Related Resources: calculators

Boussinesq Approximation for Buoyancy Formula and Calculator

Fluids Dynamics and Engineering

Boussinesq Approximation for Buoyancy Formula and Calculator

In fluid dynamics, the Boussinesq approximation (pronounced [businɛsk], named for Joseph Valentin Boussinesq) is used in the field of buoyancy-driven flow (also known as natural convection). It ignores density differences except where they appear in terms multiplied by g, the acceleration due to gravity. The essence of the Boussinesq approximation is that the difference in inertia is negligible but gravity is sufficiently strong to make the specific weight appreciably different between the two fluids. Sound waves are impossible/neglected when the Boussinesq approximation is used since sound waves move via density variations.

The advantage of the approximation arises because when considering a flow of, say, warm and cold water of density ρ₁ and ρ₂ one needs only to consider a single density ρ: the difference Δρ = ρ₁− ρ₂ is negligible. Dimensional analysis shows that, under these circumstances, the only sensible way that acceleration due to gravity g should enter into the equations of motion is in the reduced gravity g′ where:

Eq. 1

g_r = g · ( ρ₂ - ρ₁ ) . ρ

(Note that the denominator may be either density without affecting the result because the change would be of order g(Δρ/ρ)² The most generally used dimensionless number would be the Richardson number and Rayleigh number.

The mathematics of the flow is therefore simpler because the density ratio ρ₁ / ρ₂ , a dimensionless number, does not affect the flow; the Boussinesq approximation states that it may be assumed to be exactly one.

Where:

g = Acceleration due to Earth (m/s²)
ρ₂ = Final Density (kg/m³)
ρ₁ = Initial Density (kg/m³)
ρ = Single Density (kg/m³)
g_r = Reduced Acceleration due to Gravity (m/s²)

Source:

• Wikipedia
• Physical fluid dynamics . New York: Van Nostrand Reinhold Co.

Related

• Buoyancy Equation and Review
• Buoyant Force in Liquids Animation
• Fritzsche Formula for Pressure Change at Standard Conditions Equation and Calculator
• Liquid Pressure Drop in Pipe and Pipe Fittings Spreadsheet Calculator
• Pascal's Law Pascal's law or the principle of transmission of fluid-pressure
• Water Vapor Density atmosphere Equation - Water vapor in the atmosphere obeys the ideal gas law
• Head Loss Head loss is the reduction in the total head or pressure of the fluid as it moves through a fluid system.
• Friction Factor Friction factor has been determined to depend on the Reynolds number for the flow and the degree of roughness of the pipes
• Darcy's Equation Darcy's Equation Fluids Flow Equation - also called Darcy Weisbach equation.