How to identify the sign of a derived nondimensional parameter and its physical meaning? I think that the nondimensional group is ordinarily defined to be positive value in a physical problem. But in some particular case, we probably need to decide the sign of a derived dimensionless parameter. For example, I have defined nondimensional gravity is
$$G=\frac{gL^3}{\nu^2},$$
where $g$, $L$ and $\nu$ are gravitational acceleration, length scale, and kinematic viscosity. Then I derived another dimensionless group and defined it as Rayleigh number, because the similarity in form,
$$Ra=\frac{gL^3\beta\Theta}{\alpha \nu},$$
where $\beta$ and $\alpha$ are the thermal expansion coefficient and diffusivity. Note that the temperature scale $\Theta$ here is given by a fraction instead of a temperature difference in the system($\Theta=\theta_0-\theta_{ref.}$). The problem (my confusion) is that in normal definition $Ra$ (or $\Theta$) increases as $\theta_0$, however, the $\Theta$ defined by the fraction decreases as increase in $\theta_0$. My question is how can I identify the sign of my redefined $Ra$ correctly to be consistent with the usual definition? Is it appropriately if I simply add a minus sign before the $Ra$?
Thank you!
 A: Let's consider the Rayleigh-Bernard convection as a natural convective system. 
The governing equations are the incompressible Navier-Stokes equations with the Boussinesque approximation as body force accounting for bouyant forces due to temperature variations:
$$\boldsymbol{v}\cdot\boldsymbol{\nabla}\boldsymbol{v}=\nu\nabla^2\boldsymbol{v} + \beta g\left(T-T_r\right)\boldsymbol{e}_y$$
and the thermal advection-diffusion equation:
$$\boldsymbol{v}\cdot\boldsymbol{\nabla}T=\alpha\nabla^2T$$
$\boldsymbol{v}$ is the velocity field, $T$ is the temperature field, $\nu$ is the kinematic viscosity, $\alpha$ is the thermal diffusivity, $\beta$ is the coefficient of thermal expansion, $g$ is the gravitational constant, $T_r$ is the reference temperature at which $\beta$ was determined and $\boldsymbol{e}_y=\left(0,-1,0\right)$ is the unit vector in the direction of the acceleration due to gravity.
Let's non-dimensionalize to determine the relevant dimensionless numbers:
$$\tilde{y}=\frac{y}{H}\quad\tilde{\boldsymbol{v}}=\frac{\boldsymbol{v}}{U}\quad\tilde{T}=\frac{T-T_r}{\Delta T}$$
where $H$ is the length scale of the system (channel height), $U$ is the velocity scale and $\Delta T$ is the imposed temperature difference over the channel height.
The non-dimensionalized governing equations become:
$$\tilde{\boldsymbol{v}}\cdot\tilde{\boldsymbol{\nabla}}\tilde{\boldsymbol{v}}=\frac{1}{\mathrm{Re}}\tilde{\nabla}^2\tilde{\boldsymbol{v}} + \mathrm{\Pi}\tilde{T}\boldsymbol{e}_y$$
$$\tilde{\boldsymbol{v}}\cdot\tilde{\boldsymbol{\nabla}}\tilde{T}=\frac{1}{\mathrm{Pe}}\tilde{\nabla}^2\tilde{T}$$
with the relevant dimensionless numbers identified as:
$$\mathrm{Re}=\frac{UH}{\nu}\quad\mathrm{Pr}=\frac{\nu}{\alpha}\quad\mathrm{Pe}=\mathrm{Re}\mathrm{Pr}\quad\mathrm{\Pi}=\frac{\beta gH\Delta T}{U^2}$$
Note: $\mathrm{\Pi}$ at the moment is just some arbitrary dimensionless number which is not identified yet.
Unlike in forced convection, the velocity scale $U$ is not imposed but is due to the dynamics of the bouyant system. The system becomes interesting once inertial terms become similar to viscous terms such that $\mathrm{Re}\sim 1$. We can then identify:
$$U\sim\frac{\nu}{H}$$
and the non-dimensionalized governing equations become:
$$\tilde{\boldsymbol{v}}\cdot\tilde{\boldsymbol{\nabla}}\tilde{\boldsymbol{v}}=\tilde{\nabla}^2\tilde{\boldsymbol{v}} + \mathrm{\Pi}\tilde{T}\boldsymbol{e}_y$$
$$\tilde{\boldsymbol{v}}\cdot\tilde{\boldsymbol{\nabla}}\tilde{T}=\frac{1}{\mathrm{Pe}}\tilde{\nabla}^2\tilde{T}$$
$$\mathrm{Pe}=\mathrm{Pr}\quad\mathrm{\Pi}=\frac{\beta gH^3\Delta T}{\nu^2}=\frac{\mathrm{Ra}}{\mathrm{Pr}}\quad$$
where we have identified the Rayleigh number, $\mathrm{Ra}$ in the definition of $\mathrm{\Pi}$:
$$\mathrm{Ra}=\frac{\beta gH^3\Delta T}{\nu\alpha}$$
To answer your (original) questions:


*

*If $\beta$ is a constant, then the definition of $\mathrm{Ra}$ contains a temperature difference not a temperature ratio. However, for ideal gases we can identify $\beta\sim\frac{1}{T_m}$ (where $T_m$ is the averaged temperature) and then indeed it becomes a ratio. The result is still that increasing the temperature difference across the channel result in increasing the Rayleigh number.

*A sign in a dimensionless number is irrelevant information for a dimensional analysis. We want to know how different mechanisms (inertia vs viscous forces, etc.) relate to each other such that we may simplify the problem. The sign of course play a crucial role in the equations and as such is included through the unit vector $\boldsymbol{e}_y$. It is a good habit to always define the terms in dimensionless numbers as absolute values.
