Scale analysis of natural convection

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While scale analysis cannot provide the exact functions in eq. (6.36), the form of these functions can be provided (Bejan, 2004). Let us refer to Fig. 6.1 and consider the governing equations in the thermal boundary layer (y˜δt) for the entire flat plate (x˜L). The thickness of the thermal boundary layer in which the effects of the heated wall are felt is much smaller than the length of the vertical plate, i.e. {{\delta }_{t}}\ll L. For the continuity equation (6.17) to be satisfied, the scales of the two terms must be the same:

\frac{u}{L}\sim \frac{v}{{{\delta }_{t}}}

The scale of the velocity component in the y-direction is therefore:

v\sim \frac{{{\delta }_{t}}}{L}u (1)

which indicates that v\ll u for flow in the thermal boundary layer. The scale of the velocity component in the x-direction, u, is still unknown at this point.

While the left hand side of the energy equation (6.21) shows the effect of advection, the right-hand side shows the effect of diffusion. The scales of the two advective terms on the left hand side of eq. (6.21) are:u\frac{\partial T}{\partial x}\sim u\frac{\Delta T}{L},\text{ }v\frac{\partial T}{\partial y}\sim v\frac{\Delta T}{{{\delta }_{t}}}\sim u\frac{\Delta T}{L}

which indicates that the scale of the second term on the left hand side of eq. (6.21) is identical to the scale of the first term when the scale of the velocity component in the y-direction is given by eq. (6.37). The temperature difference \Delta T={{T}_{w}}-{{T}_{\infty }} in the above scale analysis represents the scale of the excess temperature, T-{{T}_{\infty }}. The scale of the right-hand side of eq. (6.21) is:

\alpha \frac{{{\partial }^{2}}T}{\partial {{y}^{2}}}\sim \alpha \frac{\Delta T}{\delta _{t}^{2}}

The scales of the two sides of the energy equation must be the same:u\frac{\Delta T}{L}\sim \alpha \frac{\Delta T}{\delta _{t}^{2}}

The above equation can be used to estimate the scale of u as follows:

u\sim \alpha \frac{L}{\delta _{t}^{2}} (2)

The scale of v can be obtained by substituting eq. (2) into eq. (1):

v\sim \frac{\alpha }{{{\delta }_{t}}} (3)

where the scale of the thermal boundary layer thickness, δt, is still unknown at this point.

The respective scales of the two inertial terms on the left-hand side of the momentum equation (6.20) are:

u\frac{\partial u}{\partial x}\sim \frac{{{u}^{2}}}{L}


v\frac{\partial u}{\partial y}\sim v\frac{u}{{{\delta }_{t}}}\sim \frac{{{u}^{2}}}{L}

The respective scales of the viscosity and buoyancy terms on the right-hand side of eq. (6.20) are:

\nu \frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}\sim \nu \frac{u}{\delta _{t}^{2}}


g\beta (T-{{T}_{\infty }})\sim g\beta \Delta T

We can very well see that three forces are at play in the boundary layer region, which are inertia, viscosity and buoyancy forces. Considering the scale of u obtained from eq. (2), the scales of these three forces are

\begin{matrix}
   \frac{{{\alpha }^{2}}L}{\delta _{t}^{4}}, & \frac{\nu \alpha L}{\delta _{t}^{4}}, & g\beta \Delta T  \\
   \text{Inertia} & \text{Viscous} & \text{Buoyancy}  \\
\end{matrix} (4)

Among these three forces, the buoyancy force is never negligible because without it natural convection would not occur. Therefore, the scale of buoyancy force can be used to measure the importance of the inertial and viscous forces. Dividing eq. (6.40) by the scale of buoyancy force, gβΔT , one obtains the following:\begin{matrix}
   \frac{{{\alpha }^{2}}}{g\beta \Delta T{{L}^{3}}}{{\left( \frac{L}{{{\delta }_{t}}} \right)}^{4}}, & \frac{\nu \alpha }{g\beta \Delta T{{L}^{3}}}{{\left( \frac{L}{{{\delta }_{t}}} \right)}^{4}}, & 1  \\
   \text{Inertia} & \text{Viscous} & \text{Buoyancy}  \\
\end{matrix}

which can be expressed in terms of dimensionless parameters as (Bejan, 2004):

\begin{matrix}
   {{\left( \frac{L}{{{\delta }_{t}}} \right)}^{4}}\text{Ra}_{L}^{-1}{{\Pr }^{-1}}, & {{\left( \frac{L}{{{\delta }_{t}}} \right)}^{4}}\text{Ra}_{L}^{-1}, & 1  \\
   \text{Inertia} & \text{Viscous} & \text{Buoyancy}  \\
\end{matrix} (5)

where

\text{R}{{\text{a}}_{L}}=\frac{g\beta ({{T}_{w}}-{{T}_{\infty }}){{L}^{3}}}{\nu \alpha } (6)

is the Rayleigh number that is related to the Grashof number by \text{R}{{\text{a}}_{L}}=\text{G}{{\text{r}}_{L}}\Pr . Therefore, the relative importance of inertia and viscous forces depends on the Prandtl number, which is a property of the fluid. Thus, if the Prandtl number is high (\Pr \gg 1), the inertia term will be negligible and the viscosity term will balance the buoyancy term, whereas if the Prandtl number is low enough (\Pr \ll 1) – as for liquid metals – then the inertia term is considerable and balances the buoyancy term in steady state. The scale analysis for high- and low-Prandtl number fluids is presented in detail below.