# Free Boundary Flow

The common feature of external convection discussed in the preceding subsections is that the flows are always near a heated (or cooled) solid wall. For some other applications, however, the thermally induced flow occurs without presence of a solid wall and is referred to as free boundary flow. When a point or line heat source is immersed into a bulk fluid, the fluid near the heat source is heated, becomes lighter and rises to form a plume [see Fig. 1 (a)]. The plume generated by a point heat source is axisymmetric, but a line heat source will create a two-dimensional plume. A thermal is a column of rising air near the

Figure 1: Examples of free boundary flows

Figure 2: Physical model of natural convection from a line heat source

ground due to the uneven solar heating of the ground surface. The lighter air near the ground rises and cools due to expansion [see Fig. 1(b)].

Let us consider now the two-dimensional free boundary flow induced by a line heat source (see Fig. 2). Since the problem is symmetric about x = 0, only half of the domain (x > 0) needs to be studied. In the coordinate system shown in Fig. 2, the continuity, momentum and energy equations are the same as those for natural convection near a vertical flat plate, eqs. $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$, $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}+g\beta (T-{{T}_{\infty }})$ and $u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{{{\partial }^{2}}T}{\partial {{y}^{2}}}$, respectively. The boundary conditions at $y\to \infty$ can still be described by eqs. $u=v=0,\text{ }y\to \infty$ and $T={{T}_{\infty }},\text{ }y\to \infty$ from External Natural Convection from Heated Vertical Plate. However, the boundary conditions at the centerline, x = 0, should be changed to:

$\frac{\partial u}{\partial y}=\frac{\partial T}{\partial y}=0,\text{ }v=0\text{ at }y=0 \qquad \qquad(1)$

which indicates that the velocity component in the x-direction and the temperature at the centerline are at their respective maximum. Gebhart et al. (1970) assumed that the difference between the centerline temperature and the bulk fluid temperature is of the following power law form:

${{T}_{0}}-{{T}_{\infty }}=N{{x}^{n}} \qquad \qquad(2)$

where the index n will be determined using the overall energy balance. The continuity, momentum, and energy equations can be transformed to the following ordinary differential equations via the same similarity variables for natural convection over a vertical flat plate in Section 6.4.1 and considering eq. (2).

${F}'''+\theta +(3+n)F{F}''-(2n+2){{({F}')}^{2}}=0 \qquad \qquad(3)$

${\theta }''+\Pr [(n+3)F{\theta }'-4n{F}'\theta ]=0 \qquad \qquad(4)$

Equations (3) and (4) are applicable to any natural convection problem that satisfies eq. (2). To get the ordinary differential equations that are specific for natural convection induced by a line heat source, proper values of N and n, as well as appropriate boundary conditions, must be specified. The energy convected across any horizontal plane in the plume is

${q}'=\rho {{c}_{p}}\int_{-\infty }^{\infty }{u(T-{{T}_{\infty }})dy} \qquad \qquad(5)$

which is identical to the intensity of the line heat source and it can be expressed in terms of similarity variables

${q}'=4\nu \rho {{c}_{p}}N{{\left( \frac{g\beta N}{4{{\nu }^{2}}} \right)}^{1/4}}{{x}^{(5n+3)/4}}\int_{-\infty }^{\infty }{{F}'\theta d\eta } \qquad \qquad(6)$

Since the line heat source is the only source of heating, the above q' should be independent from x which can be true only if n is equal to − 3 / 5. Therefore, the ordinary differential equations for natural convection over a line heat source respectively become:

${F}'''+\theta +\frac{12}{5}F{F}''-\frac{4}{5}{{({F}')}^{2}}=0 \qquad \qquad(7)$

${\theta }''+\frac{12}{5}\Pr (F{\theta }'+{F}'\theta )=0 \qquad \qquad(8)$

which are subject to the following boundary conditions:

$F(\eta )={F}''(\eta )=0,\text{ and }\theta (\eta )=1\text{ at }\eta =0 \qquad \qquad(9)$

${F}'(\eta )=\theta (\eta )=0\text{ at }\eta \to \infty \qquad \qquad(10)$

Equations (7) – (10) can be solved numerically and the results are shown in Fig. 3. In contrast to Fig. 1 from Similarity Solution for Natural Convection on a Vertical Surface that shows the velocity at the wall is zero, the velocity at the centerline is at its maximum. With known intensity of the line heat source, the constant N in eq. (2) can be obtained from eq. (6):

$N={{\left( \frac{{{{{q}'}}^{4}}}{64g\beta {{\rho }^{2}}{{\mu }^{2}}c_{p}^{4}{{I}^{4}}} \right)}^{1/5}} \qquad \qquad(11)$

Figure 3: Velocity and temperature distributions of natural convection from a line heat source (a) velocity, (b) temperature

where

$I=\int_{-\infty }^{\infty }{{F}'(\eta )\theta (\eta )d\eta } \qquad \qquad(12)$

is a function of Prandtl number. The values of I calculated at Pr = 0.7, 1.0, 6.7, and 10.0 by Gebhart et al. (1970) are 1.245, 1.053, 0.407 and 0.328, respectively.

$\frac{\partial u}{\partial r}=\frac{\partial T}{\partial r}=0,\text{ }v=0\text{ at }r=0 \qquad \qquad(13)$

$u=v=0,\text{ }T={{T}_{\infty }}\text{ at }r\to \infty \qquad \qquad(14)$

Similar to the case of line heat source, the centerline temperature can also be expressed as a power function of x as indicated by eq. (2). Introducing Stokes stream function:

$u=\frac{1}{r}\frac{\partial \psi }{\partial r},\text{ }v=-\frac{1}{r}\frac{\partial \psi }{\partial x} \qquad \qquad(15)$

and defining the following similarity variables:

$\eta =\frac{r}{x}\text{Gr}_{x}^{1/4},\text{ }\psi =\nu xF(\eta ),\text{ }\theta =\frac{T-{{T}_{\infty }}}{{{T}_{0}}-{{T}_{\infty }}} \qquad \qquad(16)$

the following ordinary differential equations are obtained:

${F}'''+\eta \theta +(F-1){{\left( \frac{{{F}'}}{\eta } \right)}^{\prime }}-\frac{1+n}{2}\frac{{{({F}')}^{2}}}{\eta }=0 \qquad \qquad(17)$

$(\eta {\theta }'{)}'+\Pr (F{\theta }'-n{F}'\theta )=0 \qquad \qquad(18)$

The intensity of the point heat source is:

$q=\rho {{c}_{p}}\int_{0}^{\infty }{u(T-{{T}_{\infty }})2\pi rdr}=2\pi \nu \rho {{c}_{p}}N{{x}^{n+1}}\int_{0}^{\infty }{{F}'\theta d\eta } \qquad \qquad(19)$

and it must be independent from x. Therefore, the value of n must be − 1 and the ordinary differential equations for natural convection over a point heat source become:

${F}'''+\eta \theta +(F-1){{\left( \frac{{{F}'}}{\eta } \right)}^{\prime }}=0 \qquad \qquad(20)$

$(\eta {\theta }'{)}'+\Pr (F\theta {)}'=0 \qquad \qquad(21)$

The velocity components in the x- and r- directions are related to the dimensionless variables by the following equations:

$u=\frac{\nu }{x}\sqrt{\text{G}{{\text{r}}_{x}}}\left( \frac{{{F}'}}{\eta } \right),\text{ }v=-\frac{\nu }{x}\text{Gr}_{x}^{1/4}\left( \frac{F}{\eta }-\frac{{{F}'}}{2} \right) \qquad \qquad(22)$

which can be substituted into eqs. (13) and (14) to obtain the following boundary conditions for eqs. (20) and (21):

${{\left( \frac{{{F}'}}{\eta } \right)}^{\prime }}=0,\text{ }F=0,\text{ }{\theta }'=0\text{ and }\theta =1\text{ at }\eta =0 \qquad \qquad(23)$

$\frac{{{F}'}}{\eta }=\theta =0\text{ at }\eta \to \infty \qquad \qquad(24)$

## References

Gebhart, B., Pera, L., and Schorr, A.W., 1970, “Steady Laminar Natural Convection Plumes above a Horizontal Line Heat Source,” International Journal of Heat and Mass Transfer, Vol. 13, pp. 161-171