On the modelling of steady turbulent fountains

Abstract

The paper deals with steady-state turbulent fountains in a homogeneous surrounding. A set of mono-dimensional conservation equations is first derived from the Navier–Stokes equations. In contrast with equations used for plumes or rising fountains, these equations reveal additional terms in order to account for the effect of the (annular) down-flow on the fountain up-flow. Large-eddy simulations are then performed and used to determine, from radial profiles, the values of the constants (associated with these additional terms) to be fitted in the model. With these constants, analytical solutions of the model for steady-state fountain are proposed and compared with previous experiments.

Appendices

Appendix 1

This appendix details the formulation of ‘the-top hat’ confined model used in this study.

Fig. 11

The different steps of the modelling leading to the corrected plume model from the Navier--Stokes equations

From the Navier–Stokes Eq. (2), (3) and (4) and after some algebra, Carazzo et al. [12] derived this new set of equations:

$$ \frac{\partial r u w^2}{\partial r} + \frac{\partial r w^3}{\partial z}= -2 r g w\eta - 2 w \frac{\partial r\overline{ w' u'}}{\partial r}\,,$$

(30)

$$ \frac{\partial r w u \eta }{\partial r} + \frac{\partial r w^2 \eta }{\partial z}= - r g \eta ^2 - \eta \frac{\partial r\overline{ w' u'}}{\partial r}\,,$$

(31)

$$ \frac{\partial r u w^3}{\partial r} + \frac{\partial r w^4}{\partial z}= -3 r g w^2 \eta - 3 w^2 \frac{\partial r\overline{ w' u'}}{\partial r}\,.$$

(32)

The main interest of these equations is that the vertical velocity w is involved in all the r-derivative terms. Then, these terms will vanish when integrated over the up-flow region since $w(z,r=b)=0$. To integrate these three equations, they also introduced the so-called shape functions of velocity, of density deficit and of shear stress

$$w=w_m(z)f(r,z);\,\eta = \eta _m(z)h(r,z)\quad \,\overline{u' w'} = - \frac{1}{2} w_m(z) ^2 j(r,z)\,.$$

Let us now introduce the following integrals which slightly differ from those given by Carazzo et al. [12]

$$ I_1= \int _0^1 r' f(r',z)h(r',z) \,dr'\, ,$$

(33)

$$ I_2= \int _0^1 f(r',z)\frac{\partial }{\partial r'} \left( r' j(r',z)\right) \, dr'\, ,$$

(34)

$$ I_3= \int _0^1 r' h(r',z)^2\, dr'\,,$$

(35)

$$ I_4= \int _0^1 h(r',z)\frac{\partial }{\partial r'} \left( r' j(r',z)\right) \,dr' ,$$

(36)

$$ I_5= \int _0^1 r' f(r',z)^2 h(r',z)\, dr',$$

(37)

$$ I_6= \int _0^1 f(r',z)^2\frac{\partial }{\partial r'} \left( r' j(r',z)\right) \, dr'\,,$$

(38)

$$ I_7= \int _0^1 r' f(r',z)^3 \,dr'\,,$$

(39)

$$ I_8= \int _0^1 r' f(r',z)^4\,dr',$$

(40)

$$ I_9= \int _0^1 r' f(r',z)^2 h(r',z)\, dr',$$

(41)

where $r' = r / b$ is the dimensionless radial coordinate.

By integrating Eq. (30), (31) and (32) with respect to $r'$ from $0$ to $1$, we obtain the following set of equations:

$$ \frac{d(b^2 w_m^3)}{dz}= -2gb^2w_m\eta _m \frac{I_1}{I_7} + b w_m^3 \frac{I_2}{I_7} \,,$$

(42)

$$ \frac{d(b^2 w_m^4)}{dz}= -3gb^2w_m^2\eta _m \frac{I_5}{I_8} + \frac{3}{2} b w_m^4 \frac{I_6}{I_8}\,,$$

(43)

$$\frac{d(b^2 w_m^2 \eta _m)}{dz}= -gb^2\eta _m^2 \frac{I_3}{I_9}+\frac{1}{2} b \eta _m w_m^2\frac{I_4}{I_9}\,.$$

(44)

These equations can be rewritten in a form similar to those given by Morton et al. [28], providing (5), (6) and (7), in which the parameters of the theoretical model are

$$\begin{aligned} c_1&= \frac{3}{2}\left( \frac{I_2}{I_7}-\frac{I_6}{I_8} \right) \, ,\quad c_2 =3\left( \frac{I_1}{I_7}-\frac{I_5}{I_8}\right) \,,\\ c_3&= \frac{4I_1}{I_7}-\frac{3I_5}{I_8},\,\quad c_4 =\frac{3I_6}{2I_8}-\frac{2I_2}{I_7}\,,\\ c_5&= \frac{2I_1}{I_7}-\frac{3I_5}{I_8}+\frac{I_3}{I_9},\,\quad c_6 = \frac{I_2}{I_7}-\frac{3I_6}{2I_8}+\frac{I_4}{2 I_9}\,. \end{aligned}$$

The main steps of the calculations presented in this appendix are summarized in Fig. 11 in the form of a diagram.

Appendix 2

In order to obtain analytical expressions for the fountain height and in the same way as done by [11], it is convenient to treat separately the two regimes corresponding to weak ($\varGamma _i \gg 1$) or forced ($\varGamma _i \ll 1$) fountains.

In the case of weak fountains, we introduce the following relations

$$ \varLambda _i = \frac{\lambda _2 \varGamma _i ^{\omega _1}}{b_i \left( \varGamma _i+1\right) ^{\omega _2}}\quad \hbox { and } \quad \xi = z \varLambda _i \, $$

in (16), then we are led to

$$ \frac{d \varGamma }{d\xi } = \varGamma ^{1-\omega _1}\left( \varGamma +1\right) ^{\omega _2+1}\,.$$

(45)

According to the fact that in the vicinity of the fountain height $H_{ss}$, $\varGamma$ tends towards $\infty$ as $\xi \rightarrow H_{ss}\varLambda _i$, the fountain function can be sought in this region in the form

$$ \varGamma _{\mathrm{approx}}=\frac{1+\omega _2}{\omega _1-\omega _2-2}\,+\frac{1}{\epsilon (\xi )}\,, $$

(46)

where $\epsilon (\xi )$ is a function which tends to zero when $\xi \rightarrow H_{ss} \varLambda _i$.

By injecting (46) in (45) and by expanding this equation with respect to $\epsilon$, we obtain at leading order

$$ \varGamma _{\mathrm{approx}} \simeq \frac{1+\omega _2}{\omega _1-\omega _2-2}+\Big (\left( \omega _2-\omega _1+1\right) \left( H_{ss}\varLambda _i-\xi \right) \Big )^{-1/\left( \omega _2-\omega _1+1\right) }\,,$$

(47)

where $H_{ss}\varLambda _i$ is the constant of integration. In order to determine this constant, we use the fact that $\varGamma _i \gg 1$ which allows us to consider that the approximated solution remains accurate even at the vicinity of the source ($\varGamma _{\mathrm{approx}}(\xi =0)\simeq \varGamma _i$). This last step leads us to the following relation :

$$ \frac{H_{ss}}{b_i}=\frac{\left( \varGamma _i+1\right) ^{\omega _2}}{\lambda _2 \varGamma _i ^{\omega _1}} \frac{\left( \varGamma _i -\displaystyle {\frac{1+\omega _2}{\omega _1-\omega _2-2}}\right) ^{-\left( \omega _2-\omega _1+1\right) }}{\left( \omega _2-\omega _1+1\right) }\,. $$

(48)

In the case of forced fountains, Eq. (17) can be split into two parts as follows

$$H_{ss}= \frac{1}{\varLambda _i}\left( \int _{0}^{\infty }\varGamma ^{\omega _1-1} \left( \varGamma +1\right) ^{-\omega _2-1}\,d\varGamma -\int _{0}^{\varGamma _i}\varGamma ^{\omega _1-1} \left( \varGamma +1\right) ^{-\omega _2-1} \,d\varGamma \right) $$

(49)

$$\quad \approx \frac{1}{\varLambda _i}\left( \int _{0}^{\infty }\varGamma ^{\omega _1-1} \left( \varGamma +1\right) ^{-\omega _2-1}\,d\varGamma -\int _{0}^{\varGamma _i}\varGamma ^{\omega _1-1} \,d\varGamma \right) \,,$$

(50)

which finally leads us to the fountain height given by (24).