1. Scale Analysis of Internal Natural Convection
Dr. Om Prakash Singh
Asst. Prof., IIT Mandi

2. Boundary layer thickens
Two-dimensional rectangular enclosure with isothermal sidewalls.

3. Governing equations
The equations governing the conservation of mass, momentum, and energy at every point in the cavity are
Note that in writing these equations, we modeled the fluid as Boussinesq-incompressible, in other words, ρ = constant everywhere except in the body force term of the y momentum equation, where it is replaced by ρ[1−β(T−T0)].

4. Scale analysis of governing equations
Immediately after t = 0, the fluid bordering each sidewall is motionless: This means that near the sidewall, the energy equation (5.4) expresses a balance between thermal inertia and conduction normal to the wall,
This equality of scales follows from recognizing T, t, and δT as the scales of
changes in T, t, and x in eq. (5.4) and velocity has not yet developed (u=v=0)
Hence,
The layer δT rises along the heated wall.

5. Scale analysis of governing equations
The velocity scale of this motion v is easier to see if we first eliminate the pressure P between the two momentum equations (5.2) and (5.3):
This new equation contains three basic groups of terms: inertia terms on the left-hand side and four viscous diffusion terms plus the buoyancy term on the right-hand side. It is easy to show that the three terms that dominate each basic group are

6. Scale analysis of governing equations
In terms of representative scales, the momentum balance (5.8) reads
The driving force in this balance is the buoyancy effect (gβ T)/δT, which is not zero. It is important to determine whether the buoyancy effect is balanced by friction or inertia. Dividing eq. (5.9) through the friction scale and recalling that δ2T ∼ αt yields
Therefore, for fluids with Prandtl number of order 1 or greater, the correct momentum balance at t = 0 + is between buoyancy and friction,

7. Scale analysis of governing equations
We conclude that the initial vertical velocity scale is
This velocity scale is valid for fluids such as water and oils (Pr > 1) and is marginally valid for gases (Pr ∼ 1).
The heat conducted from the sidewall into the fluid layer δT is no longer spent solely on thickening the layer: Part of this heat input is carried away by the layer δT rising with velocity v. Thus, in the energy equation, we see a competition among three distinct effects:

8. Scale analysis of governing equations
As t increases, the convection effect increases [v ∼ t, eq. (5.11’)], while the effect of inertia decreases in importance. There comes a time tf when the energy equation expresses a balance between the heat conducted from the wall and the enthalpy carried away vertically by the buoyant layer,
At such a time, the layer thickness is
where RaH is the Rayleigh number based on the enclosure height,

9. Scale analysis of governing equations
In addition to thermal layers of thickness δT, f, the sidewalls develop viscous (velocity) wall jets. The thickness of these jets δv from the momentum balance (5.7) for the region of thickness x ∼ δv outside the thermal layer. In this region, the buoyancy effect is minor, and we have a balance between inertia and viscous diffusion,
Hence,
In the steady state, t > tf, the fluid near each sidewall is characterized by a two-layer structure:
a thermal boundary layer of thickness δT, f and
a thicker wall jet δv,f ∼ Pr1/2δT,f. The development of this structure is shown in Fig. 5.2.

10. Fig. 5.2. Development of two-layer structure near the warm wall.

11. End