A modified integral model for negatively buoyant jets in a stationary ambient

Abstract

A simple modification is introduced into the integral model (IM) CorJet in an effort to predict better the characteristics of negatively buoyant jets (NBJ) discharged in a stationary ambient. Although this modification was developed for the CorJet model, it can be applied to every IM which employs the entrainment hypothesis. The detrainment of fluid from the main flow is taken into account by inserting a coefficient “p” into the conservation equations of volume, buoyancy and tracer mass flux. This coefficient expresses the ratio of the specific mass flux of the detrained fluid to the net specific mass flux entrained to the NBJ. The value of p is assumed constant along the jet trajectory and up to the maximum jet height, becoming zero thereafter. Results show that the modified CorJet model (MCM) predicts reasonably well experimental data from the literature and data from experiments performed in this work. The optimal value of p and therefore the detrained fluid from the main NBJ flow was found to decrease as the jet initial densimetric Froude number increases.

Appendices

Appendix 1: Entrainment coefficient in the IM CorJet

CorJet is based on the entrainment hypothesis for the closure of conservation equations. According to this hypothesis, the entrainment velocity $\hbox {u}_{\mathrm{e}^{*}}$ at the nominal jet width $\hbox {b}_{*}$ is equal to the local jet centerline velocity $\hbox {u}_{\mathrm{c}^*}$ multiplied by the entrainment coefficient $\upalpha$, i.e. $\hbox {u}_{\mathrm{e}^{*}}=\upalpha \hbox {u}_{\mathrm{c}^{*}}$ [12].

In CorJet, the entrainment coefficient $\upalpha$ is assumed to be variable taking the asymptotic values $\upalpha _\mathrm{j}$ and $\upalpha _\mathrm{p}$ in the regions where the flow is momentum (jet-like) and buoyancy (plume-like) dominated, respectively, and the values in between at the transition from jets to plumes. Equation 23 gives the expression of $\upalpha$,

$$\begin{aligned} \upalpha ={\upalpha }_\mathrm{j} +({\upalpha }_\mathrm{p}-{\upalpha }_\mathrm{j})\left( {{\hbox {F}_\mathrm{lp}^{2} }\bigg /{\left( {\frac{\hbox {F}_\mathrm{l}^{2} }{\sin \uptheta }} \right) }} \right) \end{aligned}$$

(23)

where $\upalpha _\mathrm{j}=0.055,\,\upalpha _\mathrm{p}=0.083$ and $\hbox {F}_\mathrm{l}$ and $\hbox {F}_\mathrm{lp}$ are the local densimetric Froude number and the densimetric Froude number for the pure plume, respectively. $\hbox {F}_\mathrm{l}$ and $\hbox {F}_\mathrm{lp}$ are given by Eqs. 24 and 25, where $\hbox {g}_\mathrm{c}$’ is the apparent gravitational acceleration on the jet axis given by Eq. 26.

$$\begin{aligned} \hbox {F}_\mathrm{l}&= \frac{\hbox {u}_\mathrm{c}}{\sqrt{\hbox {g}_\mathrm{c}^{\prime }\hbox {b}}}\end{aligned}$$

(24)

$$\begin{aligned} \hbox {F}_\mathrm{lp}&= \sqrt{\frac{5\uplambda ^{2}}{4\upalpha _\mathrm{p}}}=4.67\end{aligned}$$

(25)

$$\begin{aligned} \hbox {g}_\mathrm{c}^{\prime }&= \hbox {g}\left( {\frac{\uprho _\mathrm{a} -\uprho _\mathrm{c}}{\uprho _\mathrm{a}}}\right) \end{aligned}$$

(26)

As observed in Eq. 23, the second term on the R.H.S. is negative as long as the term $\hbox {F}_{\mathrm{l}}^{2}/\hbox {sin}\uptheta$ is negative. Since $\hbox {F}_{\mathrm{l}}^{2}$ is always less than zero for NBJ $(\uprho _\mathrm{a}<\uprho _\mathrm{o})$, the sinus of the local jet inclination $(\uptheta )$ determines the sign of $\hbox {F}_{l}^{2}/\hbox {sin}\uptheta$. This means that the entrainment coefficient $\upalpha$ takes values less than $\upalpha _\mathrm{j}$ along the ascending flow region of NBJ and greater than $\upalpha _\mathrm{j}$ at the descending flow region. At the maximum jet rise height, $\hbox {sin}\uptheta$ becomes equal to zero and therefore $\upalpha =\upalpha _\mathrm{j}$. It is noted that as the plume is the final stage of flow for buoyant jets, the maximum value that $\upalpha$ can take is $\upalpha _\mathrm{p}$. $\upalpha$ is also affected by $\uptheta$. As $\uptheta$ takes greater values, as is the case for large discharge angles, $\upalpha$ takes smaller values in the ascending flow region of NBJ.

The complete behavior of $\upalpha$ implemented in CorJet for NBJ discharged into stagnant ambient, is shown in Fig. 10. It is seen that in the ascending region of flow, the reduction of $\upalpha$ is described by Eq. 23. The minimum value that $\upalpha$ takes is $\upalpha =\upalpha _\mathrm{j}$-($\upalpha _\mathrm{p}-\upalpha _\mathrm{j})\cdot \hbox {sin}\uptheta$, which is dependent only on $\uptheta$ and therefore on ${\uptheta }_\mathrm{o}$. Beyond this point, $\upalpha$ is assumed to increase linearly between the ascending and descending regions of flow, reaching, eventually, the value $\upalpha _\mathrm{p}$. It is noted that Jirka [8] assumed this linearity in CorJet due to lack of precise experimental data along the jet path.

Fig. 10

Behavior of the entrainment coefficient $\upalpha$ for NBJ discharged into a stationary ambient (as adapted from [8])

Appendix 2: Expressions for local jet parameters in the ZEF

$$\begin{aligned} \hbox {u}_{\mathrm{c}^{*}}&= \frac{2\hbox {M}_{*}}{\hbox {Q}_{*}}\end{aligned}$$

(27)

$$\begin{aligned} \hbox {b}_{*}&= \frac{\hbox {Q}_*}{\sqrt{8\hbox {M}_*}}\end{aligned}$$

(28)

$$\begin{aligned} \hbox {g}_{\mathrm{c}^{*}}^{\prime }&= \frac{\hbox {J}_*}{\hbox {Q}_{\mathrm{scalar}^*}}\end{aligned}$$

(29)

$$\begin{aligned} \hbox {C}_{\mathrm{c}^*}&= \frac{\hbox {Q}_{\mathrm{c}^{*}}}{\hbox {Q}_{\mathrm{scalar}^*} }\end{aligned}$$

(30)

$$\begin{aligned} \hbox {Q}_{\mathrm{scalar}^{*}}&= \hbox {Q}_*\left( {\frac{\uplambda ^{2}}{\uplambda ^{2}+1}}\right) \end{aligned}$$

(31)

Appendix 3: Expressions for jet parameters at the end of ZOFE

$$\begin{aligned} \hbox {L}_{\mathrm{e}^{*}}&= 5\left( {1-\hbox {e}^{-2\mathrm{F}_\mathrm{o}/\mathrm{F}_\mathrm{lp}}}\right) \end{aligned}$$

(32)

$$\begin{aligned} \hbox {x}_{\mathrm{e}^{*}}&= \hbox {L}_{\mathrm{e}^{*}}\cos \uptheta ,\quad \hbox {z}_{\mathrm{e}^{*}} =\hbox {L}_{\mathrm{e}^{*}}\sin \uptheta \end{aligned}$$

(33)

$$\begin{aligned} \hbox {Q}_{\mathrm{e}^{*}}&= \sqrt{2}\end{aligned}$$

(34)

$$\begin{aligned} \hbox {M}_{\mathrm{e}^{*}}&= \hbox {M}_{\mathrm{o}^{*}}\end{aligned}$$

(35)

$$\begin{aligned} \hbox {J}_{\mathrm{e}^{*}}&= \hbox {J}_{\mathrm{o}^{*}}\end{aligned}$$

(36)

$$\begin{aligned} \hbox {Q}_{\mathrm{ce}^{*}}&= \hbox {Q}_{\mathrm{co}^{*}} \end{aligned}$$

(37)

where $\hbox {L}_{\mathrm{e}^{*}}$ is the dimensionless modified ZOFE length.