2.1. Flow settings and theoretical scaling

We consider a planar, temporal forced plume with an initial velocity and temperature fields given by $u_x=u_y=0$ and by

(2.1) $$\begin{gather} u_z(x,y,z,t=0)= U_0\, f(x/H_0), \end{gather}$$

(2.2) $$\begin{gather}\theta(x,y,z,t=0)= T_0\, f(x/H_0), \end{gather}$$

where $x$ is the cross-stream direction, $y$ is the spanwise direction, $z$ is the streamwise direction, $H_0$ is the initial width of the forced plume, $U_0$ is the initial velocity of the forced plume, $T_0$ is the difference between the temperature inside and outside the plume (which is assumed to be positive), and $f(\xi )$ (with $\xi =x/H_0$) is a symmetric, dimensionless function which has a limited support around $\xi =0$ such that $\int f(\xi ) \,{\rm d}\xi = 1$. By considering an incompressible flow and the Boussinesq approximation, the evolution of the velocity and the temperature fields is governed by

(2.3)$$\begin{gather} \frac{\partial {\boldsymbol{u}}}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla}\boldsymbol{u} +\frac{1}{\rho_0} \boldsymbol{\nabla} p - \nu \nabla^2 \boldsymbol{u} + \beta \boldsymbol{g} \theta =0, \end{gather}$$

(2.4)$$\begin{gather}\frac{\partial \theta }{\partial t} + \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} \theta - \kappa \nabla^2 \theta = 0, \end{gather}$$

where $\nu$ is the kinematic viscosity, $\kappa$ the thermal diffusivity, $\beta$ the thermal expansion coefficient and $\boldsymbol {g}=(0,0,-g)$. The dimensionless parameters which define the initial state are the Reynolds number $Re_0=U_0 H_0/\nu$, the Froude number $Fr_0=U_0/\sqrt {A_0 g H_0}$ and the Prandtl number $Pr = \nu / \kappa$, where $A_0=\beta T_0/2$ is the Atwood number. In the present work, all the simulations are done at $Pr =1$.

The system of equations (2.3) and (2.4) is solved in a cubic box of size $L$ by using periodic boundary conditions and the initial conditions (2.1) and (2.2). As a consequence, the flow is statistically homogeneous in the spanwise $(y)$ and streamwise $(z)$ directions and is statistically symmetric about the $x=0$ plane, and the average operator (hereafter denoted as $\overline {({\cdot })}$) can be composed by a spatial average in the $y$–$z$ plane and by an ensemble average over independent samples. In accordance with such statistical symmetries of the flow, the equations for the mean flow solution read

(2.5)$$\begin{gather} \frac{\partial \overline{u_z} }{\partial t} = \nu \frac{\partial^2 \overline{u_z} }{\partial x^2} -\frac{\partial \overline{u_z u_x} } {\partial x} +\beta g \bar{\theta} , \end{gather}$$

(2.6)$$\begin{gather}\frac{\partial \bar{\theta} }{\partial t}= \kappa \frac{\partial^2 \bar{\theta} }{\partial x^2} -\frac{\partial \overline{\theta u_x} }{\partial x} . \end{gather}$$

Integration over $x$ gives the equations for the total thermal content $C(t) \equiv \int _L \bar {\theta }(x,t)\, {{\rm d}\kern0.7pt x}\equiv C_0$ and the volume flux $Q(t) \equiv \int _L \overline {u_z}(x,t)\, {{\rm d}\kern0.7pt x}$, which obey

(2.7)$$\begin{gather} \frac{{\rm d} Q }{{\rm d} t} = \beta g C, \end{gather}$$

(2.8)$$\begin{gather}\frac{{\rm d} C }{{\rm d} t} = 0, \end{gather}$$

with the obvious solutions

(2.9)$$\begin{gather} Q(t)=Q_0 + \beta g C_0 t, \end{gather}$$

(2.10)$$\begin{gather}C(t) = C_0 . \end{gather}$$

Given the expressions (2.1) and (2.2) for the initial profiles, we have an explicit expression for the integration constants $C_0=T_0 H_0$ and $Q_0=U_0 H_0$. Equation (2.9) indicates the presence of two different regimes in time. Initially, the constant term $Q_0$ dominates the flow rate: this is the shear (or jet) regime. At later times, buoyancy increases the total momentum together with the flow rate $Q$: this is the buoyancy (or plume) regime. The transition between the two regimes is expected to occur when the two terms in (2.9) are of the same order, i.e. at the cross-over time

(2.11)\begin{equation} t_g = \frac{U_0}{\beta g T_0}= \frac{Fr_0^2}{2} t_0, \end{equation}

where $t_0=H_0/U_0$ is the shear time. The above relation shows that the transition time to the buoyancy regime increases quadratically with the Froude number. We remark that an equivalent cross-over time has been introduced in the case of unstably stratified shear layers by Brizzolara et al. (Reference Brizzolara, Mollicone, van Reeuwijk, Mazzino and Holzner2021).

In analogy with the initial condition, we rewrite the total thermal content and volume flux as

(2.12)$$\begin{gather} Q(t)=U(t) H_U(t), \end{gather}$$

(2.13)$$\begin{gather}C_0=T(t) H_{T}(t). \end{gather}$$

The characteristic velocity $U(t)$ and temperature $T(t)$ are defined as

(2.14)$$\begin{gather} U(t) = \frac{1}{H_U(t)} \int_L \overline{u_z}(x,t)\, {{\rm d}\kern0.7pt x}, \end{gather}$$

(2.15)$$\begin{gather}T(t) = \frac{1}{H_T(t)} \int_L \bar{\theta}(x,t) \,{{\rm d}\kern0.7pt x}. \end{gather}$$

The length scales $H_{U}(t)$ and $H_{T}(t)$ represent the width at time $t$ of the velocity and temperature profiles, respectively. Their precise definition is arbitrary, provided that is consistent with the initial value $H_{U}(t=0)=H_{T}(t=0)=H_0$. In § 3.1, we will adopt a definition based on the full width at half maximum of the profiles. By inserting (2.12) into (2.9), we obtain

(2.16)\begin{equation} U(t)\, H_U(t) = U_0 H_0 + 2 g A_0 H_0 t, \end{equation}

which relates the time evolution of $H_U(t)$ and $U(t)$. Introducing the time-dependent Reynolds number $Re(t) = U(t)H_U(t)/\nu$, the exact relation (2.16) can be rewritten as

(2.17)\begin{equation} Re(t)=Re_0 \left (1 + \frac{t}{t_g} \right ). \end{equation}

To derive an explicit prediction for the growth of $H_U(t)$, we further assume the following dimensional relation:

(2.18)\begin{equation} \frac{{\rm d}H_U(t)}{{\rm d}t}=\alpha U(t), \end{equation}

where $\alpha$ represents the entrainment coefficient of the process. The physical motivation for this simple dimensional relation is that the growth of the width of the plume is due to the entrainment process, which is driven by the turbulent flow inside the plume. The hypothesis of self-similarity of the flow implies that the typical intensity of the turbulent velocity fluctuations is proportional to the mean velocity $U(t)$. It is worth noticing that, in principle, the entrainment coefficients in the momentum-dominated and buoyancy-dominated regimes could be different. As an example, two distinct values of the entrainment coefficients have been observed in unstable stratified shear layers, depending on the regime (Brizzolara et al. Reference Brizzolara, Mollicone, van Reeuwijk and Holzner2023). Nonetheless, the assumption of a single value for the entrainment coefficient in our model has the advantage to allow the derivation of explicit predictions for the time evolution of $H(t)$ and $U(t)$.

Combining the relations (2.16) and (2.18), we obtain an explicit expression for the growth of the mixing layer,

(2.19)\begin{equation} H_U(t) = H_0 \sqrt{1 + 2 \alpha \left (\frac{t}{t_0} \right ) + \frac{2\alpha}{Fr_0^2} \left (\frac{t}{t_0}\right )^2}, \end{equation}

which can be rearranged to highlight the dependence on the cross-over time,

(2.20)\begin{equation} H_U(t) = H_0 \sqrt{1 + \alpha \, Fr_0^2 \left (\frac{t}{t_g}\right) + \frac{\alpha}{2}\, Fr_0^2 \left (\frac{t}{t_g}\right )^2}. \end{equation}

Inserting (2.20) in the exact relation (2.16), we obtain a prediction for $U(t)$:

(2.21)\begin{equation} U(t)=U_0 \frac{H_0}{H_U(t)}\left (1 + \frac{t}{t_g} \right ). \end{equation}

Moreover, inserting (2.13) into (2.10), we get $T(t) = T_0 H_0 /H_T(t)$. Assuming the the width of the temperature profile $H_T(t)$ is proportional to $H_U(t)$, so that the ratio $\gamma = H_U(t)/H_T(t)$ is constant, we can exploit the model (2.20) to obtain scaling predictions for $H_T(t)$ and $T(t)$.

From (2.11), we get that, if $Fr_0 \gg 1$, the cross-over time $t_g$ is much longer than the shear time $t_0$. In this case, from the expressions (2.19) and (2.20), it is possible to identify two different scaling regimes that follow one another over time. In the range of times $t_0 \ll t \ll t_g$, the constant term $1$ and the quadratic term $(\alpha /2) Fr_0^2 (t/t_g)^2$ in (2.20) can be neglected, and the evolution is dominated by the initial momentum. This leads to the following temporal scaling laws

(2.22)\begin{equation} \left.\begin{gathered} H_U(t) = \gamma H_T(t) \simeq \sqrt{2\alpha} H_0 \left({t/t_0} \right)^{1/2} \\ U(t) \simeq \frac{U_0}{\sqrt{2\alpha}} \left({t/t_0} \right)^{{-}1/2} \\ T(t) \simeq \gamma \frac{T_0}{\sqrt{2\alpha}} \left({t/t_0} \right)^{{-}1/2} \end{gathered}\right\}. \end{equation}

The scaling law for $T(t)$ is a consequence of the constancy of $C(t)=C_0$.

These scaling laws are clearly transient, because the quadratic term $(\alpha /2) Fr_0^2 (t/t_g)^2$ in (2.20) eventually becomes dominant for $t \gg t_g$. Therefore, at long times, we expect that the system recovers the plume regime

(2.23)\begin{equation} \left.\begin{gathered} H_U(t) = \gamma H_T(t) \simeq \sqrt{\frac{\alpha}{2}} H_g \left({t/t_g} \right) \\ U(t) \simeq \sqrt{\frac{2}{\alpha}} U_g \\ T(t) \simeq \gamma T_g \sqrt{\frac{2}{\alpha}} \left({t/t_g} \right)^{{-}1} \end{gathered}\right\}, \end{equation}

where $H_g=H_0(U_0/U_g)=H_0 Fr_0$, $U_g=(A_0 g H_0)^{1/2}$ and $T_g=T_0(U_g/U_0) = T_0/Fr_0$, also known as free-fall quantities. We will refer to (2.22) and (2.23) as the jet and the plume temporal scaling, respectively.

We note that the model (2.20) allows to define exactly the time of the transition between these regimes by intersecting the asymptotic laws for $H_U(t)$. The transition is expected to occur at $t=2t_g$, the mixing thickness and the velocity at the transition are

(2.24)\begin{equation} \left. \begin{gathered} H_U(2t_g) {}= \sqrt{1+ 4\alpha} H_0, \\ U(2t_g) {}= (3/ \sqrt{1+ 4\alpha}) U_0 . \end{gathered}\right\} \end{equation}

In addition to the prediction for the global quantities of the system $H_{U,T}$, $U$ and $T$, the assumption of self-similarity of the turbulent flow allows to derive also dimensional scaling predictions for other statistical quantities, such as the kinetic energy dissipation rate $\varepsilon _\nu = \nu \langle (\partial _j u_i)^2 \rangle$ and the r.m.s. turbulent velocity fluctuations $u' = \sqrt {2 k/3 }$, where $k = \langle \frac {1}{2} |\boldsymbol {u}-\bar {\boldsymbol {u}}|^2 \rangle$ is the turbulent kinetic energy (here and in the following, $\langle {\cdot } \rangle$ indicates the spatial average in the region $|x| \leqslant H_U(t)$). From dimensional analysis, one has $\varepsilon _\nu (t) \simeq U(t)^3/H_U(t)$, while the scaling of velocity fluctuations $u'(t)$ is assumed to be proportional to that of $U(t)$. Combining the scaling for $\varepsilon _\nu$ and $u'$, we can also derive a prediction for the Taylor-microscale Reynolds number $Re_\lambda = u' \lambda /\nu = u'^2\sqrt {15/ \nu \varepsilon _\nu }$, where $\lambda = \sqrt {15\nu /\varepsilon _\nu } u'$ is the Taylor microscale. In the momentum-dominated regime $t_0 \ll t \ll t_g$, we obtain

(2.25)\begin{equation} \left.\begin{gathered} \varepsilon_\nu(t) \simeq \frac{U_0^3}{H_0} \left({t/t_0} \right)^{{-}2} \\ Re_\lambda \simeq \sqrt{Re_0} \end{gathered}\right\} \end{equation}

and in the buoyancy-dominated regime $t \gg t_g$, we get

(2.26)\begin{equation} \left.\begin{gathered} \varepsilon_\nu(t) \simeq \frac{U_0^3}{H_0 Fr_0^4} \left({t/t_g} \right)^{{-}1} = \frac{U_g^2}{t}\\ Re_\lambda \simeq \sqrt{Re_0} \left({t/t_g} \right)^{1/2} \end{gathered}\right\}. \end{equation}

We remark that, formally, the pure jet scaling laws (2.22) can be observed only in the case of non-buoyant temporal jet ($Fr_0=\infty$). Therefore, if the initial Froude number is $Fr_0 \leqslant 1$, the jet scaling cannot be observed and the scaling (2.23) holds at all times. Moreover, at finite $Fr_0$, we expect that the jet term $\alpha Fr_0^2(t/t_g)$ in (2.20) is contaminated by both the constant term $1$ at short times $t \leqslant t_0$ and by the buoyancy term $(\alpha /2)Fr_0^2(t/t_g)^2$ for $t \geqslant t_g$, thus preventing the observation of clear jet scaling regime (2.22), unless there is a large scale separation between $t_g$ and $t_0$, i.e. $Fr_0 \gg 1$.

2.2. Direct numerical simulations

Direct numerical simulations of (2.3) and (2.4) have been carried out in a cubic periodic domain of size $L^3=( 2{\rm \pi} )^3$ with $N^3=512^3$ equally spaced gridpoints, by means of a fully parallel standard pseudo-spectral code with a 2/3 dealiasing rule. Time integration is performed using a second-order Runge–Kutta scheme, with implicit integration of linear dissipative terms. The initial conditions are given by (2.1) and (2.2) with the initial width of the forced plume fixed at $H_0=L/32$ and

(2.27)\begin{equation} f(\xi) = \frac{1 }{2} \left[ 1 + \tanh\left( \frac{1 - 2 |\xi| }{2 \sigma} \right) \right], \end{equation}

where $\xi = x/H_0$ and $\sigma =5/56$. A uniform random noise with relative amplitude $0.04$ is superimposed to the velocity and temperature fields into the inner region, to trigger the initial instability.

Six different sets of simulations have been carried out, each one with 10 different realizations of the initial random perturbation to compute statistics on appropriate sample sizes. Each set is characterized by different values of $U_0$, corresponding to different $Re_0$ and $Fr_0$, while $T_0$, $H_0$ and $\nu = \kappa$ are kept fixed. An additional set of simulations with $A_0 g=0$, corresponding to the case without buoyancy (i.e. a temporal jet with the temperature as a passive scalar) has been carried out for comparison. The complete set of parameters is listed in table 1. The characteristic scales $H_U$ and $H_T$ are computed as the full width at half maximum lengths, i.e. $\overline {u_z}(H_U,t) = \overline {u_z}(0,t)/2$ and $\bar {\theta }(H_T,t) = \bar {\theta }(0,t)/2$. This definition is consistent with the initial values $H_U(t=0)=H_T(t=0)=H_0$, and it has the advantage that the values of $H_U$ and $H_T$ remain constant (and equal to $H_0$) during the initial diffusive phase of the evolution. All the characteristic quantities reported and discussed are averaged over the 10 realizations of each set.

Table 1.

Parameters of the different sets of simulations, ordered with increasing $Fr_0$. In addition to the non-dimensional parameters $Fr_0 = U_0/U_g$ and $Re_0 = U_0 H_0/\nu$, we report the numerical values (in arbitrary units) of the dimensional parameters of the simulations: $A_0g$, $U_0$, $U_g = \sqrt {A_0 g H_0}$, $t_0 = H_0/U_0$, $t_g=U_0/A_0 g$. The values of the other parameters are common to all simulations: $N= 512$, $L=2 {\rm \pi}$, $H_0=L/32$, $T_0 = 1$, $\nu =\kappa = 2.5 \times 10^{-4}$.

Before addressing the main results of the present work, let us introduce the main flow features by considering the flow case D. The instantaneous pattern taken by the flow is shown in figure 1 through a vertical section of the instantaneous dissipation and temperature field. A strongly convoluted shape of the plume boundaries is observed that leads to entrainment of laminar flow regions up to the very core of the jet. From a statistical point of view, the flow is characterized by profiles of vertical velocity $\overline {u_z}(x,t)$, temperature $\bar {\theta }(x,t)$ and velocity fluctuations $\overline {u'}(x,t) = (\frac {1}{3}\overline {|\boldsymbol {u}-\bar {\boldsymbol {u}}|^2})^{1/2}$ that are shown in figure 2. A well-known bell shape is attained by the mean velocity and temperature field with turbulent fluctuations that extend significantly beyond the edge of the plume defined by the mean profile themselves.

Figure 1.

Vertical sections in the $x$–$z$ plane of ($a$) the dissipation field $\varepsilon _{\nu }$ and of ($b$) the temperature field from simulation D at time $t=25$.

Figure 2.

Profiles of mean velocity $\overline {u_z}$ (dashed line), temperature $\bar {\theta }$ (dotted line) and velocity fluctuation $\overline {u'}$ (dash-dotted line) for the run D at time $t=20$. Data are normalized using their centreline value. The black dotted line represents the extension of $H_U$.

We also note that the width of the temperature profile is slightly larger than that of the vertical velocity. This phenomenon is in accordance with the typical value of the turbulent Prandtl number, which is below 1 for jets and plumes (Craske, Salizzoni & van Reeuwijk Reference Craske, Salizzoni and van Reeuwijk2017).