3}$) times the specific heat $c_p$ (J kg$^{-1}$ K$^{-1}$), i.e. $a_T = k (\rho c_p )^{-1}$; $D_{out}$, $D_{in}$ are the outer and inner diameters of the annular source or ring exit; $P_{atm}/\rho _a$ is the kinematic atmospheric pressure; and z is the vertical distance from the source. Following Fischer et al. [8], a characteristic length scale $d_{eq}$ (m) is defined as $d_{eq} = 2\sqrt{A_0/\pi } = 2\sqrt{D_0 d_0}$, where $D_0 = (D_{out} + D_{in} )/2$ is the mean diameter and $d_0 = (D_{out} - D_{in} )/2$ is the width of the annular slot. It is also necessary to define the inner area of the annulus $A_{in}=\pi D_{in}^2 /4$. The above set of variables contains only dimensions of length and time, and formal DA would indicate that there are only six truly independent dimensionless groups, which may be formed from the set of variables. The six groups are: The local Rayleigh number $Ra = g_0^{\prime } z^3 \nu ^{-1} a_T^{-1}$, the initial Froude number $F_0 = w_0 / \sqrt{g_0^{\prime } d_{eq}}$, the initial Reynolds number $Re=w_0 d_{eq} \nu ^{-1}$, the dimensionless initial pressure $F_{m0}=m_0 (A_{in} P_{atm}/\rho _a)^{-1}$, the ratio $D_{out} / D_{in}$, and the dimensionless axial distance $z/d_0$ or the normalised dynamic axial distance $Z = z / (d_{eq} F_0)$. Therefore, for the present study that examines a thermal annular plume, the variations of mean flow and mixing quantities can be fully described with respect to the dimensionless independent variables $z/d_0$ or Z, $D_{out} / D_{in}$, $F_0$, Re, $F_{m0}$ and Ra. The kinematic buoyancy flux is equal to $B_0 = (\pi /4) F_0^{-2} d_{eq} w_0^3$.

The behaviour of a thermal plume is laminar for values of Rayleigh number $Ra<9\times 10^6$ and it becomes turbulent when $Ra>9\times 10^6$ [39]. Regarding the annular plume of the present study, it will be seen below that the plume is turbulent in the whole region of the flow field due to the special formation of the device and because Re is higher than or approximately 1000. Therefore, the mean flow and mixing quantities will only depend on $z/d_0$ or Z, $D_{out} / D_{in}$, $F_{m0}$ and $F_0$. However, the dimensionless pressure $F_{m0}$ at the inner boundary of the annular plume only affects the flow properties close to this boundary. In regions beyond the core, for presentation independence of the experimental data and model predictions, length variables are normalised by dividing them by $l_n=d_{eq} F_0$. Velocities and concentrations (including relative temperatures) are normalised by dividing them by $w_n=w_0 F_0^{-1}$ and $c_n=c_0 F_0^{-1}$, correspondingly. In the core region, nondimensionalisation of the length variables is made by dividing them either by $D_{out}$ or $d_0$, of velocities by $w_0$ and concentrations or relative temperatures by $c_0$. For the present experimental case, the Pèclet number $Pe=w_0 d_{eq} a_T^{-1}$ is of order 1000 ($Pe>>10$); thus, the transport due to thermal diffusion is expected to be practically insignificant compared to the transport due to convection [40].

The present study examines the flow field of an annular thermal plume experimentally, theoretically, and numerically. The experimental investigation deals with the trajectory of the self-merging of the annular plume and important characteristics in the near field, including the mean axial velocities and mean concentrations along the trajectory and the centreline in the core region and beyond. The core region of the thermal annular plume is defined as the zone right after the exit and up to the complete merging of the plume. Beyond the core region, the annular plume behaves like an equivalent round plume. The theoretical investigation concerns the development of a novel integral model, named self-merging approach (SMA). Using SMA, the self-merging trajectory, as well as the mean axial velocities and mean concentrations along the annular plume centreline, are predicted and compared to the experimental observations. From SMA predictions and their comparisons, useful conclusions are derived, which may help engineers and other scientists to better understand the phenomenon and optimise the design and operation of related effluent systems. In addition, a sensitivity analysis on the integral model’s predictions is presented and practical implications are discussed.

2 The laboratory experiments

The experiments were carried out in a room space at the Environmental Engineering Laboratory (EEL), Department of Civil Engineering, University of Patras, Greece. The experimental space was demarcated with two vertical semi-cylindrical Plexiglas walls, forming a chamber as shown in Fig. 2.

Fig. 2

Side view, plan view and required details of the experimental chamber and the thermally insulated heating box with the annular nozzle. The open arrows show the cold air route and the solid arrows the hot air route; the hot air occupies the light shadowed region of the air heating box (detail c) and comes out from the ring exit of the annular nozzle forming a turbulent annular plume

A metallic frame supported the walls leaving a gap beneath them, above the floor, and another gap over the walls up to the ceiling, allowing air interchange between the room and the chamber. The experimental setup was placed in the middle of a room of 50-m$^2$ area and 2.89-m height. The floor altitude is 73 m above sea level. The air conditioning system was turned off and the door and windows were kept closed during the measurements, to prevent flow disturbances. During the measurement periods, the room air temperature varied seasonally from 16 to 33 °C and the relative humidity from 40 to 65%. The mean diameter and the width of the ring exit were $D_0=0.12$ m and $d_0=0.01$ m, correspondingly, and the area of the ring exit was $A_0 = 37.7$ cm$^2$; the mean air temperature at the exit was 68.5 °C, the flow rate was either $\mu _0 = 1.986\times 10^{-3}$ or $2.646\times 10^{-3}$ m$^3$/s, the initial effective acceleration ranged from $g_0^{\prime } = 1.14$ to 1.78 m s$^{-2}$, the kinematic buoyancy flux from $B_0 = 2.264\times 10^{-3}$ to $4.176\times 10^{-3}$ m$^4$ s$^{-3}$ and the initial Froude number from $F_0 = 1.50$ to 2.12.

The experimental setup and its main parts are schematically presented in Fig. 3. The setup includes an air compressor electronically controlled by a Venturi meter and set to the desired flow rate. The air heating box includes four air resistors of 500-W power each, installed within the box right after each of the four air entrances linked with the air compressor through flexible PVC ducts of 50-mm diameter. The box was sufficiently insulated to prevent heat losses and avoid secondary air flows. The route of the air in the box is labyrinthine (meandering). After the air enters the box, the air flow passes through the resistors and becomes hot; then, it enters through the 14 holes of 19-mm diameter each opened around the pot underneath the annulus and exits out from the annular slot. Due to this route of air flow and since the diameter of the pot is significantly longer than that of the ring exit (detail c in Fig. 2), air vortices are generated, and the air flow becomes turbulent, as this formation behaves like the “Cooper” nozzle [41]. The power of the resistors is controlled by an electronic power controller, which senses the mean air temperature within the box and regulates the air temperature at the desired level. The air velocity transducer, model 8465 TSI, was used for velocity measurements. The temperature range of the transducer is [0–93] °C. For the velocity measurements of the present work [0.125–1.5] m/s, the error was less than 10% of the reading. However, it is important to point out that this meter cannot distinguish the velocity direction and it can correctly measure the flow velocity when it is directed parallel to the sensor needle. It is observed that in regions away from the plume centreline, where there are vortices and/or entrainment, the measured velocity magnitudes were found higher than the real ones. In these regions, this effect was partly reduced by installing the transducer within a cardboard hollow cylinder of 2-cm diameter and 2-cm height, as proposed by Stefanidou et al. [42] and shown in the detail a of Fig. 2. A pen laser was placed permanently next to the sensor to mark on the horizontal surface of the air heating box the distance of the sensor location from the vertical z axis and a small spirit level was used to achieve the verticality of the sensor. The sensor was mounted on a support system that could be moved horizontally and vertically. A video camera was used to record measurement values displayed on the transducer screen; then, the measurements were downloaded on a computer for further processing.

Fig. 3

Main parts of the experimental setup: 1. Air compressor; 2. Venturi measuring system and air discharge display; 3. air heating regulator; 4. air heating box with an annular nozzle; 5. velocity transducer; 6. air velocity display; 7. video camera for recording measurement values and transfer to computer for processing; 8. Two series of 14 thermistors; 9. Signal receivers, transfer, and communication with the computer for processing

Two series of totally 14 J-type thermocouples for temperature measurements were fixed on a horizontal metal rod at the vertical symmetry plane of the annular plume. The system was able to move in the vertical direction sliding on the two vertical metal brackets and could be fixed at the desired level required for the measurements. The signals of measurements were received from electronic devices and transferred to the computer for processing. The air temperatures at the exit and the background were measured using the HH147 OMEGA Data Logger Thermometer with four thermocouples of K type with a resolution of 0.1 °C in the range $(-100, 200)$ °C and accuracy ± (0.1% reading $+0.7$ °C). At two background locations of the chamber around the middle height level, as well as at several locations and levels of the room space, the temperature was measured at several characteristic instances during the measurements, using common thermometers. A schematic presentation of the axial velocity and temperature measurement system is depicted in Fig. 3, while additional details on measurements are provided in “Appendix 2”.

3 Theoretical considerations

3.1 Basic assumptions and governing equations

A thermal plume coming from a horizontal annular source with initial velocity $w_0$, temperature $T_0$ and density $\rho _0$ is considered. The fluid is discharged in a motionless and uniform ambient environment of temperature $T_a$ and density $\rho _a$ in steady state conditions. The internal region of the plume, after the ring exit, is sealed off from the outer environment. Due to symmetry conditions, the entrainment from the inner core region of the annular plume is self-nulled. It will be shown that the outer entrainment in conjunction with the inner entrainment starvation and the flow potential induce the Coandǎ effect, which contributes to plume contraction and reattachment at $P_0$ (Figs. 1, 4). Furthermore, a vortex ring is induced by the shear stresses imposed by the velocity field of the annular plume in the inner core region. Three coordinate systems are defined: The basic Cartesian (O, x, y, z); the cylindrical $(O, r, \varphi , z)$; and the local 2D orthogonal curvilinear coordinate system $(O_l, \psi , \xi )$, where z is vertical and $\xi$ follows the curvilinear axis of the annular plume, which is defined as the locus of maximum velocities; the $\xi$ axis begins from the slot centre of the ring exit and ends up at the merging point $P_0$ at the plume centreline; $O_l$ is the origin of the local $\psi$-axis, which is perpendicular to the $\xi$-axis; the plane $(\psi , \xi )$ is vertical.

The equations of volume, momentum, and tracer conservation are derived based on the usual assumptions and approximations, namely (a) the Boussinesq approximation made for small initial density differences, (b) the Prandtl’s boundary-layer-type approximation, (c) the neglect of the molecular viscosity terms, and (d) the absence of swirl. In addition, second-order terms for the turbulence contribution to momentum and buoyancy fluxes are included [43,44,45]. The present theoretical work is going to examine the phenomenon in steady-state conditions under a novel approach, which dares to simulate and predict among others the self-merging of the annular plume. Considering that the phenomenon is axisymmetric with respect to the vertical axis z, the steady-state partial differential equations (PDE) are synoptically written below to fit to the coordinate systems mentioned above:

continuity

$$\begin{aligned} \frac{\partial (r^i hu)}{\partial r_i}+\frac{\partial (r^i w)}{\partial z_i}=0; \end{aligned}$$

(1)

$\psi$-momentum

$$\begin{aligned}{} & {} \frac{\partial [(r \sin \varphi )^i hu^2]}{\partial r_i} + \frac{\partial [(r \sin \varphi )^i (uw-\tau _{rz_i}/\rho _0)]}{\partial z_i} - r^i \left( w^2+w^{\prime 2}+ \frac{p_d}{\rho _0}\right) \frac{\textrm{d} \theta }{\textrm{d} z_i}\nonumber \\{} & {} \quad = - g^{\prime }_0 r^i hc \cos \theta - \frac{\partial [(r \sin \varphi )^i h p_d]}{\rho _0 \partial r_i}; \end{aligned}$$

(2)

$z_i$-momentum

$$\begin{aligned}{} & {} \frac{\partial (r^i huw)}{\partial r_i} + \frac{\partial [ r^i (w^2+w^{\prime 2}+ p_d /\rho _0)]}{\partial z_i}+(r \sin \varphi )^i \left( uw-\frac{\tau _{rz_i}}{\rho _0}\right) \frac{\textrm{d} \theta }{\textrm{d} z_i}\nonumber \\{} & {} \quad = g^{\prime }_0 r^i hc \sin \theta + \frac{\partial (r^i h \tau _{rz_i})}{\rho _0 \partial r_i}; \end{aligned}$$

(3)

tracer

$$\begin{aligned} \frac{\partial (r^i huc)}{\partial r_i} + \frac{\partial [r^i (wc+w^{\prime }c^{\prime })]}{\partial z_i} = -\frac{\partial (r^i hu^{\prime }c^{\prime })}{\partial r_i}. \end{aligned}$$

(4)

In PDE (1)–(4), $i=0$ for the 2D system; and $i=1$ for the normal cylindrical system; u and w are the mean velocity components in the directions either $r_0 \equiv \psi$ and $z_0 \equiv \xi$ of the 2D orthogonal curvilinear system or $r_1 \equiv r$ and $z_1 \equiv z$ of the normal cylindrical system; $h=1+\psi \textrm{d} \theta /\textrm{d} \xi$ and $h = 0$ the scale factor of the 2D and cylindrical coordinate systems, correspondingly; $\theta$ is the local inclination of $\xi$-axis; $\tau _{rz_i}=-\rho u^{\prime }w^{\prime }$ is the mean turbulent shear stress acting on the $rz_i$ plane; $w^{\prime 2}$, $u^{\prime }w^{\prime }$, $w^{\prime }c^{\prime }$, $u^{\prime }c^{\prime }$ are the mean local fluxes due to turbulence fluctuations of w, u, c; $p_d$ is the mean dynamic pressure.

It is useful to briefly describe the superposition method (SM), which is used for comparisons with SMA in Sect. 4. SM is an integral method, which can be applied in groups of interacting either pure jets or plumes [44, 46]. SM application presents several advantages for both pure jets and plumes. Regarding pure jets, the fluxes of momentum and tracer are conserved, and the PDE of momentum becomes linear with respect to the squared mean axial velocities. Regarding plumes, the fluxes of the centreline gradient of the mean-flow kinetic energy and the tracer or buoyancy are conserved and the PDE of mean flow kinetic energy becomes linear with respect to the third power of the mean axial velocities. More recently [47, 48], SM was extended to apply to finite area sources (like plane, round, or annular plumes), assuming that these sources are composed by an infinite number of point or line sources. This extension is based on the above mentioned linearity of the PDE of mean flow kinetic energy along with the conservation of the tracer or buoyancy flux.

3.2 Modelling the self-merging of the annular plume

The annular-plume merger phenomenon can be approximated by considering the merging of two symmetrical infinitesimal sectors (sector plumes or side-plane plumes) of the annular plume. This approximation can be easily explained if we pay attention to Fig. 4. A plan view of the annular source is shown in Fig. 4a. Another simple and intuitive explanation could be just unrolling a belt. If we draw the vertical planes containing two diameters, we can separate two symmetrical infinitesimal sectors of the ring exit and the flow field of two plane plumes corresponding to the same angle $\varphi \rightarrow 0$. A strictly mathematical justification of the self-merging of the annular plume to resemble that of two-plane plumes is based on the method of conformal mapping and is described in “Appendix 3”. This method enables to determine the region of validity of SMA, which is for $d_0/D_0 \le 0.31$ or, equivalently, the range $1< D_{out}/D_{in} < 2$. The following conditions are valid: (a) the annular source has a relatively low value of $d_0/D_0$, practically $d_0/D_0 \le O(0.1)$; b) all flow properties at both cross-sectional areas of each sector are correspondingly equal and also the vertical planes defining the sectors constitute symmetry planes; c) the fluxes corresponding to these sector plumes are e(ual to those of the annular plume after dividing them by the local equivalent plume perimeter, thus referring to a 2D-flow field; (d) the sector plumes entrain fluid only from their outer boundaries, since the inner space between them is entirely closed and the inner entrainment is interchanging among their inner boundaries; the inner entrainment induced by the vortex ring near the exit is detrained by the same vortex ring and circulates in the vortex region; thus, the inner entrainment may be ignored; (e) a pressure deficit exists between the external ambient fluid and the annular core and this deficit draws the annular plume towards the centreline. The annular plume contracts to a circular plume quickly and does not maintain its ring nature for an extended time. Thus, the sector plume trajectory is controlled by the combined effect of the outer entrainment and the reciprocal attraction between the two sector plumes.

Fig. 4

Sketches explaining the equivalence between a two symmetrical infinitesimal sectors of the ring exit and b two plane plumes originating from identical slots of width $d_0$ and equal initial velocity, temperature and density to the annular plume; entrainment velocity $u_{B0}$ at outer boundary $B_{w0}$, flow potential $\vec {v}_{ai}$ due to infinite number of sinks $S_i$ along annular plume centreline Oz, reattachment point $P_0$, stagnation point $P_1$, and point $P_0^{\prime }$ where centreline velocity is maximised are also shown

Therefore, the two symmetrical infinitesimal sector plumes shown in Fig. 4a are considered equivalent to two plane plumes having the same initial characteristics with those of the annular plume shown in Fig. 4b. These plane plumes entrain fluid from the surroundings and their volume flux is increasing with z because of plume expansion and satisfaction of the continuity equation. They are also subjected to a flow potential [49, 50] induced by a series of sinks $S_i (i=1, N)$ of volume strength $\delta \mu _1$ located on the annular plume centreline Oz (Fig. 4b). Batchelor [49] has shown that velocities imposed at a point $O_l$ of the plane plume axis by each sink $S_i$ can be superimposed and give the total velocity vector $\vec {v}_a = \sum _{i=1}^{N} \vec {v}_{ai}$ shown in Fig. 4b. The sinks are considered in the middle of the elemental plume volumes [51, 52]. For a three-dimensional space, the velocity $\vec {v}_{ai}$ is calculated by the formula:

$$\begin{aligned} \vec {v}_{ai}=\frac{\delta \mu _1 \vec {s}_i}{4 \pi s_i^3} \end{aligned}$$

(5)

where $\vec {s}_i$ is a vector of the distance of point $O_l$ from the sink $S_i$ $(s_i= \mid \vec {s}_i \mid )$. According to general practice, the volume strength is calculated by the continuity equation. However, for the present case of the annular plume, an analytical expression for the volume strength is utilised. Since the two plane plumes are identical, computations are performed in the one on the right and the results are valid for the whole annular-plume infinitesimal sectors. A potential flow velocity $\vec {v}_a$ with components $u_a$ and $w_a$ in the directions $\psi$ and $\xi$, correspondingly, is calculated at the trajectory points of the plane plume at each level z (Fig. 4b). The velocity components are involved in the solution of the ordinary differential equations (ODE), which will be derived afterwards. In the present case, the velocity $u_a$ is always negative for the plume on the right. Under the approximation explained above, equations (1)–(4) describe the 2D plane plumes if $i=0$ and the round plume if $i=1$. For the plane plume on the right, the following boundary conditions must be satisfied:

$$\begin{aligned} \left. \begin{array}{ll} \displaystyle u=0, w=w_m + w_a, c=c_m, \tau _{\psi \xi } =0, u^{\prime }c^{\prime } =0, p_d \approx 0 \; \text{ on } \psi =0;\\ \displaystyle u=-u_{B0}+u_a, w=w_a, c=0, \tau _{\psi \xi } =0, u^{\prime }c^{\prime } =0, p_d =0 \; \text{ on } \psi =B_{w0};\\ \displaystyle u=u_a, w=w_a, c=0, \tau _{\psi \xi } =0, u^{\prime }c^{\prime } =0, p_d =0 \; \text{ on } \psi =-B_{w0}. \end{array}\right\} \end{aligned}$$

(6)

For the equivalent round plume, the boundary conditions must be satisfied:

$$\begin{aligned} \left. \begin{array}{ll} \displaystyle u=0, w=w_m, c=c_m, \tau _{r \xi } =0, u^{\prime }c^{\prime } =0, p_d \approx 0 \; \text{ on } r=0;\\ \displaystyle u=- u_{B1}+u_a, w=0, c=0, \tau _{r \xi } =0, u^{\prime }c^{\prime } =0, p_d =0 \; \text{ on } r= B_{w1}; \end{array}\right\} \end{aligned}$$

(7)

where $u_{B1}$ is the entrainment velocity of the equivalent round plume; $B_{w1}$ is the half width of the round plume, which is defined as $B_{w1}=n_{B1} b_{w1}$; $b_{w1}= K_{w1} z$; $B_{w0}$ is the half width of the plane plume, which is defined as $B_{w0}=n_{B0} b_{w0}$; $b_{w0}= K_{w0} \xi$; for plane plumes $n_{B0}\cong 2$ and $K_{w0}=0.132$ and for round plumes $n_{B1}\cong \sqrt{2}$ and $K_{w1}=0.11$ are the spreading rate coefficients for the velocity field [43]; $u_{B0}$ is the entrainment velocity of the inclined plane plume at the nominal external boundary. This velocity is calculated by solving the equivalent round plume and using the volume flux increment $\delta \mu _1$ divided by the infinitesimal surface of the round plume boundary $2\pi B_{w1} \delta z$ that corresponds to the z-cross-section. Thus, the velocity applied in the plane plume reads:

$$\begin{aligned} u_{B0}= \frac{\delta \mu _1/\delta z}{2 \pi B_{w1}}. \end{aligned}$$

(8)

Equation (8) replaces the continuity equation regarding the plane plume. It is also assumed that the transverse velocity and concentration profiles at a cross-section of the plane plume beyond the core region ($z>7d_0=0.07$ m) may be approximated by the Gaussian-type profiles of the form:

$$\begin{aligned} \frac{f}{f_m}=\exp (-n^2) \end{aligned}$$

(9)

which are valid from $n=0$ to $n=n_{Bi}$, where f denotes either w or c; $f_m$ denotes the centreline values either $w_m$ or $c_m$; $n= \psi /b_{w0}$ for the plane plume and $n=r/b_{w1}$ for the round plume are the corresponding dimensionless distances from the plane plume trajectory and annular plume centreline. The corresponding half widths for the velocity and concentration field are $b_{w0}$, $b_{w1}$ and $b_{c0}= \lambda _0 b_{w0}$ and $b_{c1}=\lambda _1 b_{w1}$, where $\lambda _0$ and $\lambda _1$ are the spreading rate ratios for concentration over the velocity. Note that the nominal widths are defined at the transverse distance $\psi$ or r, where the axial velocity becomes equal to $e^{-1}$ of the $w_m$ or $c_m$.

3.3 Analytical expression for the strength of sinks

The strength $\delta \mu _1$ of the sink located at the level z of the annular plume centreline is considered equal to the volume contribution corresponding to the infinitesimal slice $\delta z$ of the annular plume. Since the mean flow properties are well known regarding the round plume [43], it is straightforward to derive $\delta \mu _1$ through the volume flux $\mu _1 = \pi b_{w1}^2 w_m$, after differentiating with respect to z and substituting the centreline velocity for round plumes, $w_m=w_0 A_1 F_0^{-2/3} (z/d_{eq} )^{-1/3}$. Then, after some manipulations, the following normalised relationship in dimensionless form is obtained:

$$\begin{aligned} \frac{\delta S}{\delta Z}=\sigma _1 Z^{2/3} \end{aligned}$$

(10)

where ${\delta S} = (\delta \mu _1/\mu _{01}) F_0^{-1}$, $A_1=3.71$, $\sigma _1 = \frac{20}{3}K_{w1}^2 A_1 = 0.3$. Equation (10) can be easily integrated from 0 to Z and the following relationship is derived:

$$\begin{aligned} \frac{S}{S_0}=1+ \sigma _2 F_{0} Z^{5/3} \end{aligned}$$

(11)

where $S/S_0= \mu _1/\mu _{01}$ is the bulk dilution; $S_0 = F_0^{-1}$; $\sigma _2 = \frac{3 \sigma _1}{5} = 0.18$ and $\mu _{01}= \pi d_{eq}^2 w_0 /4$ is the kinematic volume flux of the annular plume at the exit. The normalised variations of the strength ${\delta S}/{\delta Z}$ and dilution $S/S_0$ are shown in Fig. 5 with respect to the normalised vertical distance Z. Relationships (10) and (11) can be used for direct calculation of the strength of sinks and volume flux. The most important reason is the convergence of the procedure and the very good stability of computations in the core region of the annular plume. This is significant for getting results with a satisfactory accuracy from the early beginning of z, as the self-merging of the annular plume occurs very close to the plume exit.

Fig. 5

Variation of the normalised strength of sinks $\delta S/\delta Z$ located at the centreline of the annular plume (solid line); and variation of the dilution $S/S_0$ (dashed line) for $F_0=1.63$, both as functions of the normalised centreline distance Z

3.4 The side-plane plume trajectory prediction

The trajectory of the plane plume, as forced by the flow potential, the entrainment from the external boundary, and the entrainment starvation at the internal boundary, can be predicted by SMA. SMA solves the integral equations of $\xi$- and $\psi$-momentum along with the tracer conservation equation under the approximations and assumptions described above. The complete forms of the integral equations are the ordinary differential equations (ODE):

$\xi$-momentum

$$\begin{aligned} \frac{\textrm{d} m}{\textrm{d} \xi }=\zeta \sin \theta - (u_{B}+ u_{-B})w_a - u_a \mu \frac{\textrm{d} \theta }{\textrm{d} \xi }; \end{aligned}$$

(12)

$\psi$-momentum

$$\begin{aligned} \frac{\textrm{d} \theta }{\textrm{d} \xi }=\frac{\zeta \cos \theta + (u_{B}+u_{-B})(u_{B}-u_{-B}- u_a) + n_{B0} K_{w0} w_a( u_{B}-u_{-B})}{m+ 2w_a \mu + B_{w0} [2w_a^2+(u_{B}+u_{-B})(u_{B}-u_{-B}+n_{B0} K_{w0} w_a)]}; \end{aligned}$$

(13)

tracer conservation

$$\begin{aligned} \frac{\textrm{d} \beta }{\textrm{d} \xi }=-w_a \frac{\textrm{d} \zeta }{\textrm{d} \xi }; \end{aligned}$$

(14)

where $u_B$ and $u_{-B}$ are the entrainment velocities at the boundaries on the right and left, correspondingly. Regarding the side-plane plume on the right, the outer entrainment velocity is $u_B = u_{B0}$, while the inner entrainment velocity is assumed $u_{-B} = 0$. For the solution of the vertical round plume, the final forms of ODE are:

continuity

$$\begin{aligned} \frac{\textrm{d} \mu }{\textrm{d} z}= 2 \pi B_{w1} u_{B1}; \end{aligned}$$

(15)

z-momentum

$$\begin{aligned} \frac{\textrm{d} m}{\textrm{d} z}=\zeta; \end{aligned}$$

(16)

tracer conservation

$$\begin{aligned} \frac{\textrm{d} \beta }{\textrm{d} z}= 0; \end{aligned}$$

(17)

where $\mu$, m, $\zeta$ and $\beta$ are the fluxes of volume, momentum, weight deficit and buoyancy, defined in Table 1. For the side plane plume, ${\textrm{d} \zeta }/{\textrm{d} \xi }= 0$, because $c_m = B_{p0} F_0^{2/3} (\xi /d_0)^{-1}$ [43], while for the equivalent round plume, ${\textrm{d} \zeta }/{\textrm{d} z}= - g_0^{\prime }(\pi /3) \lambda _1 b_{w1}^2 c_m$, as derived from the related fluxes given in Table 1.

Table 1 Kinematic fluxes, their determination applying the Gaussian profiles (9) and their initial values at the source of area $A_0$, for a side plane plume $(i=0)$ and the equivalent round plume $(i=1)$

The self-merging trajectory of the annular plume has been measured experimentally and predicted by applying SMA. The alternative model SM could be also used to predict the self-merging of the annular plume, but this needs entirely different manipulations, which deviate from the present scope. SMA is a Fortran Power Station code prepared in the context of the present study. It solves the system of integral equations (12)–(17) along with the fluxes and their initial values given in Table 1. It utilizes the 4th order Runge–Kutta algorithm, and the results of the solution are direct. The solution proceeds as follows:

• Iteration $k=1$. Equations (15)–(17) for the round plume equivalent to the annular plume are solved for a prescribed marching step $\delta z$ and initial conditions. The strength $\delta \mu _1$ of the sink, the volume flux $\mu _1$, the mean axial velocity and mean concentration are determined at z. The strength $\delta \mu _1$ and the volume flux $\mu _1$ are calculated by (10) and (11), correspondingly.

• Iteration $k=2$. Equations (12)–(14) for the plane plume on the right are initially solved with entrainment from the external boundary only and zero velocity components by the flow potential, $u_a=w_a=0$. The entrainment velocity is taken equal to that of the equivalent round plume computed in previous iteration $k=1$ and the plane plume trajectory is calculated. Using the same $\delta z$, the marching step for the plane plume is $\delta \xi = \delta z/\sin \theta$, where $\theta$ is the local inclination angle of the plane plume trajectory. Then, the values of $u_a$ and $w_a$ induced by all the sinks assigned along the round plume centreline, are calculated at the trajectory points of the plane plume defined by $\delta \xi$ steps.

• Iterations $k=3 - 4 -5$. The computations are repeated for the side plane plume, considering for each iteration the renewed values of the trajectory inclination $\theta$ and the velocity components $u_a$, $w_a$ from the previous iteration, due to slight changes of the plane plume trajectory, up to get the appropriate convergence. The effect of flow potential ceases for the trajectory points with $r_m \le B_{w0} \sin \theta$, where $r_m$ is the horizontal distance of the trajectory points from the z axis. This action is taken because the transverse velocity and concentration profiles of the two plane plumes at this distance start overlapping on each other. The profile overlapping may lead to erroneous reattachment location, since SMA does not take this effect into account.

4 Results and discussion

4.1 Core region of the annular plume

4.1.1 Trajectory prediction of the equivalent plane plume

Figure 6 depicts the trajectory of the side plane plume of the annular plume, as predicted by SMA in comparison with measurements. The complete convergence of the trajectory of the plane plume is obtained for $k=3$ or up to 5. For instance, for the initial conditions $w_0 = 0.527$ m/s, $T_0 = 68.5$ °C, $T_a = 23$ °C for dry air, using $\delta \xi = 5\times 10^{-5}$ m, which is the smallest step to get practically a limit independent of $\delta \xi$, the accuracy in millimeters is obtained at the 5th iteration. The limit obtained for a marching step equal to $10^{-4}$ m differs 1.0% of the above final limit. Using a desktop computer with processor Intel®Xeon®CPU X5650 2.67 GHz, RAM 12GB and 64bit operating system, the computational time needed for the solution of the equivalent round plume, with the smallest step, for 22,500 steps and then, calculating the side plane plume for 8 iteration cycles, is only 102.8 s. This is one of the best advantages of the integral method.

Fig. 6

Trajectory prediction of side plane plume of the annular plume by SMA (red solid line) in comparison to experimental data; open circles show the mean location of the trajectory measured at the axes $x, -x, y, -y$, which correspond to the maxima of velocities of upward direction, and bars show the min/max values; dotted black line is equation (18), which interpolates linearly the data given and intersects the annular-plume centreline at $z_{P0}=16.1 d_0$

A linear interpolation of measurements with a coefficient of determination $R^2=0.8713$ is described by equation (18) shown with dotted line in Fig. 6.

$$\begin{aligned} r_m=r_{m0} \left( 1-\frac{z}{z_{P0}}\right) \end{aligned}$$

(18)

where $r_{m0}=5.71 d_0$ and $z_{P0}=16.1 d_0$. The measurements represent the mean location of the trajectory at the four locations $x, -x, y, -y$ of each cross-section, as well as the min and max values. It is observed that they show appreciable scatter mainly for $z \ge 7d_0$, which will be explained below. The trajectory for the complete merging predicted by SMA is also shown in Fig. 6 in comparison to the experimental measurements. The accuracy obtained by SMA is very good, since the predicted trajectory agrees well with the measured mean location of the trajectory (open circles). The reattachment point $P_0$, where the trajectory intersects the centreline z-axis, as defined in Figs. 1 and 4, is predicted by SMA to be at $z=16.0d_0$. This result is very close to the experimentally measured value.

4.1.2 Variation of the centreline velocities

The variation of the dimensionless mean axial velocity $w_m/w_0$, measured along the annular plume centreline dimensionless distance $z/d_0$ in the core region, is shown in Fig. 7. Two sets of measurements are shown: One set with mean exit velocity $w_0=0.527$ m/s, which includes the runs V1.1-1.6 with the statistical data of raw measurement values given in “Appendix 2” (Table 6), and another set with $w_0=0.702$ m/s that includes the run V1.7 with the statistical data of raw measurements given in “Appendix 2” (Table 7). Table 2 includes the locations and values of the characteristic quantities of the annular plume in the core region along with useful remarks. It is observed that the location of the minimum velocity (downward maximum velocity) has increased by 27.6% for a 33.2% initial velocity increase. This location of the core axis has the lowest total pressure. This means that it is at the neighbourhood level of the vortex centre. An increase of 33.2% of the initial velocity caused a 10.9% increase of the absolute value of downward maximum velocity. Nevertheless, the location of the stagnation point $P_1$ remained practically unaffected by the variation of the initial velocity (1.6% upward shift). The location of the stagnation point was at $z_{P1}=5.424 d_0$, for the case of $w_0=0.527$ m/s, which is in good agreement with the experimentally observed location $z_{P1}\cong 5.5d_0$ obtained using a very thin nylon strip (see movie1 along with the description in “Appendix 1”). The location of the maximum centreline velocity was shifted upwards by 19.3% for a 33.2% increase of the initial velocity, while the value of the maximum velocity remained practically unaffected (2.7% reduction only).

Fig. 7

Measured centreline variation of the dimensionless axial velocities $w_m/w_0$ of the annular plume in the core region with respect to the dimensionless distance $z/d_0$. The maximum velocity occurs at the point $P_0^\prime$ shown in Fig. 4b; the core region includes the vortex ring and the self-merging; symbol size is approximately equal to confidence interval for $a=0.001$; runs V1.1-1.6 correspond to $w_0=0.527$ m/s and V1.7 to $w_0=0.702$ m/s

Table 2 Values of characteristic quantities of the annular plume in the core region

The internal recirculating region of the present case is found within the region $0\le z< D_0/2$, in accordance with the experimental observations described by Chigier and Beer [53] and Ko and Chan [20]. Paying attention to Figs. 6 and 7, the results show that the locations of the reattachment point and the maximum centreline velocity do not coincide, as it happens in annular jets [19, 21, 53] and in a pair of identical slot jets [54], where one or two orders higher initial velocities were used than the velocities of the present work. Regarding the case with $w_0=0.527$ m/s, Fig. 6 indicates that the reattachment of the annular plume occurred at the longitudinal distance $z_{P0}=16.1d_0$, which contradicts the result shown in Fig. 7, where the maximum centreline velocity occurred at the considerably shorter distance of $7d_0$ (point $P_0^\prime$ in Fig. 4b). Careful and scrutinised observation of the phenomenon and the measurements depicted in Fig. 8 in contrast to Figs. 6 and 7 explains the above difference and proves that the correct reattachment location is the centreline distance $z_{P0}=16.1 d_0$. This decision is based on the following reasoning:

Fig. 8

Isovelocity contours of the velocity ratio $w/w_m$ of the annular plume at the cross-sections $z/d_0$: a 3.75; b 5; c 7.5; d 15 and e 30; drawing f shows schematically the unstable core trajectories of the annular plume. The initial velocity was $w_0=0.527$ m/s and the measurements were performed at several radial distances on the perpendicular axes x and y of each cross-section

As shown in Fig. 8, the pink and white coloured regions represent the regions where $w/w_m$ is approaching or being equal to 1. These regions are observed at remote radial distances for $z/d_0= 3.75$ and 5 (Fig. 8a, b), while they are gradually approaching each other for $z/d_0= 7.5$ and 15, although they remain distinct (Fig. 8c, d). This means that reattachment has not occurred yet. These regions are observed unified in a restricted area for $z/d_0= 30$ (Fig. 8e). Therefore, it is concluded that the reattachment occurs beyond the longitudinal distance $z/d_0= 15$, which agrees with the measurements shown in Fig. 6, where the reattachment occurs at $z/d_0 = 16.1$. The four pink areas read weird in Fig. 8a, b, because of measuring at locations $(\mid x \mid , 0, z)$, $(-\mid x \mid , 0, z)$, $(0, \mid y \mid , z)$ and $(0, -\mid y \mid , z)$ only. Thus, it is implied that the above differences are owed to the low initial momentum compared to buoyancy $(F_0<2)$ and the vortex ring induced in the region $0\le z/d_0 < 5.5$. Such a buoyant flow presents instabilities and the velocity trajectories of the annular plume along with the reattachment point $P_0$ oscillate and flicker like a candle flame, as schematically shown in Fig. 8f. It is noted that a plane jet attaching to an offset parallel wall is oscillating due to the infinitesimal periodic pressure fluctuations [55]. Unlike the plane jet, the most possible reasons that an annular plume is oscillating are the low momentum compared to buoyancy in conjunction with flow and temperature asymmetries induced by the vortex ring. Thus, the sensor placed at $P_0$ measured the centreline axial velocity with considerable fluctuations. On the contrary, the same sensor placed at $P_0^\prime$, between the stagnation point $P_1$ and the point $P_0$, where the axial velocities present restricted scatter compared to axisymmetric values (see also Figs. 6, 7 and 8a, b), measured the centreline maximum velocity.

For initial velocity $w_0=0.527$ m/s, the centreline values of maximum velocities measured in the region $z/d_0= 7$ to 8 are about 20% higher than the corresponding values measured at $z/d_0= 16.1$. Increasing the initial velocity, the location of the maximum velocity is shifted and approaches more and more to $P_0$, due to succeeding rather better stability conditions. An upward shift of the reattachment point occurs as well. To explain if this shift happened for $w_0 = 0.702$ m/s, axial velocity measurements at several radial distances along x and y axes of the cross-sections $z/d_0 = 16, 18, 20$ and 22 were made. The results are presented in isovelocity diagrams (Fig. 9a–d). It is evident from the diagrams that the most contracted 0.90-, 0.95- and 0.99-isovelocity contours happened in the cross-section $z/d_0 = 18$ (Fig. 9b), while the cross-section $z/d_0 = 20$ (Fig. 9c) is second in hierarchical order regarding shrinkage. However, the cross-section $z/d_0 = 18$ shows two separate 0.95-isovelocity contours in contrast to the cross-section $z/d_0 = 20$, where the velocity peak is found within the area circled by 0.99-isovelocity contour. It is noted that a similar event is observed little before the reattachment with $w_0=0.527$ m/s (Fig. 8d). Thus, it may be concluded that reattachment of the annular plume occurs at $z/d_0 \cong 20$. SMA predicts it at $z/d_0 = 20.7$, which is very close to that detected. Ko and Chan [20] have correlated the position of reattachment with the dimensionless parameter $F_{m0} = \rho _{a} m_0/(A_{in} P_a)$, where $m_0$ is the initial kinematic momentum flux, $A_{in}$ is the area included by the internal diameter of the annulus, and $P_a$ is the atmospheric pressure in the internal region of the annulus near the closed bottom.

Fig. 9

Isovelocity contours of the velocity ratio $w/w_m$ of the annular plume at the cross-sections $z/d_0$: a 16; b 18; c 20 and d 22. The initial velocity was $w_0=0.702$ m/s and the measurements were performed at several radial distances on the perpendicular axes x and y of each cross-section

The static pressure (total pressure) at the centre of the vortex ring of either an annular jet [20] or a pair of parallel plane jets [54], is lower than the pressure at other core regions and directly affects the centreline point of the horizontal symmetry plane of the vortex ring, where the downward velocity is also maximised. Since pressure is correlated with initial momentum, an analogous result is expected to occur in annular plumes. Thus, the vortex centre will be at about the same level with the maximum value of downward centreline velocity, that is at $z_{vc}/d_0 = 2.742$ for $w_0=0.527$ m/s and at $z_{vc}/d_0 = 3.5$ for $w_0=0.702$ m/s. The experimental measurements of the present work concerning annular plumes showed that the vertical location of the vortex centre, $z_{vc}$, increases with the increase in the initial velocity and, consequently, with initial momentum, contrary to what takes place in annular jets [20]. The differences may be explained by considering that only momentum dominates the behaviour of annular jets, while the behaviour of annular plumes is rather dominated by buoyancy than momentum. Regarding annular jets, the higher the initial momentum, the higher the pressure drop, the more the contraction, and the more the shrinkage of the vertical location of the vortex ring. Regarding annular plume, the low values of Froude number indicate initially low momentum values, while the flow is accelerating due to initially high buoyancy and this effect contributes to a momentum increase. Therefore, the lower pressure occurs somewhat downstream than the case without flow acceleration; the centre of the vortex ring follows the same behaviour.

Careful observation of the isovelocity diagrams (Fig. 8) detects that the black coloured region of negative velocities in diagram Fig. 8a ($z/d_0=3.75$) is more contracted than that in Fig. 8b ($z/d_0=5$). This is expected since cross-section Fig. 8a is closer to the region of lower pressures than cross-section Fig. 8b. This is a very useful observation and it could be implemented as an indirect method of pressure estimation in plumes, where the total pressure is very small and not easily measurable. It is noted that a contraction also occurs immediately above the source of a lazy plume [56], which may be attributed to a pressure drop. The radial distance of the vortex ring is independent of the dimensionless pressure $F_{m0}$ and, according to Ko and Chan [20], it is roughly at $r_{vc}/D_{out} \cong 0.18$. The upper level of the vortex ring should be a little below the observed location of the stagnation point (Table 2).

4.1.3 Important characteristics

Ko and Chan [20] have successfully correlated the locations of the reattachment point, the vortex centre and the stagnation point with the dimensionless pressure $F_{m0}$. This pressure is responsible for the outer entrainment in this region [20]. As shown in Fig. 10, the variation of $F_{m0}$ may be divided in three regions: The low-pressure region ($F_{m0} < 0.1$), the intermediate-pressure region ($0.1 \le F_{m0} \le 3$) and the high-pressure region ($F_{m0} > 3$). It is evident that for $F_{m0} > 3$, the dimensionless location of the reattachment point, $z_{P0}/D_{out}$, is highly dependent on $F_{m0}$, while for $F_{m0}<3$, this dependence is weak, and, for $F_{m0}<0.1$, $z_{P0}/D_{out}$ may depend on other parameters, like the buoyancy flux or the Froude number, as in the present cases of annular plumes. In the following the above locations are presented in relation to $F_{m0}$.

Fig. 10

Dimensionless reattachment point $z_{P0}/D_{out}$ of the present experiments (red open circles) in correlation to the dimensionless pressure $F_{m0}$, regarding annular plumes with initial velocities $w_0=0.527$ and 0.702 m/s, compared to measurements available in the literature concerning annular jets with initial velocities in the range from 8 to 50 m/s

Regarding the vertical location of the vortex centre, the results of experimental values found in the literature are depicted in Fig. 11 along with the corresponding observations of the present study for comparison. In this figure, the vertical location of the vortex centre $z_{vc}$ is normalised by the exit width $d_0$ of the annulus, because in this manner a better presentation of the correlation is managed. For the correlation of $z_{vc}/d_0$ with $F_{m0}$, the three regions defined above are considered. In the high-pressure region, the experimental measurements show that the higher the momentum, the higher the absolute value of the pressure (pressure drop) prevailing in the core region near the closed bottom of the annulus. The higher the absolute pressure, the more intense the outer entrainment, and the shorter the distance from the source of the vortex centre. In the intermediate-pressure region, the vortex centre location seems to be rather independent of momentum, whereas in the low momentum region, the vortex centre distance from the exit increases with pressure increase, that is, increase of momentum or decrease of $A_{in}$.

Fig. 11

Vertical dimensionless distance $z_{vc}/d_0$ of the vortex centre from the exit (red open circles) based on the location, where the downward velocity on the symmetry axis gets its maximum value, in comparison to the experimental values available in the literature; the values of $z_{vc}/d_0$ are correlated with the dimensionless pressure $F_{m0}$

Regarding the dimensionless location of the stagnation point, $z_{P1}/d_0$, a similar diagram to Fig. 11 is shown in Fig. 12, where $z_{P1}/d_0$ is correlated to $F_{m0}$. Paying attention to Fig. 12, a similar conclusion can be drawn as discussed previously for the distance of the vortex centre from the exit, with the difference that, for the low-pressure region, the stagnation point location is practically independent of $F_{m0}$. It is noted that, regarding the case of initial velocity $w_0=$ 0.527 m/s, the measurement reliability of the stagnation point location by the transducer has been visually verified using the video movie1 provided in “Appendix 1”. Unfortunately, no experimental studies were found in the literature concerning annular plumes with low values of $F_{m0}$ to compare them with the corresponding results of the present work. The only relevant work found is the computational study by Wang et al. [27], which has investigated the diameter ratios $D_{out}/D_{in} = 5/4, 5/3, 5/2$ and 5/1 of an annular plume with initial velocity $w_0=1.2$ m/s and air temperature 95 °C at the source exit, as well as other combinations of the initial velocity with higher temperatures. For the case $D_{out}/D_{in}= 5/4 = 1.25$, which resembles the experiments of the present work, where $D_{out}/D_{in}= 1.181$ and SMA is valid, Wang et al. [27] have computed the location of the reattachment point at the axial distance $z/D_{out}=2.6$. SMA predicts this location at $z/D_{out}=1.275$.

Fig. 12

Vertical dimensionless distance of the stagnation point $z_{p1}/d_0$ based on the observed and measured locations (red open circles), where the velocity on the symmetry axis changes direction from upward to downward, in comparison to the measurements available in the literature; the values of $z_{p1}/d_0$ are correlated with the dimensionless pressure $F_{m0}$

4.1.4 A new correlation of the reattachment location

Based on the overall information given above, it is possible to apply DA to correlate the location of the reattachment point, $z_{P0}$, with the initial kinematic momentum $m_0$, the annular source area $A_0$, a buoyancy parameter like the effective gravitational acceleration, $g_0^\prime$, and the annulus diameter ratio, $D_{out}/D_{in}$. The above dimensional independent variables include two dimensions and, thus, they can form only two dimensionless independent variables by their suitable combination. These variables may be the initial Froude number $F_0$ and the annulus diameter ratio $D_{out}/D_{in}$. After pertinent manipulations, the normalised reattachment distance, $Z_{P0} = (z_{P0}/d_{eq} ) F_0^{-1}$ is correlated to the normalised dimensionless independent variable $(D_{out}/D_{in} ) F_0^{-1/3}$. Due to the lack of available measurements in the literature, the two cases measured in the present study are used, regarding $z_{P0}$ values along with computed values for other cases in the region where SMA is valid; these values are given in Table 3. Data show that $Z_{P0}$ is slightly dependent on $(D_{out}/D_{in} ) F_0^{-1/3}$; thus, it may be considered as practically independent.

Table 3 Input and output values of parameters of SMA for the predicted cases plotted in Fig. 13

Fig. 13

Normalised dimensionless distance $Z_{P0}$ of reattachment point of annular plume correlated with the dimensionless quantity $(D_{out}/D_{in})F_0^{-1/3}$; black symbols represent the measured values, while green, blue and red symbols the calculated values using SMA

The horizontal dashed line in Fig. 13 represents the mean value of the computed values also depicted for the initial velocities $w_0=$ 0.527, 0.702 and 1.054 m/s. It is shown that the measured values of the reattachment point locations for the initial velocities $w_0=$ 0.527 and 0.702 m/s lie very close to the mean value of the data. Therefore, for annular plumes, a simple relationship is derived for the location of the reattachment point:

$$\begin{aligned} Z_{P0} = \frac{z_{P0}}{d_{eq}}F_0^{-1} = 1.38 \pm 0.05 \end{aligned}$$

(19)

Equation (19) is valid for $d_0/D_0 \le O(0.1)$, which is approximately equivalent to the region $1<D_{out}/D_{in} \le 2$. This region agrees with that estimated by applying the method of conformal mapping described in “Appendix 3”. The result obtained indicates that the dimensionless reattachment distance depends mainly upon the initial Froude number and the equivalent diameter of the annular plume.

4.1.5 Trajectory of maximum temperatures or concentrations

The trajectory of the maximum relative temperatures or concentrations measured along the annular plume in the core region is shown in Fig. 14a. The statistical data of raw measurement values, where the values of dimensionless quantities of Fig. 14 are based on, are given in “Appendix 2” (Table 10). The black open symbols (rhombus) represent the locations of measurements along the annular plume centreline, which shows some eccentricity with respect to centreline due to asymmetries originating probably from the vortex ring and low momentum. The red open circles represent the locations of the maximum values measured at the corresponding cross-sections. The red and black symbols are measurements with 32 to 33 °C of background temperature, while the blue triangles are measurements with 20 °C of background temperature. It is shown that the trajectory defined by the maximum relative temperatures or concentrations intersects the centreline at $z/d_0 \cong 6$, which is a very short distance from the annular source exit and within the vortex ring area. This is a result of the vortex ring, which recirculates part of the exit hot air to the plume centreline and then downward up to the bottom of the internal core region. Consequently, the centreline air temperatures become high enough, approaching the ring exit values, as this is shown in Fig. 14b. This figure also shows that the centreline concentrations and subsequently the temperatures remain approximately at the same level along the core (2 °C temperature decay from $z/d_0=5$ and up to 16).

Fig. 14

a Measured trajectory of dimensionless maximum temperatures or concentrations; and b maximum and centreline values of concentrations, with respect to the dimensionless distance $z/d_0$ (red open circles, black open rhombus and blue open triangles); the symbol size is approximately equal to the confidence interval for $a=0.001$

Another noteworthy observation is that the trajectory of the relative maximum temperatures or concentrations presents a steeper inclination towards the symmetry axis z than the corresponding for the maximum velocities shown in Fig. 6, effectively suppressing the nonuniformity of the temperature field within the core region. Additional measurements of temperature at the locations $(r/d_0, z/d_0)$ $(0,-0.3)$ and $(5.0,-0.3)$ at the bottom of the internal region, using the HH147 Omega thermometer, recorded mean temperatures of 53.4 °C and 54.4 °C, respectively, which correspond to the concentrations $c = 0.691$ and 0.712, for $F_0=1.86$. These values are within the range of the values measured in the core region. The physical interpretation of this finding is attributed to the recirculation of a part of the exit hot air by the vortex ring. The equivalent round plume to the annular plume has a source and core region of shorter dimensions without a vortex ring. Thus, the dilutions in the core region of the annular plume become higher than those of the round plume at the same axial distance.

4.2 Near field beyond the core region

4.2.1 Centreline velocity variation

The variation of the mean axial velocity along the centreline of the annular plume is predicted by SMA. Furthermore, the following analytical relationship (20) [48], which is based on the superposition of kinetic energy fluxes, if the source of the equivalent round plume is considered that consists of point sources, is shown in Fig. 15 in comparison to experimental data.

$$\begin{aligned} W=A_1 Z^{-1/3} \left[ \frac{4}{3} \left( \frac{b_{w1}}{d_{eq}}\right) ^2 \left\{ 1- \exp \left[ - \frac{3}{4} \left( \frac{b_{w1}}{d_{eq}}\right) ^{-2} \right] \right\} \right] ^{1/3} \end{aligned}$$

(20)

where $W=(w_m/w_0 ) F_0$. The centreline velocity measurements, where W and Z are based on, are given in “Appendix 2” (Tables 11 and 12). The variation of the normalised mean axial velocity W along the annular plume centreline calculated by SMA and by SM is shown in Fig. 15.

Fig. 15

Variation of the normalised axial velocities $W=(w_m/w_0 ) F_0$ along the centreline of annular plume with respect to the normalised distance $Z=(z/d_{eq} ) F_0^{-1}$. The experimental measurements (symbols) are compared to SMA and SM predictions; the symbol size is approximately equal to the confidence interval for $a=0.001$

The runs V1.1c-V1.6c concern the annular plume with initial velocity $w_0=0.527$ m/s, while V1.7c concerns the annular plume with $w_0=0.702$ m/s. It is evident that the effect of nozzle geometry maintains in the core region up to $Z \cong 5$. Beyond this distance, the annular plume behaves like a round plume having an equivalent diameter $d_{eq}$. Neither SMA nor SM can predict the centreline velocities in the reverse flow region occupied by the vortex ring. SMA significantly overestimates the centreline velocities in the region $Z<4$ corresponding to $z<49 d_0$, which is affected by the core region instabilities, as explained previously. Contrary to SMA, exempting the region $0<Z<2$, where SM underestimates the velocity, for $Z \ge 2$ SM shows good agreement with the measurements. SMA predictions are significantly improved for $Z>5$, where both model predictions are practically in agreement.

4.2.2 Centreline temperature or concentration variation

The variation of the mean relative temperature or mean concentration along the centreline of the annular plume is predicted by SMA. Figure 16 shows the centreline variation of the normalised mean concentrations $C=(c_m/c_0 ) F_0$ with respect to the normalised centreline distance Z. Open circles represent the normalised mean concentrations calculated by the temperature measurements given in “Appendix 2” (Table 13). The red solid line is SMA prediction regarding the equivalent round plume of the annular plume. This method predicts very well the experimental results for $Z \ge 4$, when the velocity and temperature or concentration profiles tend to become Gaussian. For $Z<4$, SMA prediction overestimates significantly the experimental data.

Fig. 16

Variation of the normalised mean concentrations $C=(c_m/c_0 ) F_0$ along the centreline of annular plume with respect to the normalised distance $Z=(z/d_{eq} ) F_0^{-1}$ for the case with $w_0=0.527$ m/s. Circles and squares show the experimental measurements; red solid line is SMA prediction and black dotted line is SM prediction; the size of symbols includes the interval limits for $a=0.001$

If the source of the equivalent round plume is considered as composed by point sources, after applying SM by superposing the buoyancy fluxes of all point plumes, the following relationship is taken for the plume centreline [47]:

$$\begin{aligned} {\overline{WC}}= \frac{\pi }{4}\lambda _{B1}^{-1} F_{0}^2 \left[ \textrm{erf} \left( \frac{d_{eq} (1+\lambda _1^2 )^{1/2}}{2 \lambda _1 b_{w1}} \right) \right] ^2 \end{aligned}$$

(21)

where $\lambda _{B1}\approx 1.15$, $\lambda _{p1}\approx 1.04$, with the buoyancy flux expressed in the normalised form, ${\overline{WC}}={\overline{wc}}/(w_0 c_0 ) F_{0}^2$. If equation (21) is divided by equation (20) for the centreline mean axial velocity, the analytical relationship for the mean centreline concentrations is derived:

$$\begin{aligned} C=\frac{{\overline{WC}}}{W} \end{aligned}$$

(22)

The variation of the normalised concentration along the centreline, expressed by equation (22) is shown in Fig. 16 in comparison to the experimental measurements and SMA prediction. In contrary to SMA prediction, SM predicts excellently almost all the region, exempting only the region $Z<2$, where the experimental data is overestimated. As shown in Figs. 15 and 16, the annular plume after complete merging obtains the behaviour of the equivalent round plume, regarding the centreline mean axial velocities and mean concentrations. A similar observation can be deduced for the centreline dilution, which is defined as $S_m=1/c_m$. Comparing the centreline dilutions achieved in the region beyond the core of the annular plume ($Z>5$), the annulus-type diffuser, which is equivalent to a rosette-type diffuser, attains equal dilutions to the rosette-type diffuser and the equivalent round plume with equal flow rates. Comparing the centreline dilutions achieved in the core region of the annular plume ($0 \le Z \le 1.6$) with the bulk dilutions achieved by a single round plume at the same centreline distances from the plume exit [38, 43], it is concluded that the annular plume succeeds 15 times on average higher centreline dilutions than the corresponding bulk dilutions of the round plume. These high dilutions may be attributed to the existence of the vortex ring, the outer entrainment increase due to the pressure drop inside the core and the longer mean diameter of the ring exit. Therefore, the annular plumes seem very promising for wastewater disposal in shallow water bodies.

4.3 Sensitivity analysis

4.3.1 Effect of strength of sinks

Figure 17 shows the significant differences between the values of $\delta \mu /\mu _0$ and $\mu /\mu _0$ calculated by the new relationships (10) and (11), and the corresponding values calculated by the continuity equation (15); $\mu _0=1986 \,{\textrm{cm}}^{3}/\textrm{s}$. Figure 17a shows that the strength $\delta \mu$ calculated by (10) starts from a zero value and infinite derivative value at the plume exit and proceeds with gradually increasing values tending to infinity and decreasing derivative values tending to zero at infinity. The strength calculated by the continuity equation again starts from a zero value, but then increases abruptly and proceeds with gradually increasing values approaching the previous strength behaviour at infinity. The volume flux $\mu$ calculated by (11) starts from $\mu _0$ and a zero value of the derivative at the exit and proceeds with gradually increasing values tending to infinity. On the contrary, Fig. 17b shows that the volume flux $\mu$ calculated by the continuity equation starts from a zero value and a higher derivative value than the previous one and proceeds with gradually increasing values approaching the behaviour of the previous volume flux at infinity. The case of calculating the strength of sinks and the volume flux by the continuity equation constitutes the reason that the solution crashes even if under-relaxation is applied. Instead, the use of relationships (10) and (11) ensures an excellent, smooth convergence at the correct limit $z_{P0}=0.16$ m, as shown in Fig. 18. The physical realism of the model and its ability to accurately predict laboratory measurements constitute additional factors to this excellent convergence. The numerical results by SMA indicated that the contribution of the flow potential to the self-merging of the annular plume is very small (only 1%), while the main contribution (99%) is owed to the outer entrainment combined with inner entrainment starvation.

Fig. 17

Variation of the dimensionless a strength of sinks $\delta \mu /\mu _0$, and b dilution $\mu /\mu _0$, with respect to the dimensionless centreline distance $z/d_0$, for $w_0=0.527$ m/s and $F_0=1.63$. Comparisons between corresponding values calculated by the new relationships (10) and (11) (solid lines) and values computed by the continuity equation (dashed lines) of the equivalent round plume to the annular plume

Fig. 18

Variation of the limits of convergence of SMA, regarding the reattachment distance $z_{P0}$, using strength of sinks and dilution values calculated by relationships (10) and (11) for $w_0=0.527$ m/s and $F_0=1.63$; the final limit is $z_{P0}=0.16$ m obtained for $\delta \xi =0.00005$ m and sinks in a 3D space

4.3.2 Effect of relative humidity on model predictions

Using the perfect gas law, the thermal volume expansion of air is $\beta = T_a^{-1}$ (${\textrm{K}}^{-1}$). This relationship is practically satisfactory for conditions of the standard atmosphere ($p=101325$ Pa, $T_a =288.15$ K)) and for adiabatically temperature variation of dry air volume with height in the troposphere. For these conditions, the website (last accessed on July 28, 2023) provided by NASA [57] is used to calculate the air densities with 6-digit accuracy. In such conditions the effective gravitational acceleration defined either through temperatures or densities results in almost the same value, as the following equation is valid:

$$\begin{aligned} g_0^\prime = g \frac{T_0-T_a}{T_a} \cong g \frac{\rho _a - \rho _0}{\rho _0} \end{aligned}$$

(23)

Using (23), $\beta$ can be well defined through the relationship:

$$\begin{aligned} \beta \cong \frac{\rho _a - \rho _0}{\rho _0 (T_0-T_a)} \end{aligned}$$

(24)

Equations (23) and (24) are invalid for wet air conditions, because the moisture contained in the air affects the air density, which depends also upon the relative humidity (RH). According to EEL meteorological station, in the room of experiments, the RH varied from 40 to 65% during the measurement periods. It is interesting to estimate the relative errors made in model predictions regarding trajectory of self-merging, centreline axial velocities and concentrations for dry air conditions instead of the wet air conditions occurring during the experiments:

Let average $RH=55$%, average room temperature $T_a=23$ °C, ring exit temperature $T_0=68.5$ °C and room altitude $H=73$ m above sea level. For wet air conditions, the website (last accessed on July 28, 2023) provided by BARANI DESIGN Technologies s.r.o. [58] is used, which provides a density calculator, calculating the density with 4 digit accuracy as a function of air temperature, air relative humidity and barometric pressure or, alternatively, altitude above sea level. The corresponding air densities for air conditions given above are $\rho _a=1.175 \,\textrm{kg}/{\textrm{m}}^{3}$ and $\rho _0=0.962 \,\textrm{kg}/{\textrm{m}}^{3}$, which give $g_0^\prime =g(\rho _a - \rho _0)/\rho _0 = 2.172\, \textrm{m}/{\textrm{s}}^{2}$. The corresponding density values for dry air conditions are $\rho _a=1.18353 \,\textrm{kg}/{\textrm{m}}^{3}$ and $\rho _0=1.02569\,\textrm{kg}/{\textrm{m}}^{3}$; thus $g_0^\prime =1.50963 \,\textrm{m}/{\textrm{s}}^{2}$, while $g_0^\prime =1.50719 \,\textrm{m}/{\textrm{s}}^{2}$, based on temperatures. Therefore, the mean effect on the Froude numbers ratio (dry $F_{0d}$ over wet $F_{0w}$) regarding predictions is $F_{0d}/F_{0w}=1.2$.

Using the dry air conditions instead of the wet air conditions given above, the model predictions and estimated relative errors are shown in Table 4. The relative error using dry air conditions instead of wet ones for SMA predictions of the reattachment point $P_0$ is − 21.2%. For the centreline mean velocity predictions at $z=60d_0$, the relative errors are 9.6% and 11.4%, and for the concentrations − 10.2% and − 12.9%, regarding SMA and SM, correspondingly. The relative errors of model predictions for dry air conditions against the measured values are given in Table 5. The relative error for the reattachment point is 0.6% regarding SMA. The corresponding errors for the centreline mean velocities at $z=60d_0$ are − 14.7% and 4.6%, while for the mean concentrations − 8.6% and 4.8%, regarding SMA and SM.

Table 4 Relative errors of predictions using dry air conditions instead of wet air conditions

Table 5 Relative errors of predictions using dry air conditions against measured values

5 Summary and conclusions

The present study concerns the experimental, theoretical, and numerical investigation of the mean flow and mixing properties of a thermal annular plume in the core region and beyond it in the near field. Two sets of experimental runs were performed. One complete set with a flow rate of $1986 \,{\textrm{cm}}^{3}/\textrm{s}$ and another one with a flow rate of $2646 \,{\textrm{cm}}^{3}/\textrm{s}$ for only axial velocity measurements. The air velocity transducer TSI 8465 was used for velocity measurements along with thin nylon strips to detect the velocity direction. A series of 14 thermocouples was used for temperature measurements.

In the core region, experiments on annular plumes with initial velocities $w_0=0.527$ and 0.702 m/s were performed. The mean flow and mixing characteristics at the areas along the trajectory of maximum values and along the centreline of the core were measured and predicted where possible. A vortex ring, the self-merger, the points of stagnation and reattachment and a nearly uniform temperature were detected in the core region. The self-merger is the result of the pressure deficit existing between the external ambient fluid and the annular core. This deficit draws the annular plumes to the centreline (Coandǎ effect). Beyond the core region, the mean axial velocities and mean temperatures for each set were measured and predicted at several axial distances along the plume centreline.

An original integral model for the prediction of the self-merging trajectory of the annular plume, briefly named SMA, is developed. SMA has two main points of interest that deserve to be highlighted: (a) The validity that two anti-diametrical plane plumes are equivalent to the annular plume in the core region is justified by the method of conformal mapping, which also helps to determine the applicability limits of SMA. SMA predictions are acceptable for diameter ratio of annular plume $d_0/D_0 \le O(0.1)$, which is approximately equivalent to the range $1< D_{out}/D_{in} < 2$. (b) Calculation of sink strength and volume flux in the self-merging region requires the external analytical functions (10) and (11) proposed herein. In general, SMA predictions are compared with the experimental data and the accuracy is satisfactory.

For $w_0=0.527$ m/s, the reattachment location $P_0$ is experimentally detected at $z_{P0}=16.1 d_0$, while SMA predicts it at $z_{P0}=16.0 d_0$. For $w_0=0.702$ m/s, $P_0$ is experimentally detected at $z_{P0} \cong 20 d_0$, while SMA predicts it at $z_{P0}=20.7 d_0$. Both predictions are very close to the experimental values. The present experiments indicated that the location of the maximum upward velocity at the centreline of annular plumes does not coincide with the reattachment location, in contrast to the annular jets with high initial velocities. This fact is attributed to the waving and flickering of the self-merging trajectories of the annular core due to low momentum and disturbances induced by the vortex ring. Another finding is that the outer entrainment and the entrainment starvation from the internal boundaries of the equivalent 2D plumes contribute to the convergence limit by 99% and only 1% is owed to the flow potential.

With the aid of dimensional analysis, the reattachment point, $z_{P0}$, is successfully correlated with the dimensionless independent variables of the initial Froude number $F_0$ and the annulus diameter ratio $D_{out}/D_{in}$. It is found that the normalised dimensionless reattachment distance $Z_{P0}=(z_{P0}/d_{eq} ) F_0^{-1}$ is practically constant, $Z_{P0}=1.38 \pm 0.05$ in the region $1< D_{out}/D_{in} \le 2$, meaning that $z_{P0}$ is mainly proportional to $F_0$ and $d_{eq}$.

Experimental observations reveal that a vortex ring is generated within the core region, occupying the area up to the level $z \cong 5.5 d_0$ for both sets examined. The vorticity of the annular plume propagates a downward velocity at the plume centreline. The velocity direction was detected using a very thin nylon strip, which was initially directing downwards for $0 \le z < 5.5 d_0$ and changed abruptly to the upward direction for $z \ge 5.5 d_0$ (see description on movie1 in “Appendix 1”). The centreline velocity was measured, and its minimum (downward maximum) value − 0.926 m/s occurred at the level $z=2.742 d_0$ for the set with $w_0=0.527$ m/s and − 1.027 m/s at $z=3.5 d_0$ for $w_0=0.702$ m/s. This centreline location should get the lowest atmospheric pressure. The location $z \cong 5.5 d_0$, where the centreline velocity changes direction, defines the stagnation point. The experimental measurements of the mean axial velocity at the plume centreline show that the stagnation point is located at $z_{P1}=5.42 d_0$ for the set with $w_0=0.527$ m/s and at $z_{P1}=5.51 d_0$ for $w_0=0.702$ m/s. The good stability of the stagnation point is a matter of related stability of the vortex ring due to the solid bottom, which prevents the downward motion of the vortex ring.

The axisymmetric mode of SMA is used to predict the centreline mean axial velocities and mean concentrations of the annular plume. The predictions are compared with the experimental measurements and are in very good agreement for $Z \ge 5$, regarding velocities and concentrations, while they overestimate the velocities and concentrations for $Z<5$ and $Z<4$, correspondingly. In addition, the above centreline quantities are predicted using the superposition method (SM) and the results are also compared with the experimental measurements and SM predictions and are in excellent agreement with the experimental data for $Z \ge 2$.

The research ends up with some implications regarding the sensitivity of critical parameters used in the models. The use of suitable equations in SMA is mainly understood by the smoothness of convergence towards the measured intersection point in predicting the self-merging of the annular plume and, additionally, by the physical realism of the model and its ability to accurately predict laboratory measurements. Another critical point is the use of air densities based on the standard atmosphere. The sensitivity analysis shows that the use of dry air densities instead of densities of wet air with an average relative humidity 55% may result to errors − 21.2% in SMA predictions of the reattachment point $P_0$. The errors made in the centreline velocity predictions at $z=60 d_0$ are 9.6% and 11.4%, and for the concentrations − 10.2% and − 12.9%, regarding SMA and SM, correspondingly. The relative error of SMA prediction for dry air conditions against the measured value for the reattachment point $P_0$ is 0.6%. The corresponding errors for the centreline mean velocities and mean concentrations at $z=60 d_0$ are − 14.7% and − 8.6% for SMA, while 4.6% and 4.8% for SM predictions.

For practical purposes, it is implied that the annular plume constitutes the limiting case of the equivalent rosette-type diffuser, which may be used in the design of systems for wastewater disposal in the ocean. In the case of discharges in deep water bodies, the annulus-type diffuser, which is equivalent to a rosette-type diffuser, succeeds at the water surface approximately equal dilutions to the rosette-type diffuser and the equivalent round plume for equal flow rates. In shallow water bodies, the annulus-type diffuser may achieve about 15 times (on average) higher centreline dilutions than the corresponding bulk dilutions of the round plume. This result is due to the better mixing in the core region owed to the vortex ring, the increase of the outer entrainment due to the pressure drop inside the core and the longer diameter of the ring exit.

The present research may help engineers, model developers and project designers dealing with wastewater disposal systems into water bodies or air-pollutant and cooling-tower emissions into the atmosphere, to scientifically support decision-making and management practices.

Appendices

Appendix 1: Description of the video movie1.mp4

The velocity transducer is unable to detect the velocity direction. Therefore, a visual method was used to detect the centreline location where axial velocities change direction. For the basic case of initial velocity $w_0=0.527$ m/s, the procedure was recorded in the video named movie1.mp4 and it can be observed frame-by-frame. Initially, a black cardboard was installed behind the flow field, about 20 cm away from the plume centreline and a narrow strip of white cardboard marked every 1 cm was placed 2 cm behind the plume centreline, both in the vertical direction. Care was paid to fixing the marked strip so that the zero level was accurately at the level of the ring exit. Then, a video camera was fixed on a tripod about 50 cm in front of the plume centreline and at the same level as the location of the phenomenon to be recorded. Putting the device in normal operation, a very thin nylon strip tethered in a stretched thread was moving at the plume centreline vertically from up to down. It is observed that the thin nylon strip was directing upwards over the level of 5.5 cm and at this level changed abruptly the direction, which was maintained all the way downward. This video is accompanying the paper.

Appendix 2: Measurement details—Tables

Prior to the experimental measurements, the hot air production system was activating. The measurements always started twenty minutes after turning the setup on, to allow the temperature stabilization in the air heating box. At the cross-sections of the nozzle exit, the temperatures were recorded for 84 min, using the HH147 OMEGA thermometer with four thermistors. The air temperature variation was determined at each cycle of the electronic controller. The thermistors were placed at the four cross-sectional locations $x=-x=y=-y=6$ cm of the nozzle exit. Each thermistor recorded the mean value of temperature per 1 min time interval. The result is that the average value and standard deviation of the mean air temperature level at the exit were kept at $68.5 \pm 3.81$ °C and the cycle duration was about 3 min and 55 s on the warm summer days or approaching 3 min during cold days of the year. It is important to notice that the room was thermally well insulated, while the renewal of the room air was extremely slow during the experiments because all openings were kept closed. The room and chamber temperature at the beginning of the measurements was uniform. The increase of the background mean temperature of the room air from the beginning to the end of each-day measurements (duration of 7.5 h, approximately) was less than 2.0 °C, which causes a gradual reduction of the effective gravitational acceleration $g_0^{\prime }$ up to 5%, approximately. Similarly, continuous temperature measurements of the background air environment inside the experimental chamber were recorded at the levels 0.25 m, 0.75 m, 1.25 m, and 1.75 m above the floor, in order to check the degree of the air uniformity during the experimental measurements. The air temperature at the level of 1.75 m of the chamber background, compared to the temperature at 0.25 m, was found less than 1 °C higher. The average relative humidity inside the room during the measurement hours was 55%. A 1 °C-temperature increase causes a weak stratification of air density (density slope $-2.67\times 10^{-3}$ kg/m$^4$). According to the annular plume characteristics and applying the experimental results by Ogino et al. [59], it is estimated that the plume would be trapped by the density interface and to outflowed horizontally at the level 4.36 m above the floor, which is higher than the ceiling level being at 2.89 m. Consequently, it was not expected nor observed any plume entrapment during the measurements conducted. It is noted that the maximum level for the measurements was 2.34 m above the floor. If the system of the room and the buoyant source is considered as a filling box, it is useful to determine the characteristic time needed for the air volume of the room to pass through the system of the air compressor-heating box. According to Baines and Turner [60] and Caulfield and Woods [61], based on the data given above, this time is longer than 15 h for both flow rates used. As the experiments took place in the cylindrical chamber, the plume, a little lower than the ceiling and over the chamber, spread horizontally and mixed with the air of the room mainly outside the chamber. The chamber was supplied with the cold air of the room from the radial entrance gap near the floor (Fig. 2). The air temperature in the gap remained slightly less ($\le 0.5$ °C) than the mean temperature of the background air of the chamber due to the slight stratification of the room air. The time needed for steady-state temperature in the room air is estimated to be more than 11 h [62]. Thus, the plume characteristics remained practically unaffected by the filling box effect.

The velocity was measured at several levels along the vertical z-axis in the annular plume centreline. Velocity measurements were also made in both x and y directions of several cross-sections of the main flow. For performing the latter measurements, the air velocity transducer was mounted within a cardboard hollow cylinder, as proposed by Stefanidou et al. [42]. The main experiments were carried out with the flow rate $\mu _0 = 1986$ cm$^3$/s, which provided mean velocity $w_0 = 52.7$ cm/s at the exit. In addition, another set of axial velocity measurements was carried out with flow rate $\mu _0 = 2646$ cm$^3$/s providing a mean exit velocity $w_0 = 70.2$ cm/s. To get a reliable mean axial velocity value for each measurement in a prescribed point of the flow field, the recording duration was set at least to 6 min, providing 36 values of 10-s time step with 10-s averaging time for each value. Reliability checks of the velocity transducer performed at the plume centreline showed that the error was less than 10%. Tables 6 and 7 include the statistical data of the experimental axial velocity measurements at the plume centreline in the core region for $w_0=0.527$ m/s and 0.702 m/s, correspondingly, along with the measurement conditions. Tables 8 and 9 include the corresponding measurements beyond the core region. Table 10 includes the data for the trajectory of merging. For the verification of the measurement reliability of velocities, several runs of measurements were repeated.

The temperatures were measured at several levels z. At each level, simultaneous temperature measurement was made at the whole cross-section. The computer recorded one instant temperature value per second at each thermistor location, simultaneously for all thermistors. The temperature measurement procedure was very simple. Initially, the system of thermistors was located and fixed at the required level for measurements. Before turning the system of the hot air production on, the system of thermistors is turned on and was recording the room air temperature for about 20 min. This measurement was used to set a common calibration level for all thermistors. The time interval for reliable temperature measurements was at least 35 min for a cross-section. For each thermistor location, the number of values, the average, the min/max values, and the standard deviation have been calculated. Table 11 includes the statistical data regarding the temperature measurements in the core region along the plume centreline, accompanied by the measurement conditions, and Table 12 the maximum temperatures and their location, for $w_0=0.527$ m/s. Table 13 includes the temperatures along the plume centreline beyond the core region for $w_0=0.527$ m/s. It is noted that the temperature measurements were repeated some months later for reliability verification.

Table 6 Statistical data of centreline axial velocity measurements in the core region of the annular plume with $w_0=0.527$ m/s

Table 7 Statistical data of centreline axial velocity measurements in the core region of the annular plume with $w_0=0.702$ m/s

Table 8 Statistical data of the centreline axial velocity measurements beyond the core region of the annular plume with $w_0=0.527$ m/s

Table 9 Statistical data of the centreline axial velocity measurements beyond the core region of the annular plume, with $w_0=0.702$ m/s

Table 10 Statistical data of the trajectory location in the core region of the annular plume with $w_0=0.527$ m/s

Table 11 Statistical data of the centreline temperature measurements in the core region of the annular plume with $w_0=0.527$ m/s

Table 12 Statistical data of the measurements of maximum temperatures and their location at several cross-sections of the core region of the annular plume with $w_0=0.527$ m/s

Table 13 Statistical data of the centreline temperature measurements beyond the core region of the annular plume with $w_0=0.527$ m/s

Appendix 3: Mathematical justification of the self-merging of the annular plume to resemble that of two-plane plumes

The assumption made in Sect. 3.2, saying that the two infinitesimal sectors of the core region of the annular plume may be considered as two plane plumes, is supported by the method of conformal mapping [63, 64]. In the complex plane, which is defined by the real axis x and the imaginary axis y, it is plotted a strip $S={z=x+iy \mid b \le x \le a,-\pi \le y \le \pi }$, where $a=\log (R_1)$ and $b=\log (R_2)$; $R_1$, $R_2$ are the lengths of the outer and inner radius of the cross-section of the core region of the annular plume at level $z>0$. It is assumed that a horizontal entrainment velocity $u_{B0}$ is applied in the outer boundary on the right of strip S, as shown in Fig. 19a. The complex function $\zeta =e^z$ maps the strip S to the annulus $A={\zeta =\eta +i \psi \mid R_1, R_2 }$, and the field of horizontal entrainment velocities $u_{B0}$ to the radial velocities $u_{B1}$ applied in the outer boundary of the annular plume, as shown in Fig. 19b. It is noted that this is a one-to-one mapping. The same function maps the source strip $S_0={z=x+iy \mid b_0 \le x \le a_0,-\pi \le y \le \pi }$ to the annular source $A_0={\zeta =\eta +i \psi \mid D_{out}/2, D_{in}/2 }$, where $a_0=\log (D_{out}/2)$ and $b_0=\log (D_{in}/2)$; $D_{out}$, $D_{in}$ are the lengths of the outer and inner diameter of the cross-section of the annular source. The field point P(x, y) is mapped to $P(\eta ,\psi )$ in Cartesian or $P(r,\theta )$ in polar coordinates. The outer boundaries of cross-sections are shown in Fig. 19 with brown dashed line and large dots, which are numbered from 1 to 8, while the inner boundaries with blue dashed line and the mean diameter of the plume source is plotted by a dot-dash-dot red line. The red point $A(\frac{a_0-b_0}{2},0)$ of the strip at the source centreline is mapped to the red point $A(\frac{D_{out}-D_{in}}{2},0)$ at the centre of the annular slot. The black points at the source centreline of the strip, $B_1(\frac{a_0-b_0}{2},\frac{L_s}{2})$ and $B_2(\frac{a_0-b_0}{2},-\frac{L_s}{2})$, are mapped to the same black point $B_1(\frac{D_{out}-D_{in}}{2},\pi ) \equiv B_2(\frac{D_{out}-D_{in}}{2},-\pi )$ at the centre of the annular slot. The large brown dots numbered from 1 to 8 at the strip boundary on the right are mapped to the large brown dots at the outer boundary of the annular plume following one-to-one the corresponding numbering.

Fig. 19

Conformal mapping of strip a along with the field of the entrainment velocities $u_{B0}$ to annulus b along with the corresponding field of the radial entrainment velocities $u_{B1}$

The boundary conditions (like entrainment) are $2\pi$-periodic functions of the angle $\theta$ of the polar coordinate system $O(r,\theta )$ shown in Fig. 19b. Thus, the solution made in the 2D problem with entrainment extended along the boundary on the right of the strip is mapped on the outer boundary of the annulus. Experimental evidence has shown that the 2D behaviour of a slot diffuser of a finite length is maintained up to a level approximately equal to the diffuser length [46, 65]. Since the 2D behaviour of each plane plume of slot width $d_0$ should be extended at least up to the reattachment location $z_{P0}$, the length $L_s$ of a finite slot should be at least $L_s \approx z_{P0}$ or $L_s/z_{P0}=O(1)$. Thus, as $z_{P0}/d_0=O(10)$, it is implied that $L_s/d_0=O(10)$; for the strip shown in Fig. 19a representing the slot source of the plane plume with width $(a_0-b_0)$ and length $L_s=2\pi$, the strip aspect ratio is $L_s/(a_0-b_0)=2\pi /\log (D_{out}/D_{in})=O(10)$. Solving the latter equation with respect to $D_{out}/D_{in}$, one gets $D_{out}/D_{in} \approx 1.9$ or, equivalently, $d_0/D_0 \le 0.31$; $d_0$, $D_0$ are the slot width and the mean diameter of the annular source, correspondingly. This means that a solution for well predicting $z_{P0}$ using plane plumes is efficient for annular plume sources with diameter ratios in the range $1< D_{out}/D_{in} < 2$. Based on the above justification, it is concluded that the assumption made in Sect. 3.2 is valid for the cases studied herein.