It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number (Re).
Usually the density decreases due to an increase in temperature and causes the fluid to rise.
The transition to turbulent flow occurs in the range 108 < GrL < 109 for natural convection from vertical flat plates.
At higher Grashof numbers, the boundary layer is turbulent; at lower Grashof numbers, the boundary layer is laminar, that is, in the range 103 < GrL < 106.
There is an analogous form of the Grashof number used in cases of natural convection mass transfer problems.
A critical value exists for the Rayleigh number, above which fluid motion occurs.
[2] The first step to deriving the Grashof number is manipulating the volume expansion coefficient,
One involves the energy equation while the other incorporates the buoyant force due to the difference in density between the boundary layer and bulk fluid.
This discussion involving the energy equation is with respect to rotationally symmetric flow.
This analysis will take into consideration the effect of gravitational acceleration on flow and heat transfer.
This equation expands to the following with the addition of physical fluid properties:
From here we can further simplify the momentum equation by setting the bulk fluid velocity to 0 (
This relation shows that the pressure gradient is simply a product of the bulk fluid density and the gravitational acceleration.
The next step is to plug in the pressure gradient into the momentum equation.
To find the Grashof number from this point, the preceding equation must be non-dimensionalized.
This means that every variable in the equation should have no dimension and should instead be a ratio characteristic to the geometry and setup of the problem.
where: The dimensionless parameter enclosed in the brackets in the preceding equation is known as the Grashof number: Another form of dimensional analysis that will result in the Grashof number is known as the Buckingham π theorem.
This method takes into account the buoyancy force per unit volume,
due to the density difference in the boundary layer and the bulk fluid.
With reference to the Buckingham π theorem there are 9 – 5 = 4 dimensionless groups.
In forced convection the Reynolds number governs the fluid flow.
But, in natural convection the Grashof number is the dimensionless parameter that governs the fluid flow.
Using the energy equation and the buoyant force combined with dimensional analysis provides two different ways to derive the Grashof number.
However, above expression, especially the final part at the right hand side, is slightly different from Grashof number appearing in literature.
Following dimensionally correct scale in terms of dynamic viscosity can be used to have the final form.
In a recent research carried out on the effects of Grashof number on the flow of different fluids driven by convection over various surfaces.
[4] Using slope of the linear regression line through data points, it is concluded that increase in the value of Grashof number or any buoyancy related parameter implies an increase in the wall temperature and this makes the bond(s) between the fluid to become weaker, strength of the internal friction to decrease, the gravity to becomes stronger enough (i.e. makes the specific weight appreciably different between the immediate fluid layers adjacent to the wall).
The effects of buoyancy parameter are highly significant in the laminar flow within the boundary layer formed on a vertically moving cylinder.
It can be concluded that buoyancy parameter has a negligible positive effect on the local Nusselt number.
This is only true when the magnitude of Prandtl number is small or prescribed wall heat flux (WHF) is considered.