Natural Convection in Encloses: Scale Analysis

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Figure 1 Two-dimensional natural convection in rectangular enclosure.

Two-dimensional natural convection in a rectangular enclosure with two differentially heated sides and insulated top and bottom surfaces (see Figure 1) will be considered. The fluid is assumed to be Newtonian and incompressible. The system is initially at a uniform temperature of zero and therefore, the fluid is motionless ( $u = v = 0$ ). At time zero the two sides are instantaneously heated and cooled to $Δ T / 2$ and $- Δ T / 2$ , respectively. The transient behavior of the system during the establishment of the natural convection (Patterson and Imberger, 1980; Bejan, 2004) is the subject of this analysis.

It is assumed that the fluid is single-component and that there is no internal heat generation in the fluid. Therefore, the governing equations for this internal convection problem can be obtained by simplifying the governing equations for natural convection:

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$	(1)

$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\nu \left( \frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}} \right)$	(2)

$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{\rho }\frac{\partial p}{\partial y}+\nu \left( \frac{{{\partial }^{2}}v}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}v}{\partial {{y}^{2}}} \right)-g[1-\beta (T-{{T}_{0}})]$	(3)

$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \left( \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}} \right)$	(4)

Immediately after imposing the temperature difference ( $t = 0 +$ ), the fluid is still motionless ( $u = v = 0$ ), hence the energy equation (4) reflects the balance between the thermal inertia and the conduction in the fluid. The scales of the two terms enclosed in the parentheses on the right-hand side of eq. (4) are $\Delta T/\delta _{t}^{2}$ and $Δ T / H 2$ , respectively. Since ${{\delta }_{t}}\ll H$ , one can conclude that

${{\partial }^{2}}T/\partial {{y}^{2}}\ll {{\partial }^{2}}T/\partial {{x}^{2}}$

The balance of scales for eq. (4) then becomes:

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

Thus, the scale of the thermal boundary layer thickness becomes:

${{\delta }_{t}}\sim{\ }{{(\alpha t)}^{1/2}}$	(5)

To estimate the scale of the velocity, one can combine eqs. (2) and (3) by eliminating the pressure to obtain:

$\frac{\partial }{\partial x}\left( \frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y} \right)-\frac{\partial }{\partial y}\left( \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} \right) =\nu \left[ \frac{\partial }{\partial x}\left( \frac{{{\partial }^{2}}v}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}v}{\partial {{y}^{2}}} \right)-\frac{\partial }{\partial y}\left( \frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}} \right) \right]+g\beta \frac{\partial T}{\partial x}$	(6)

where the left-hand side represents the inertia terms, and the right-hand side represents the viscosity and buoyancy terms. The scales of these three effects are shown below

$\begin{align} & \text{Inertia Viscosity Buoyancy} \\ & \text{ }\frac{v}{{{\delta }_{t}}t}\text{ }\qquad\nu \frac{v}{\delta _{t}^{3}}\text{ }\qquad\frac{g\beta \Delta T}{{{\delta }_{t}}} \\ \end{align}$	(7)

To examine the relative strength of each effect, one can divide the above expression by the scale of viscosity effect to obtain

$\begin{align} & \text{Inertia Viscosity Buoyancy} \\ & \text{ }\frac{1}{\Pr } \qquad\qquad1 \qquad\frac{g\beta \Delta T\delta _{t}^{2}}{\nu v} \\ \end{align}$

where eq. (5) was used to simplify the inertia term. For the fluid with $\Pr >1$ , the momentum balance at $t = 0 +$ requires a balance between the viscosity and buoyancy terms:

$1\sim{\ }\frac{g\beta \Delta T\delta _{t}^{2}}{\nu v}$

Substituting eq. (5) into the above expression and rearranging the resultant expression, the scale of vertical velocity at the initiation of the natural convection is obtained as follows:

$v\sim{\ }\frac{g\beta \Delta T\alpha t}{\nu }$	(8)

Let us turn our attention now to the energy equation (4). The first term on its left-hand side is the inertia term, and its scale is $Δ T / t$ . The second and third terms on the left-hand side are the advection terms and both of them have scale $v Δ T / H$ . The right-hand side of eq. (4) is the conduction term, and its scale is $\alpha \Delta T/\delta _{t}^{2}$ . As time increases, the effect of the inertia term weakens, hence the effect of advection becomes stronger. This trend continues until a final time, tf, when the energy balance requires balance between the advection and conduction terms, i.e.,

$v\frac{\Delta T}{H}\sim{\ }\alpha \frac{\Delta T}{\delta _{t,f}^{2}}\sim{\ }\frac{\Delta T}{{{t}_{f}}}$

Thus, the scale of tf becomes

${{t}_{f}}\sim{\ }{{\left( \frac{\nu H}{g\beta \Delta T\alpha } \right)}^{1/2}}$	(9)

The scale of thermal boundary layer thickness at time t_f is:

${{\delta }_{t,f}}\sim{\ }{{(\alpha {{t}_{f}})}^{1/2}}\sim{\ }H\text{Ra}_{H}^{-1/4}$	(10)

At time t_f, natural convection in the rectangular enclosure reaches steady-state and the thickness of the thermal boundary layer no longer increases with time.

As the fluid in the thermal boundary layer near the hot wall is heated and rises due to buoyancy force, it drags the fluid outside of the thermal boundary layer upward forming a wall jet. Similar to the thermal boundary layer thickness, the wall jet thickness also increases with time until t = t_f when the maximum wall jet thickness, δ_v,f, is reached (see Fig. 6.19). Outside the thermal boundary layer, the buoyancy force is absent and the thickness of the wall jet can be determined by balancing the inertia and viscosity terms in eq. (6):

$\frac{v}{{{\delta }_{v}}t}\sim{\ }\nu \frac{v}{\delta _{v}^{3}}$

which can be rearranged to obtain:

${{\delta }_{v}}\sim{\ }{{(\nu t)}^{1/2}}\sim{\ }{{\Pr }^{1/2}}{{\delta }_{t}}$	(11)

Figure 2 Two-layer structure near the heated wall.

For t > t_f , steady-state has been reached, and the wall jet thickness is related to the thermal boundary layer thickness by ${{\delta }_{v,f}}\sim{\ }{{\Pr }^{1/2}}{{\delta }_{t,f}}$ . Similarly, the condition to have distinct vertical wall jets or momentum boundary layers is $δ v, f < L$ , or equivalently:

$\frac{H}{L}<Ra_H^{1/4}Pr^{-1/2}$	(12)

When the vertical wall jet encounters the horizontal wall, it will turn to the horizontal direction and become a horizontal jet. This horizontal jet will contribute to the convective heat transfer from the heated wall to the cooled wall:

${{{q}'}_{conv}}\sim{\ }\rho {{v}_{f}}{{\delta }_{T,f}}{{c}_{p}}\Delta T$ <center>

Considering eqs. (8) and (10), the above scale of convective heat transfer becomes:

<center> ${{{q}'}_{conv}}\sim{\ }k\Delta T\text{Ra}_{H}^{1/4}$

When a warm jet is formed at the top and a cold jet is formed at the bottom, there will be a temperature gradient along the vertical direction. The heat conduction due to this temperature gradient is:

${{{q}'}_{cond}}\sim{\ }kL\frac{\Delta T}{H}$

The condition under which the horizontal wall jets can maintain their temperature identity is when the heat conduction along the vertical direction is negligible compared to the energy carried by the horizontal jets:

$kL\frac{\Delta T}{H}<k\Delta T\text{Ra}_{H}^{1/4}$

or equivalently

$\frac{H}{L}>\text{Ra}_{H}^{-1/4}$

The characteristics of various heat transfer regimes are summarized in the following table.

Characteristics of natural convection in a rectangular enclosure heated from the left side

Regimes	I: Conduction	II: Tall Systems	III: Boundary layer	IV: Shallow systems
Condition of occurrence	$Ra H < 1$	$H/L>Ra_H^{1/4}$	$Ra_H^{-1/4}<H/L<Ra_H^{1/4}$	$H/L<Ra_H^{-1/4}$
Flow pattern	Clockwise circulation	Distinct boundary layer on top and bottom walls	Boundary layer on all four walls. Core remains stagnant	Two horizontal wall jets flow in opposite directions.
Effect of flow on heat transfer	Insignificant	Insignificant	Significant	Significant
Heat transfer mechanism	Conduction in horizontal direction	Conduction in horizontal direction	Boundary layer convection	Conduction in vertical direction
Heat transfer	$q'˜ k H Δ T / L$	$q'˜ k H Δ T / L$	$q'˜(k / δ T, f) H Δ T$	$q'˜(k / δ T, f) H Δ T$

References

Bejan, A., 2004, Convection Heat Transfer, 3rd ed., John Wiley & Sons, New York.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Patterson, J., and Imberger, J., 1980, “Unsteady Natural Convection in a Rectangular Cavity,” J. Fluid Mech., Vol. 100, pp. 65-86.

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Natural Convection in Encloses: Scale Analysis

From Thermal-FluidsPedia

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