# Dimensionless parameters for natural convection

While the Reynolds number was used as a dimensionless parameter to characterize the flow for the case of forced convection, it cannot be used to characterize the natural convection because the characteristic velocity is not available. To identify the appropriate dimensionless parameters for description of natural convection, one defines the following dimensionless variables:

 $X=\frac{x}{L},\text{ }Y=\frac{y}{L},\text{ }U=\frac{u}{{{u}_{0}}},\text{ }V=\frac{v}{{{u}_{0}}},\text{ }\theta =\frac{T-{{T}_{\infty }}}{{{T}_{w}}-{{T}_{\infty }}}$ (1)

where u0 is a reference velocity that is unknown at this point. Equations (6.17), (6.20) and (6.21) can be respectively nondimensionalized as follows (Incropera et al., 2007):

 $\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$ (2)
 $U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=\frac{1}{{{\operatorname{Re}}_{L}}}\frac{{{\partial }^{2}}U}{\partial {{Y}^{2}}}+\frac{g\beta ({{T}_{w}}-{{T}_{\infty }})L}{u_{0}^{2}}\theta$ (3)
 $U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{1}{{{\operatorname{Re}}_{L}}\Pr }\frac{{{\partial }^{2}}\theta }{\partial {{Y}^{2}}}$ (4)

where

 ${{\operatorname{Re}}_{L}}=\frac{{{u}_{0}}L}{\nu }$ (5)

is the Reynolds number based on the reference velocity. To simplify the dimensionless governing equations, one can choose the reference velocity to be:

 ${{u}_{0}}=\sqrt{g\beta ({{T}_{w}}-{{T}_{\infty }})L}$ (6)

so that the factor before dimensionless temperature, θ, in eq. (3) becomes unity. With this choice of reference velocity, the Reynolds number becomes:

 ${{\operatorname{Re}}_{L}}=\frac{\sqrt{g\beta ({{T}_{w}}-{{T}_{\infty }}){{L}^{3}}}}{\nu }$ (7)

In natural convection problems, we define Grashof number, GrL, to be

 $\text{G}{{\text{r}}_{L}}=\operatorname{Re}_{L}^{2}=\frac{g\beta ({{T}_{w}}-{{T}_{\infty }}){{L}^{3}}}{{{\nu }^{2}}}$ (8)

as the dimensionless number which represents the ratio of the buoyancy force and the viscosity force acting on the fluid. Equations (3) and (4) then become:

 $U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=\frac{1}{\text{Gr}_{\text{L}}^{\text{1/2}}}\frac{{{\partial }^{2}}U}{\partial {{Y}^{2}}}+\theta$ (9)
 $U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{1}{\text{Gr}_{\text{L}}^{\text{1/2}}\Pr }\frac{{{\partial }^{2}}\theta }{\partial {{Y}^{2}}}$ (10)

The role of the Grashof number for a natural convection problem is similar to the role of Reynolds number for a forced convection problem. The Prandtl number, which reflects the ratio between momentum and thermal diffusions, is another parameter that affects natural convection. Therefore, it is expected that the average Nusselt number for natural convection, ${{\overline{\text{Nu}}}_{L}}=\bar{h}L/k$, will be a function of the Grashof number and the Prandtl number:

 ${{\overline{\text{Nu}}}_{L}}=f(\text{G}{{\text{r}}_{L}},\Pr )$ (11)

The objective of this chapter is thus to identify the appropriate forms of the above function for various geometric configurations and physical conditions.

## References

Incropera, F.P., DeWitt, D.P., Bergman, T.L., Lavine, A.S. 2007, Fundamentals of Heat and Mass Transfer, 6th ed., John Wiley & Sons, New York.