Since forced convection has an imposed velocity field, the same can not be assumed for natural convection. In this case the CFD is more complex and another normalized parameters have to be considered.

Natural convection

The natural convection has a very big Richardson number Ri, which means that the motion is solely driven by buoyancy forces. There is no imposed velocity. Actually, buoyancy is what induces velocity in convective flows. The parametric flow is characterized by the Cover Figure, which illustrates the hot and the cold walls. Hot spots of fluids emerge from the hot wall and take the shape of light chips. The same is true for the opposite wall, the cold one. The cold spots of fluids tend to fall. The upper picture from the Cover Figure is about air, while the bottom one is about liquid, because Pr < 1. The structure of the temperature field becomes an agent. The effect of the Prandtl number Pr is the same as in the forced convection.

The problem of the natural convection

The problem of the natural convection can be explained by an example. Considering two holes and no motion on the path between these two. The characteristic velocity of this flow is zero, because there is no velocity scale imposed, thus there is no Re. The velocity scale depends only on the buoyancy. To address this aspect, it is necessary to introduce a characteristic velocity, this is given by the buoyancy. This velocity scale is the free fall velocity. These are the general equations:

∇∙u = 0

∂u/∂t + u∙∇u = -∇p* + (1/Re)∇²u + Ri∙θ∙z

∂θ/∂t + u∙∇θ = [1/(Pr∙Re)]∙∇²θ

Where:

Re = uL/ν ; Pr = ν/α ; Ri = gβΔTL/U²

and the free fall velocity is given by:

U_ff = √(gβΔTL)

Hence, Ri is the square of the free fall velocity divided by the square value of the imposed velocity. However, for the natural convection, this term is zero. Hence, to compute the characteristic velocity, the only quantity is the free fall velocity. Hence, for the natural convection U_ff is the characteristic velocity. This means that when dealing with the natural convection, the Oberbeck-Boussinesq equations are modified.

∇∙u = 0

∂u/∂t + u∙∇u = – ∇p* + (1/√Gr)∙∇²u + θz

∂θ/∂t + u∙∇θ = [1/(Pr∙√Gr)]∇²θ

Where:

Gr = gβΔTL³/ν² = Re² ; Nu = f(Pr,Gr)

As can be seen, if the free fall velocity is used to normalize the equation, this derives to the Grashof number Gr. The reason for this procedure is, because it is not possible to normalize with the reference velocity U, since this is zero. Hence, by using the free fall velocity, the system orientation re-writes to this Grashof number and this plays the role of the free fall Reynolds number (Re_ff). Therefore, for natural convection it is not possible to compute Re, it must be computed the free fall velocity and this allows to calculate Gr, which is the really important parameter for the natural convection. There is also the Prandtl number, the buoyancy term, that by using U_ff in the normalized form results in buoyance term θ_z. Hence, in the natural convection regime, the efficiency of the exchange is computed by the Nusselt number (Nu) as function of Pr and Gr. This one is a kind of Re, while Pr is based on buoyancy. Gr is computed based on the free fall velocity, thus used, a priori, in the natural convection process. In some applications, instead of using Gr, it is used the Rayleight number.

Ra = gβΔTL³/(ν∙α) = Gr∙Pr

Hence, the system becomes:

∇∙u = 0

∂u/∂t + u∙∇u = – ∇p* + √[(Pr/Ra)]∙∇²u + θ∙z

∂θ/∂t + u∙∇θ = [1/√(Pr∙Ra)]∙∇²θ

Nu = f(Pr,Ra)

The information provided by the Rayleigh number (Ra) is essentially the same as Gr. However, it is possible to correlate these since Ra = Gr∙Pr. When Ra is considered instead of Gr, it is possible to notice that 1/Re become √(Pr∙Ra), while the diffusivity becomes 1/√(Pr∙Ra). Hence it is just a change of variable, since Ra plays the role of GR and the heat exchange is a function of Pr and Ra.

References

• This article was based on the lecture notes written during the Industrial Aerodynamics course attended at Dallara Academy.