Fluid inflow from a source on the base of a channel

Abstract

A spectral method is detailed to simulate the viscous Boussinesq flow of a plume of heavy fluid emanating from a line source on the bottom of a channel. A small initially semi-circular “bubble” grows for small time, rising up to some height before it starts to flow outwards horizontally. We discuss the results for different values of Reynolds number, flow rate and density differential, considering the viscous and inertial effects.

Appendix: The Boussinesq solution equations

This derivation is described in full by [3], but for completeness, we include it here. The vorticity equation (8) in view of (18) and (19) becomes

$$\begin{aligned} \begin{aligned}&\sum _{m=1}^{M}\sum _{n=1}^{N} \bigtriangleup _{mn}^{2}A_{mn}'(t) \sin \left( \frac{m \pi x}{L}\right) \sin \left( \frac{n \pi y}{h}\right) \\&\quad =\left[ u \frac{\partial \omega }{\partial x} + v \frac{\partial \omega }{\partial y}- \frac{\partial \rho }{\partial x} \right] -\frac{1}{Re} \sum _{m=1}^{M} \sum _{n=1}^{N}\bigtriangleup _{mn}^{4} A_{mn}(t) \sin \left( \frac{m \pi x}{L}\right) \sin \left( \frac{n \pi y}{h}\right) . \end{aligned} \end{aligned}$$

(28)

The constant $\bigtriangleup _{mn}$ is defined in Eq. (18). Multiplying by $\sin \left( \frac{k \pi x}{L}\right) \sin \left( \frac{l \pi y}{h}\right)$ and integrating over the two-dimensional domain $-L \le x \le L$, $0 \le y \le h$, the orthogonality relations then yield

$$\begin{aligned} \begin{aligned} A_{kl}'(t)&=\frac{1}{Re} \Delta _{kl}^{2}A_{kl}(t)-\frac{2}{\Delta _{kl}^{2}Lh}\int _0^{h} \int _{-L}^{L}\left[ u \frac{\partial \omega }{\partial x} + v \frac{\partial \omega }{\partial y}+ \frac{\partial \rho }{\partial x} \right] \sin \left( \frac{k \pi x}{L}\right) \sin \left( \frac{l \pi y}{h}\right) \hbox {d}x\,\hbox {d}y, \end{aligned} \end{aligned}$$

(29)

for $k={1,...,M}$ and $l={1,...,N}$. The second system of differential equations is obtained by considering the density perturbation function $\bar{\rho }$. However, we observe that singular behaviour near the source requires the extra factor G(x, y) or $G^*(x,y)$ in the representation (19) for $\bar{\rho }$, and as a result, the orthogonality of the trigonometric functions no longer produces a decoupled system. Substituting our expression for $\bar{\rho }$ into equation (6) gives

$$\begin{aligned} G(x,y)\left\{ B_{00}'(t)+\sum _{n=1}^{N}B_{0n}'(t) \cos \left( \frac{n \pi y}{h}\right) +\sum _{m=1}^{M}\sum _{n=1}^{N}B_{mn}'(t) \sin \left( \frac{m \pi x}{L}\right) \sin \left( \frac{n \pi y}{h}\right) \right\} +\left( u \frac{\partial {\bar{\rho }}}{\partial x}+v \frac{\partial \bar{\rho }}{\partial y} \right) =0. \nonumber \\ \end{aligned}$$

(30)

In the absence of the G(x, y) term, we could isolate the $B_{00}'$ coefficient simply by integration. However, this is not the case, but we proceed with this technique regardless to give the first of the coupled differential equations to be

$$\begin{aligned} \begin{aligned} L_{00} B_{00}'(t)+\sum _{n=1}^{N}L_{0n} B_{0n}'(t)+\sum _{m=1}^{M}\sum _{n=1}^{N}L_{mn} B_{mn}'(t)- \int _{0}^{h} \int _{-L}^{L}G(x,y)\left( u \frac{\partial \bar{\rho }}{\partial x}+v \frac{\partial {\bar{\rho }}}{\partial y} \right) \hbox {d}x\,\hbox {d} y, \end{aligned} \end{aligned}$$

(31)

where

$$\begin{aligned}&L_{00}=\int _{0}^{h} \int _{-L}^{L} G(x,y) \hbox {d}x\,\hbox {d}y, \end{aligned}$$

(32)

$$\begin{aligned}&L_{0n}=\int _{0}^{h} \int _{-L}^{L} G(x,y) \cos \left( \frac{n \pi y}{h}\right) \hbox {d}x\, \hbox {d}y, \end{aligned}$$

(33)

and

$$\begin{aligned} \begin{aligned} L_{mn}=\int _{0}^{h} \int _{-L}^{L} G(x,y) \cos \left( \frac{m \pi x}{L}\right) \sin \left( \frac{n \pi y}{h}\right) \hbox {d}x\,\hbox {d}y, \end{aligned} \end{aligned}$$

(34)

and similarly

$$\begin{aligned} \begin{aligned} M_{l00} B_{00}'(t)&+\sum _{n=1}^{N}M_{l0n} B_{0n}'(t) +\sum _{m=1}^{M}\sum _{n=1}^{N}M_{lmn} B_{mn}'(t) \\&- \int _{0}^{h} \int _{-L}^{L}\left( u \frac{\partial \bar{\rho }}{\partial x}+v \frac{\partial {\bar{\rho }}}{\partial y} \right) \cos \left( \frac{l \pi y}{h}\right) \hbox {d}x\,\hbox {d}y, \end{aligned} \end{aligned}$$

(35)

for $l=1,2,\dots ,N$ where

$$\begin{aligned}&M_{l00}=\int _{0}^{h} \int _{-L}^{L} G(x,y) \cos \left( \frac{l \pi y}{h}\right) \hbox {d}x\,\hbox {d}y \end{aligned}$$

(36)

$$\begin{aligned}&M_{l0n}=\int _{0}^{h} \int _{-L}^{L} G(x,y)\cos \left( \frac{n \pi y}{h}\right) \cos \left( \frac{l \pi y}{h}\right) \hbox {d}x\,\hbox {d}y, \end{aligned}$$

(37)

and

$$\begin{aligned} \begin{aligned} M_{lmn}=\int _{0}^{h} \int _{-L}^{L} G(x,y)\cos \left( \frac{m \pi x}{L}\right) \sin \left( \frac{n \pi y}{h}\right) \cos \left( \frac{l \pi y}{h}\right) \hbox {d}x\,\hbox {d}y. \end{aligned} \end{aligned}$$

(38)

The final group of equations is similarly obtained by multiplying the density equation by the basis functions and integrating, so that

$$\begin{aligned} \begin{aligned}&N_{kl00} B_{00}'(t)+\sum _{n=1}^{N}N_{kl0n} B_{0n}'(t) +\sum _{m=1}^{M}\sum _{n=1}^{N}N_{klmn} B_{mn}'(t) \\&= \int _{0}^{h} \int _{-L}^{L}\left( u \frac{\partial \bar{\rho }}{\partial x}+v \frac{\partial {\bar{\rho }}}{\partial y} \right) \cos \left( \frac{k \pi x}{L}\right) \sin \left( \frac{l \pi y}{h}\right) \hbox {d}x\,\hbox {d}y, \end{aligned} \end{aligned}$$

(39)

for $k=1,2,\dots ,M, \ l=1,2,\dots ,N$ and where

$$\begin{aligned}&N_{kl00}=\int _{0}^{h} \int _{-L}^{L} G(x,y) \cos \left( \frac{k \pi x}{L}\right) \sin \left( \frac{l \pi y}{h}\right) \hbox {d}x\,\hbox {d}y, \end{aligned}$$

(40)

$$\begin{aligned}&N_{kl0n}=\int _{0}^{h} \int _{-L}^{L} G(x,y) \cos \left( \frac{k \pi x}{L}\right) \cos \left( \frac{n \pi y}{h}\right) \sin \left( \frac{l \pi y}{h}\right) \hbox {d}x\,\hbox {d}y, \end{aligned}$$

(41)

and

$$\begin{aligned} \begin{aligned} N_{klmn}=\int _{0}^{h} \int _{-L}^{L} G(x,y) \cos \left( \frac{m \pi x}{L}\right) \cos \left( \frac{k \pi x}{L}\right) \sin \left( \frac{n \pi y}{h}\right) \sin \left( \frac{l \pi y}{h}\right) \hbox {d}x\,\hbox {d}y. \end{aligned} \end{aligned}$$

(42)

For convenience, we now give the right-hand sides of each of these expressions their own notation, so that

$$\begin{aligned}&R_{00}=-\int _{0}^{h} \int _{-L}^{L}\left( u \frac{\partial \bar{\rho }}{\partial x}+v \frac{\partial {\bar{\rho }}}{\partial y} \right) \hbox {d}x\,\hbox {d}y, \end{aligned}$$

(43)

$$\begin{aligned}&R_{0l}=-\int _{0}^{h} \int _{-L}^{L}\left( u \frac{\partial \bar{\rho }}{\partial x}+v \frac{\partial {\bar{\rho }}}{\partial y} \right) \cos \left( \frac{l \pi y}{h}\right) \hbox {d}x\,\hbox {d}y, \ l=1,2,\dots ,N \end{aligned}$$

(44)

and

$$\begin{aligned} \begin{aligned} R_{kl}=-\int _{0}^{h} \int _{-L}^{L}\left( u \frac{\partial {\bar{\rho }}}{\partial x}+v \frac{\partial {\bar{\rho }}}{\partial y} \right) \cos \left( \frac{k \pi x}{L}\right) \sin \left( \frac{l \pi y}{h}\right) \hbox {d}x\,\hbox {d}y, \end{aligned} \end{aligned}$$

(45)

for $k=1,2,\dots ,M$, $l=1,2,\dots ,N$