2.1 The experimental setup

Experiments were conducted in a 551 mm-high tank of horizontal section $588~\text{mm}\times 300~\text{mm}$ , filled with demineralized water at room temperature (see figure 1). The front and back sides of the tank were made of 20 mm-thick transparent plastic (polymethyl methacrylate, or PMMA), whereas 5 mm-thick aluminium plates at the sidewalls conducted the heat from water in the adjacent compartments to the test section. In the present experiments, one lateral sidewall (volume $300~\text{mm}\times 20~\text{mm}\times 551~\text{mm}$ ) was filled with hot water which was kept at a constant temperature by means of a thermostatic bath with a maximal power of 2.2 kW and a precision of $0.1\,^{\circ }$ C. This water was circulating at a rate of $0.43~\text{l}~\text{s}^{-1}$ . For insulation, the other side as well as the bottom were covered with PMMA and PVC, respectively, whereas the top boundary was covered with extruded polystyrene foam (see figure 1 a). Since the aluminium wall was in contact with two liquid baths at different temperatures, its own temperature was situated in between and changed gradually as the tank warmed up. Although this slowed down the total heating time of the test section, and therewith the different phases of the flow evolution, this change was too slow to modify the flow dynamics. Before starting the experiment, it was verified that the fluid in the test section was at rest. At the start of the experiment, hot fluid from the thermostatic bath was injected into the (empty) side compartment. To compensate for the initial transient effects of heating the system, the fluid injected at the start of the experiment was preheated to a few degrees above the desired temperature ( $T_{c}$ ).

Experiments were conducted at the initial temperatures of $T_{0}=21.4\,^{\circ }$ C in the test section and at $T_{c}=53.6\,^{\circ }$ C in the lateral compartment, the latter being controlled by a probe. This was the maximum temperature difference supported by the experimental device, which was sufficient to get a significantly turbulent plume. The resulting wall temperature was approximately $39\,^{\circ }$ C (see appendix A for details on the calculation). Therefore, the Rayleigh number and the Prandtl number for our experiments are, respectively,

where ${\it\beta}$ , ${\it\nu}$ and ${\it\kappa}$ are the fluid thermal expansion, kinematic viscosity and thermal diffusivity, respectively (taken at $30\,^{\circ }$ C), $g$ is the gravitational acceleration and $H$ the height of the cavity. The compartment temperature was kept constant in the present paper, and a dozen tests were conducted for this same temperature difference, with very similar results obtained in all cases.

Figure 1. Sketch of the experimental setup with (a) the heating system and (b) the PIV/LIF image acquisition system. (a) Front view. (b) General setting.

To monitor the temperature and calibrate the T-LIF images in the test section, the temperature was measured near the rear wall in the middle of the tank with seven to 11 platinum temperature probes (see figure 1 a) with a precision between $0.2\,^{\circ }$ C and $0.5\,^{\circ }$ C. To visualize the particles and the fluorescent dye, the flow was illuminated by a vertical laser sheet shining through a transparent slit at the bottom. A solid continuous laser (Spectra-Physics) of 532 nm operating at 4.5 W was used and generated a beam of 2.3 mm thickness which was transformed into a light sheet by means of an oscillating mirror. Considering the relatively slow velocities in the convection plumes, an oscillating mirror is preferable to a cylindrical lens since it generally provides a more uniform sheet. To ensure identical lighting on every image and minimal exposition of the dye to the laser light, the mirror oscillations were triggered in phase with the image acquisition so that the laser beam passed an integer number of times over the entire view during each exposure time. For temperature measurements with LIF, Rhodamine B was used as fluorescent dye. Details are discussed below (§ 2.2). For PIV measurements, the water was seeded with polyamide tracer particles with a mean diameter of $30~{\rm\mu}\text{m}$ and a density of $1016~\text{kg}~\text{m}^{-3}$ .

The flow was recorded simultaneously by two 12-bit  $1024\times 1024$ pixel Dalsa CCD cameras at a frame rate of 1 Hz. One camera only recorded the particle positions for PIV measurements by taking bursts of three successive images, and the other camera was equipped with a band filter with wavelengths passing between 565 nm and 585 nm and recorded the fluorescent dye for the T-LIF measurements. To compute the velocity fields for each burst, two pairs of images, representing a short and a long time interval, were processed using the PIV cross-correlation algorithm Uvmat (Matlab toolbox http://servforge.legi.grenoble-inp.fr/projects/soft-uvmat). Thus, we obtained a field containing 17 vectors per square centimetre with an average error estimated at 6 % in the plume region.

All the variables employed in this paper will be described within a Cartesian coordinate system ( $x,z$ ) with $z$ being the upward vertical direction (see figure 1 b).

2.2 Details of the T-LIF technique

Rhodamine B is a temperature sensitive dye in which the emission sensitivity to temperature changes with wavelength. Dissolved in water, the emission peak is around 575 nm, and the sensitivity to temperature is at its maximum around 570 nm (see Bruchhausen, Guillard & Lemoine Reference Bruchhausen, Guillard and Lemoine2005). An optimal choice for the bandpass filter is found at 565 to 585 nm. To minimize absorption of incident light along the laser path, the Rhodamine B concentration was chosen to be as low as possible. Since the dimensions of the experiment are significantly bigger than the dimensions used in previous T-LIF applications (Coolen et al. Reference Coolen, Kieft, Rindt and van Steenhoven1999; Bruchhausen et al. Reference Bruchhausen, Guillard and Lemoine2005; Petracci, Delfos & Westerweel Reference Petracci, Delfos and Westerweel2006), we used a relatively lower concentration here, i.e. $3.5\times 10^{-5}~\text{g}~\text{l}^{-1}$ .

In order to avoid permanent local intensity variations due to light absorption or optical aberrations (see figure 2 a), the images were normalized with two reference fields taken for two homogeneous temperatures $T_{lo}$ and $T_{hi}$ (see Coolen et al. Reference Coolen, Kieft, Rindt and van Steenhoven1999):

This equation allows us to find a unique polynomial relation between light intensity and temperature. To obtain this relation, each experiment has been calibrated in advance, using the temperature data from the probes in the tank. An example of this procedure is given in figure 2(b). The final result (see figure 3 a,b) has a precision of approximately $\pm 0.2\,^{\circ }$ C (figure 3 c) on a field of $3200~\text{cm}^{2}$ corresponding to $1024\times 1024$  pixels (see figure 3), which is relatively high compared with previous T-LIF experiments reported in the literature.

Figure 2. T-LIF calibration with (a) raw image (axes in centimetres) taken at constant temperature and showing fluorescence intensity variations due to absorption along beam paths and optical reflections. These defaults are corrected by normalization (see text). (b) Normalized intensity $I_{n}$ at the position of the vertically distributed temperature probes plotted against their measured temperature. The 11 resulting curves are all fitted by the same second-order polynomial (in this case, $I_{n}=6.91\times 10^{-4}T^{2}-1.01\times 10^{-1}T+2.86$ ).

Figure 3. Examples of T-LIF results. Grey levels represent temperature in degrees Celsius, axes are in centimetres. (a) Temperature field at the beginning of the experiment ( $t=50$  s). (b) Temperature field when stratification is established ( $t=500$  s). Remaining striations that are due to temporal variations in laser light did not affect the results. (c) Stratification profiles taken at different times, averaged over 10 s with probe measurements ( $+$ ) and T-LIF measurements (solid lines).

Note that other methods referred to as ‘two-colour’, based on two dyes (e.g. Coppeta & Rogers Reference Coppeta and Rogers1998) or two spectral bands of the same dye (e.g. Bruchhausen et al. Reference Bruchhausen, Guillard and Lemoine2005), may be used to correct for variations in the illumination. However, the dimensions of our experiment make them very difficult to apply here because of light absorption and re-absorption phenomenon, and, given achieved accuracy of the simpler one-colour method, the expected gain should not be significant.

3.1 Plume and stratification evolution

At the start of the experiment the heated wall causes a thin hot plume which rises along the boundary and hits the top of the tank (figure 4 a). A vortex motion forms in the top left-hand corner and, while moving to the right along the top boundary, entrains ambient cold fluid at rest (figures 3 a and 4 b). When it hits the right wall, it moves vertically downwards against buoyancy, transferring its energy from inertia into mixing. A gravity-current-like motion results and returns the mixed fluid until it hits the left wall, whereafter the newly formed warm upper layer makes a slumping wave-like motion (figure 4 c–e). This overturning and slumping motion has also been observed for plumes at the centre of a cylindrical tank, with the flow evolution scenario depending on the tank aspect ratio (Kaye & Hunt Reference Kaye and Hunt2007).

Figure 4. Sequence of images showing the onset of the turbulent plume and the establishment of the stratification. Grey levels indicate the temperature and arrows the velocity field. Axes are in centimetres, and temperature is in degrees Celsius. (a–f) Fields at the beginning of the experiment (velocity scale $1~\text{cm}:0.1~\text{cm}~\text{s}^{-1}$ ). (g–i) Fields at later times (velocity scale $1~\text{cm}:0.05~\text{cm}~\text{s}^{-1}$ ).

In the next stage of the flow evolution, the wall plume continues to supply hot fluid to this warm upper layer. This layer is now of a slightly lower temperature than that in the plume. In the next stage, instead of a vortex, the plume motion gives rise to a jet-like motion along the top boundary. With the thickening of the upper layer, the local buoyancy decreases and this jet also decreases in intensity as can be inferred from the vectors in figure 4(g–i). The continuous supply of hot fluid above formerly injected fluid causes the descent of the interface. Eventually, after approximately 40 min of heating, the warm upper layer reaches the bottom, and the interior is entirely stratified in temperature (figure 4 g–i).

In the subsequent final stage, the plume entrains fluid from the stratified ambient and gradually decreases the interior temperature gradient. The temperature profiles shown in figures 3(c) and 5 reveal the details of the evolution of the stratification and the gradual heating of the interior. These temperature profiles remain very similar in shape up to approximately 2000 s, i.e. before the warm layer has reached the lower boundary. The temperature profile near the bottom is subsequently distorted because of the laminar character of the plume in this region.

Figure 5. Ambient temperature profiles at different times showing the evolution of stratification.

From the above evolution, we can distinguish three dynamically different zones: (A), (B) and (C) (see figure 5). Zone (A) at the top ( $z>50$  cm) is characterized by a strong stratification in a thin top layer that is continuously supplied by hot and hardly mixed plume fluid. This warm top layer (A) forms at the very beginning and was also present in the experiments with salt plumes of Cooper & Hunt (Reference Cooper and Hunt2010). Closer inspection of the plume motion below shows that this is due to the thin laminar flow adjacent to the vertical wall. This flow has been modelled in detail by Wells & Worster (Reference Wells and Worster2008). The middle zone (B) ( $20~\text{cm}<z<50~\text{cm}$ ) is stratified, and exchanges fluid with the wall plume. In the lower layer, zone (C), the plume is thin and laminar. As a consequence, the exchange of fluid and the mixing of the layers is weak. Since the heated layers are pushed downwards by continuity, the temperature gradient in this region increases in time. Eventually, zones (A), (B) and asymptotically also (C) will become homogeneous and obtain the temperature of the heated sidewall.

To study the turbulent properties of the plume, we consider the Reynolds decomposition $u_{i}=\overline{u_{i}}+u_{i}^{\prime }$ , with the bar denoting a time average over 200 s and the prime representing fluctuations. The Reynolds tensor is defined as $\overline{u_{x}^{\prime }u_{z}^{\prime }}$ and is displayed in figure 6(a–c) at three different instants in time (i.e. $t=80$ , 400 and 1600 s), with the highest values at the edge of the plume, indicating turbulent mixing. A clear transition is visible at a height in the range of 22 cm to 25 cm. This transition corresponds to a change in flow regime from laminar to turbulent, as can be evaluated using the local Grashof number defined as

where ${\rm\Delta}T$ is the temperature difference between the temperature imposed at the wall, and the temperature outside the plume. For ${\rm\Delta}T\approx 15\,^{\circ }$ C, the typical transition value of $10^{9}$ is actually reached at $z_{crit}\approx 24$  cm (Bejan & Lage Reference Bejan and Lage1990). Indeed a clear turbulent activity can be noticed in figure 6(a–c) above this height. This transition from laminar to turbulent flow in the boundary layer is noticeable in all measurements.

The turbulence and consequently the mixing starts in zone (B). During the flow evolution it also continues to mix when the warm layer has increased in thickness (see figures 6 a–c in relation to 6 d–f). In contrast, close to the wall, the plume is laminar and heated fluid is transported upward without much mixing and causes the high temperature gradient near the top in zone (A). The large temperature gradient perpendicular to the wall is clearly noticeable in figure 6(d–f).

Figure 6. (a–c) Reynolds tensor $\overline{u_{x}^{\prime }u_{z}^{\prime }}$ ( $\text{cm}^{2}~\text{s}^{-2}$ ) and (d–f) temperature ( $^{\circ }$ C) in the plume region at $t=80$ , 400 and 1600 s after the start of the forcing. Axes are in centimetres.

3.2 Ambient flow characteristics

More detailed information about the flow can be obtained from the local Richardson number, defined as

where ${\it\rho}(x,z)$ and $U(x,z)$ are the local density and velocity, respectively. In general, the Richardson number reveals gravitational instabilities ( $\partial {\it\rho}(x,z)/\partial z>0$ ) and shear or Kelvin–Helmholtz instabilities ( $0<Ri<1/4$ ). Gravitational instabilities are very localized to instantaneous and intermittent flow reversals during the initial flow stage (see figure 4 b). Therefore, here we focus on shear instabilities that are displayed for visualization purposes on a logarithmic scale of the absolute Richardson number in figure 7). The details of the mixing are highlighted by the low (dark) values of the Richardson number. In particular, the mixing due to the continued plume motion along the top boundary is important during the upper layer formation and its growth in time (see figure 7 a–f). After the upper layer formation in the initial flow stage, the mixing occurs principally between the rising plume and the interior, whereas the mixing between the upper and lower layer is negligible (see figure 7 e–g).

Figure 7. Absolute value of Richardson number $Ri=|-(g/{\it\rho})(\partial {\it\rho}/\partial z)/(\partial U/\partial z)^{2}|$ at different times corresponding to figure 4 using a logarithmic scale ( $Ri>1$ in white zones). Black spots in the unstratified zone (C) ( $(g/{\it\rho})(\partial {\it\rho}/\partial z)\approx 0$ ) are due to noise. Axes are in centimetres.

The streamlines and stratification in figure 8 show the averaged flow evolution of the circulation during the warm layer deepening. As in the case of a laminar plume (see Worster & Leitch Reference Worster and Leitch1985, figure 7), the hot fluid injection along the top boundary causes the entire stratification to move downwards. The difference with laminar flow is apparent from the turbulent exchange of the warm layer with the plume motion (top-left quarter), and becomes more pronounced with time from the circulation pattern in figure 8(b,c). This exchange and consecutive mixing in the plume is responsible for the presence of the front between the upper and lower layer. This front is indeed absent in the case of a laminar plume. In the lower layer, by continuity, the descending interface forces the fluid in the lower layer into the plume. This effect becomes more pronounced once the lower layer is thinner and the motion remains horizontal (figure 8 c).

Figure 8. Velocity (arrows) and temperature (grey levels) fields averaged over 200 s, with streamline patterns. Axes are in centimetres, temperature is in degrees Celsius. Time averages cover the spans (a) $t=100{-}300$  s, (b) $t=700{-}900$  s and (c) $t=2300{-}2500$  s.

Figure 9. Averaged quantities in the plume. (a) Typical vertical velocity and temperature profiles (at $z\approx 40$  cm and $t=150$  s). The dotted lines show the top-hat profile used in the model (§ 4). (b) Plume width $b$ as a function of $z$ at different times. (c) Plume characteristic velocity $w$ as a function of $z$ at the same times.

3.3 Wall plume details for the modelling

Figure 9(a) shows the velocity and temperature profiles in the plume. To reduce the scatter, the data (recorded with a frequency of 1 Hz) are averaged over a period of 30 s. These velocity and temperature profiles conserve their general shape at all heights. To determine the characteristic quantities of the plume which are used in the model, i.e. the plume width $b(z)$ , the characteristic plume velocity $w(z)$ and the characteristic plume temperature $T(z)$ , we have used conservation of volume, momentum and energy (buoyancy) fluxes. They can be written for all $z$ as

and

where $u_{z}(x,z)$ and $T_{xz}(x,z)$ are the local vertical velocity and temperature, respectively, and $T_{e}(z)$ is the ambient temperature. To simplify notation, $b(z)$ , $w(z)$ and $T(z)$ will be written as $b$ , $w$ and $T$ , respectively, in the following.

The profiles of $b$ and $w$ (in figures 9 b and 9 c, respectively) clearly show the laminar–turbulent transition in the wall plume at $z\approx 24$  cm in accordance with figure 6(a–c). For $2~\text{cm}<z\lesssim 24$  cm the plume width remains almost constant while velocity increases linearly, and it expands in the zone for $24~\text{cm}<z<45$  cm (see also figure 6 a–c). The increase in plume width is not balanced by an equivalent gain in volume flux, so that the velocity $w$ decreases. At the upper boundary the vertical volume flux goes to zero, and the plume is deviated horizontally. Since the vertical velocity also changes to zero, the evaluation of $b$ becomes meaningless, as is apparent at $z>45$  cm (figure 9 b).

3.4 The entrainment coefficient

The entrainment coefficient is the key parameter for the modelling described below and will be determined empirically. It is evaluated with the equation of volume conservation (4.1) introduced in next section:

The experimental values for $bw$ and $w$ are accurately obtained from PIV measurements. Using a linear regression between $\text{d}(bw)/\text{d}z$ and $w$ , a unique value for ${\it\alpha}$ is calculated at each time step for the entire wall plume and therefore includes the contributions of both the homogeneous and the stratified zones of the turbulent plume (see appendix B for details). The values are plotted in figure 10(a) against time. It starts around 0.08 when the environment is still homogeneous, a value which is close to that of a buoyant jet (e.g. Ellison & Turner Reference Ellison and Turner1959; Kaminski, Tait & Carazzo Reference Kaminski, Tait and Carazzo2005) or a fountain (e.g. Baines Reference Baines2002). The time-averaged value ${\it\alpha}=0.019$ is in good agreement with the entrainment value obtained by Cooper & Hunt (Reference Cooper and Hunt2010), which was obtained from the position of the density front. Although their value is close to the present one for $t>1000$  s, there is a non-negligible decrease of a factor close to 10 during the establishment of the stratification over the depth of the tank.

Figure 10. (a) Experimental entrainment coefficient ${\it\alpha}$ calculated with (3.5) for the entire plume against time, with the average value (dash-dotted line) and exponential fit (dashed line) with $t_{0}$ taken at 30 s for the onset time of the plume. (b) Entrainment coefficient, ${\it\alpha}$ , for the turbulent part of the plume (red), the laminar part (green) and total plume (blue) with the respective exponential fits. The data have been averaged over 20 s.

The consequent raise in temperature of the interior also decreases the buoyancy flux of the plume. The mean Reynolds stress term therewith decreases in time by approximately a factor of four, but the height of the transition between laminar and turbulent regimes characterized by the Grashof number hardly moves (see figure 6). This indicates that the turbulence activity of the plume decreases in time, but that it remains nevertheless turbulent. As mentioned above, in shear flows the entrainment coefficient in a stratified fluid is relatively small compared with that in a homogeneous fluid (see Fernando Reference Fernando1991). Since variations in time of buoyancy and velocity in the plume remain limited, it is the increase in stratified layer-depth in the ambient which decreases the entrainment coefficient.

Figure 10(b) differentiates the evolution of ${\it\alpha}$ in the turbulent (B) and laminar zones (A). (Note that the mechanism of entrainment in the laminar case is essentially different, but the definition (3.5) can still be used in this zone.) The initial rapid decrease of ${\it\alpha}$ in the turbulent zone is due to the increasing stratified layer-depth, with the minimal value reached when eventually the interior is entirely stratified (see figure 7 i). In the laminar zone, ${\it\alpha}$ remains constant at the beginning as the density of the ambient does not change, and then it decreases after approximately 600 s, when the front reaches the laminar part of the plume.

The sampling of the present data are not sufficient to calculate the local entrainment coefficient. However, figure 5 shows that the global temperature gradient in the turbulent zone (B) remains approximately constant (see figure 10 b after 1200 s). The entrainment coefficient within this stratified zone is also found approximately constant. The two constant values for ${\it\alpha}$ , for the homogeneous and the stratified regions, allow the decrease of ${\it\alpha}$ observed in the turbulent zone to be accurately reproduced. The details of this calculation are shown in appendix B.

4.1 The turbulent conservation equations

We consider an adiabatic prismatic box of height $H$ , width $L$ (along $y$ ) and horizontal cross-sectional area $A$ . At one sidewall, a distributed buoyancy source emits a buoyancy flux per unit area ${\it\phi}$ . The plume is considered to be fully turbulent and, in accordance with Morton et al. (Reference Morton, Taylor and Turner1956) theory, we assume the Boussinesq approximation, and the similarity of the velocity and buoyancy profiles with height. Furthermore, the velocity profile is assumed to be top-hat and the rate of entrainment proportional to the velocity at that height. All water properties except density are assumed to be constant with temperature.

The width of the plume, its vertical velocity and relative buoyancy are respectively denoted as $b$ , $w$ and ${\it\Delta}$ ; ${\it\Delta}$ can be expressed as ${\it\Delta}=g({\it\rho}_{e}-{\it\rho})/{\it\rho}_{1}=g{\it\beta}(T-T_{e})$ . Subscripts ‘ $e$ ’ and ‘1’ refer to ambient and reference fluid, respectively, whereas variables with no subscript refer to the plume. The reference variables may conveniently correspond to the characteristics of the fluid at $t=0$ . Let us also introduce the ambient buoyancy ${\it\Delta}_{e}=g({\it\rho}_{e}-{\it\rho}_{1})/{\it\rho}_{1}=g{\it\beta}(T_{1}-T_{e})$ . The conservation equations for volume, momentum and buoyancy deficiency in the plume are written as

where ${\it\varepsilon}$ accounts for dissipation due to wall shear stress that we will neglect in the following.

Unlike in Cooper & Hunt (Reference Cooper and Hunt2010), the buoyancy flux, ${\it\phi}$ , is variable due to the heat flux, ${\it\varphi}$ , across the wall. Their relation per unit area across the wall is given by

where $C_{p}$ is the specific heat capacity of water at constant pressure. The heat flux, ${\it\varphi}$ , has to be defined here as a function of the temperature imposed at the wall $T_{w}$ and the temperature of the ambient fluid $T_{e}$ . Then we write

where $h_{w}$ is the heat transfer coefficient which can be expressed according to Tsuji & Nagano (Reference Tsuji and Nagano1988) as

where the constant $K_{w}=0.120$ and $k$ is the thermal conductivity of the fluid. Since the buoyancy term for the ambient was written as ${\it\Delta}_{e}=g{\it\beta}(T_{1}-T_{e})$ , where $T_{1}$ is the reference temperature, one can introduce a wall buoyancy, ${\it\Delta}_{w}$ , defined as ${\it\Delta}_{w}=g{\it\beta}(T_{w}-T_{1})$ , so that (4.2) becomes

where $A_{w}={K_{w}}^{3}({\it\kappa}^{2}/{\it\nu})$ is the effective diffusion constant quantifying the buoyancy flux from the wall. Re-writing the conservation equations in terms of volume flux $Q=bw$ , momentum flux $M=bw^{2}$ and buoyancy flux $F=bw{\it\Delta}$ , the equations (4.1) reduce to

In ‘filling-box’ models the characteristic time scales of the ambient are assumed to be much larger than those in the plume. The heat equation in the ambient fluid is then written as

where $U$ is the vertical velocity of the ambient. Following Baines & Turner (Reference Baines and Turner1969)’s assumption for which the plume width is negligible compared with the horizontal cross-section of the box (i.e. $b\ll A/L$ ), $U$ can be expressed as $U=-QL/A$ .

We non-dimensionalize equations (4.6) and (4.7) with the scalings

and obtain for the conservation equations

Here the wall buoyancy term, ${\it\delta}_{w}$ , is

and the Rayleigh and Prandtl numbers introduced are defined as

For the heat equation (4.7), we obtain in non-dimensional form

where $HL/A$ is the box aspect ratio. The second term on the right-hand side expressing molecular diffusion in the ambient is neglected in the following.

With equations (4.9) and (4.12), we can now determine the position of the first disturbed zone in the ambient, corresponding to the lower limit of the interface between the upper and lower layer (see Worster & Huppert Reference Worster and Huppert1983). For an initially uniform ambient, i.e. $\partial {\it\delta}_{e}/\partial {\it\zeta}=0$ , for the vertical position of the front, denoted ${\it\zeta}_{fr}$ , one obtains

where ${\it\delta}_{e}^{fr}$ is the initial ambient buoyancy. This can be integrated using the condition ${\it\zeta}_{fr}({\it\tau}=0)=1$ to give

Note that the front position in time depends directly on the entrainment coefficient ${\it\alpha}$ .

4.2 Hybrid model including the laminar part

As we have seen in the experimental results, the wall plume is laminar over approximately one-third of its height. This requires a particular treatment, especially for configurations with a lower Grashof number than the present one. Below, the variables with a hat are used for the modelling of the laminar zone. In order to measure its influence, similarity solutions are used for the stream function ${\it\psi}$ and the temperature perturbation $T-T_{e}$ , given a stratification profile proportional to some power of $z$ (see Worster & Leitch Reference Worster and Leitch1985). Defining ${\it\psi}$ such that $u_{x}=-\partial {\it\psi}/\partial z$ and $u_{z}=\partial {\it\psi}/\partial x$ , these similarity solutions are written in terms of non-dimensional functions $\widehat{F}(\hat{{\it\xi}})$ and $\widehat{G}(\hat{{\it\xi}})$ as

The constants ${\it\lambda}_{T}$ and $\hat{m}$ characterize the density profile and satisfy the relation

(For the Rayleigh number Worster & Leitch (Reference Worster and Leitch1985) use a characteristic temperature scale denoted ${\rm\Delta}T$ , which is obtained from the instant density profile. To keep the Rayleigh number fixed, here ${\rm\Delta}T$ is replaced by ${\it\lambda}_{T}(T_{w}-T_{1})$ , where the density variation is represented by the coefficient ${\it\lambda}_{T}$ .) The ambient is homogeneous when $\hat{m}=0$ , and stably stratified for $\hat{m}>0$ . The two functions $\widehat{F}$ and $\widehat{G}$ are based on the similarity variable

They satisfy the following two differential equations derived from the boundary-layer equations for steady convection along a plate:

which are subject to the boundary equations

Using the non-dimensional variables (4.8), one can deduce the integral variables $q$ , $m$ and $f$ for the laminar part ( ${\it\zeta}<{\it\zeta}_{crit}$ ):

where $\widehat{K_{Q}}$ , $\widehat{K_{M}}$ and $\widehat{K_{F}}$ are integrals calculated from the non-dimensionalized functions $\widehat{F}$ and $\widehat{G}$ as, respectively,

These values rely on the constant $\hat{m}$ which characterizes the stratification. Deducing the expression of the characteristic width and velocity, we obtain $b\propto {\it\zeta}^{(1-3\hat{m})/4}$ and $w\propto {\it\zeta}^{(1-3\hat{m})/2}$ . Both agree well with observations shown respectively in figures 9(b) and 9(c). In the case of an ambient stratification composed of a lower homogeneous layer and an upper stratified layer, a similarity solution in the upper layer is still assumed unlike the influence of the lower layer. The set of equations (4.22) is then used two times, but with different values for $\hat{m}$ and ${\it\lambda}_{T}$ : the values for $\hat{m}$ are zero in the homogeneous layer and positive above, and the values for ${\it\lambda}_{T}$ are noted ${\it\lambda}_{T}^{h}$ and ${\it\lambda}_{T}^{s}$ , respectively. The similarity solutions obtained in the upper layer have to be corrected, since the fluxes from the unstratified lower layer are different from those that the stratified solution predicts. Thus the similarity variables $\hat{{\it\xi}}$ , $\widehat{F}$ and $\widehat{G}$ are adjusted in the upper layer by applying the coefficients ${\it\lambda}_{{\it\xi}}$ , ${\it\lambda}_{F}$ and ${\it\lambda}_{G}$ , respectively, which are determined with the continuity conditions of the front. The continuity of the $x$ -scale is written from (4.17) as

giving

Note that the constant $C_{{\it\delta}}$ embeds all the parameters necessary to describe the entire density profile. By a similar reasoning, the continuity of the characteristic velocity $w$ , written as

gives

A third equality is derived from the continuity of the characteristic temperature inside the plume, which is proportional to the quantity ${\it\delta}-{\it\delta}_{e}$ . Taking into account the density jump in the ambient, equal to ${\it\delta}_{e}^{fr}-{\it\delta}_{e}^{0}$ , we write

and obtain

This development and the assumption of similarity for a partially stratified ambient have been successfully validated against experimental velocity fields (Caudwell Reference Caudwell2015). This theory also allows the expression of the local value of ${\it\alpha}$ in the laminar zone:

The overall value can then be computed from this expression and appears to compare well with the curve of figure 10(b) (not shown).

4.3 Temperature at the wall

In some cases the temperature $T_{w}$ is known from measurements. Since it was not measured independently in the present experiments, we determine it from the temperature $T_{c}$ in the side compartment and a global heat transfer coefficient $h_{g}$ . This coefficient accounts for the transfer of heat across the aluminium wall from the thermostatic fluid to the working fluid. Because the heat flux is conserved throughout the vertical section, one may write

providing an expression for the temperature at the wall as a function of $T_{c}$ , $T_{e}$ and the global heat transfer coefficient $h_{g}$ . The resulting function can reasonably be reduced to a simple linear relation, given a limited range of variation for $T_{e}$ (from $21\,^{\circ }$ C to  $41\,^{\circ }$ C). Details are given in appendix A. The averaged value of $h_{g}$ has empirically been found to be equal to $305~\text{W}~\text{m}^{-2}~\text{K}^{-1}$ . Since the thermal diffusivity of aluminium is 1000 times higher than that of water, we assume that ${\it\delta}_{w}$ is uniform within the plate, so that we obtain the relation

where $\langle {\it\delta}_{e}\rangle$ is the averaged value of the ambient buoyancy over the entire height. It is important to note that the variations of $\langle T_{e}\rangle$ are very slow compared with the dynamics of the flow, so that the variations of ${\it\delta}_{w}$ have no significance for the flow description.

4.4 Numerical procedure

Four models, summarized in table 1, have been run to investigate the influence of both the laminar–turbulent modelling and the variation of ${\it\alpha}$ in time (source code available at http://servforge.legi.grenoble-inp.fr/projects/soft-wpsm). A first model, Turb, solves the equations (4.9) for a constant entrainment coefficient, using the average value of the plot represented in figure 10(a). A second model, Hybrid, takes into account the laminar part based on the similarity solution and uses a constant ${\it\alpha}$ (mean value) for the turbulent zone. The third model, Turb  ${\it\alpha}$ , uses a variable ${\it\alpha}$ following the exponential fit shown in figure 10(a). The fourth model, Hybrid  ${\it\alpha}$ , includes both the laminar and the turbulent layer as in Hybrid, but has a variable entrainment coefficient for the turbulent layer based on the fit shown in figure 10(b).

To solve the system of equations (4.9) together with (4.12) we employ a numerical method similar to the one presented by Germeles (Reference Germeles1975). Under the condition that the velocities within the plume are much larger than in the ambient, the numerical model solves the plume equations (4.9) at each time step. The slow movement of the stratification in the ambient follows (4.12), i.e. each layer moves downward by a small amount. In the model, the plume equations are discretized along ${\it\zeta}$ and solved by a Runge–Kutta method of fourth and fifth order, assuming that the ${\it\delta}_{e}$ and $f$ profiles follow ‘staircase’ functions, i.e. they are constant on each step in space.

At each time step, ${\it\delta}_{w}$ is adjusted following (4.32). When the laminar zone is modelled, the value of ${\it\zeta}_{crit}$ is based on the instantaneous Grashof number. Once the stratification affects the laminar zone, the constants $\hat{m}$ , $\widehat{K_{Q}}$ , $\widehat{K_{M}}$ and $\widehat{K_{F}}$ are redefined at each iteration. Following (4.16), $\hat{m}$ is determined by the plot of $\ln (1+{\it\delta}_{e}/{\it\delta}_{w})$ against $\ln (Ra_{H}{\it\zeta}^{3})$ which has to be a straight line over the concerned zone for the similarity laws to be valid. It is verified that this is indeed the case.

The ambient is homogeneous at $t=0$ . For the fully turbulent model, the volume, momentum and buoyancy fluxes, $q$ , $m$ and $f$ , respectively, are assumed to be equal to zero at the bottom boundary of the box. When the laminar zone is modelled, the equations (4.22) for $0\leqslant {\it\zeta}\leqslant {\it\zeta}_{crit}$ are used to determine these quantities, and the calculation of the turbulent zone starts with these values at each iteration. Water properties, such as thermal expansion ${\it\beta}$ , kinematic viscosity ${\it\nu}$ , thermal diffusivity ${\it\kappa}$ and thermal conductivity $k$ are kept constant and evaluated at $30\,^{\circ }$ C.

Figure 11 presents profiles for the ambient temperature, plume volume flux $Q$ and plume buoyancy flux $F$ . Results are close to those of Cooper & Hunt (Reference Cooper and Hunt2010), except for the ambient temperature evolution which is reduced in our case as expected owing to the different boundary conditions. The volume and buoyancy fluxes are very similar, demonstrating that imposing a constant heat flux or a constant temperature is equivalent for a certain period of time, with the exception of the final temperature which is established within the cavity.

Figure 11. Comparison between the model with buoyancy flux ${\it\phi}$ imposed (solid lines) (Cooper & Hunt Reference Cooper and Hunt2010) and constant wall temperature imposed (Turb, dashed lines). Variables are (a) ambient temperature $T_{e}$ , (b) plume volume flux $Q$ and (c) plume buoyancy flux $F$ . Both started with the same buoyancy flux ${\it\phi}={A_{w}}^{1/3}{{\it\Delta}_{w}}^{4/3}$ , with ${\it\Delta}_{w}$ defined in accordance with the parameters of § 3.4.

5.1 The ambient stratification

All models predict the ambient stratification reasonably well. This confirms former experimental studies reported in the literature which successfully validate the MTT theory against the measured evolution of the stratification. Major discrepancies occur for the sharpness of the density front in models Turb and Hybrid for constant ${\it\alpha}$ . Small differences in the modelling also appear on all graphs near the top boundary due to the presence of a horizontal laminar layer in the experiment as shown in figure 12(b,d,f,h) (explained at the end of § 3.3).

As expected, the Hybrid model improves the agreement with the density profiles for small $z$ . In particular, figure 12(d,h) shows that the front attains the bottom of the cavity asymptotically, whereas it never does for the MTT modelling. This is an essential difference between the laminar and turbulent models (see Worster & Leitch Reference Worster and Leitch1985).

When including the variation of ${\it\alpha}$ with time, the difference between the Hybrid and the Turb models becomes more pronounced, especially at the front. A high value for ${\it\alpha}$ at $t=0$ indeed reduces the density step. The variation of ${\it\alpha}$ also improves the overall agreement between the experimental and theoretical density profiles.

5.2 The plume

The simultaneous information about the density and the velocity in a field allows for the measurement of volume flux $Q$ , momentum flux $M$ and buoyancy flux $F$ in the plume (see figure 13). These are not accessible from classical experiments. Generally, above $z=42$  cm the deviation of the plume results in large differences with the models for $Q$ and $M$ (first and second columns), especially at early times. In particular, the initial horizontal motion near the top, after the onset of the plume and the consequent entrainment, biases the comparison with the models (see figure 4 d) so that the best comparison with theoretical models is obtained for $t>100$  s. This artefact is absent for $F$ (third column) since the buoyancy flux is set to zero at the top boundary. The introduction of a zero vertical velocity is not possible in the one-dimensional model because of the conservation of momentum. Furthermore, we notice that the volume and momentum fluxes, shown in the first and second columns of figure 13, respectively, have the same order of magnitude as found in the experiments.

The hybrid model, Hybrid, introduces an important change in the slopes of the momentum and volume fluxes in particular (see figure 13(e,j,k) at $z\approx 24$  cm) corresponding to the laminar–turbulent transition. It shows much better agreement with the experiments for small $z$ and improves the modelling of the turbulent region as well.

Although the variation of ${\it\alpha}$ alone (model Turb  ${\it\alpha}$ ) is of benefit to the modelling of $Q$ (figure 13 g), it does not significantly change $M$ (figure 13 h) and makes $F$ even worse (figure 13 i). Hybrid  ${\it\alpha}$ models the volume flux $Q$ (figure 13 j) much better, and also shows a reasonable agreement for $M$ (figure 13 k). Therefore, the best results are obtained when both the laminar zone and the variable entrainment coefficient are included.

Strong discrepancies remain nevertheless for $M$ and $F$ in all cases. The former are most likely due to the shear stress along the wall, which is neglected in all cases. The poor agreement found for the buoyancy flux can be explained by the technical difficulty to evaluate it precisely in the experiment; because of the shape of the temperature profile within the boundary layer (see figure 9 a), local lack of resolution or inaccurate positioning of the $x$ -origin may result in significant underestimations of the buoyancy. In addition, all fluid parameters except density are kept constant with the temperature, even though the mean temperature evolves in time. In particular, $F$ shows a strong dependency on the Prandtl number in the laminar regime (see (4.22)).

In conclusion, the introduction of the laminar zone mainly benefits the modelling of the plume and the lower part of the eventually established stratification. On the other hand, the introduction of a variable ${\it\alpha}$ is important to properly model the overall density profile.