Boussinesq approximation (buoyancy)
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In
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
, the Boussinesq approximation (, named for
Joseph Valentin Boussinesq Joseph Valentin Boussinesq (; 13 March 1842 – 19 February 1929) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat. Biography From 1872 to 1886, he was appoi ...
) is used in the field of buoyancy-driven flow (also known as
natural convection Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convect ...
). It ignores density differences except where they appear in terms multiplied by , the acceleration due to gravity. The essence of the Boussinesq approximation is that the difference in
inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
is negligible but gravity is sufficiently strong to make the
specific weight The specific weight, also known as the unit weight, is the weight per unit volume of a material. A commonly used value is the specific weight of water on Earth at , which is .National Council of Examiners for Engineering and Surveying (2005). ''Fu ...
appreciably different between the two fluids.
Sound waves In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
are impossible/neglected when the Boussinesq approximation is used since sound waves move via density variations. Boussinesq flows are common in nature (such as atmospheric fronts, oceanic circulation,
katabatic wind A katabatic wind (named from the Greek word κατάβασις ''katabasis'', meaning "descending") is a drainage wind, a wind that carries high-density air from a higher elevation down a slope under the force of gravity. Such winds are sometim ...
s), industry ( dense gas dispersion, fume cupboard ventilation), and the built environment (natural ventilation,
central heating A central heating system provides warmth to a number of spaces within a building from one main source of heat. It is a component of heating, ventilation, and air conditioning (short: HVAC) systems, which can both cool and warm interior spaces. ...
). The approximation is extremely accurate for many such flows, and makes the mathematics and physics simpler.


The approximation

The Boussinesq approximation is applied to problems where the fluid varies in temperature from one place to another, driving a flow of fluid and
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
. The fluid satisfies
conservation of mass In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass can ...
, conservation of
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
and
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
. In the Boussinesq approximation, variations in fluid properties other than density are ignored, and density only appears when it is multiplied by , the gravitational acceleration. If is the local velocity of a parcel of fluid, the
continuity equation A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
for conservation of mass is : \frac + \nabla\cdot\left(\rho\mathbf\right) = 0. If density variations are ignored, this reduces to The general expression for conservation of momentum of an incompressible, Newtonian fluid (the
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
) is :\frac + \left( \mathbf\cdot\nabla \right) \mathbf = -\frac\nabla p + \nu\nabla^2 \mathbf + \frac\mathbf, where (nu) is the
kinematic viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
and is the sum of any
body force In physics, a body force is a force that acts throughout the volume of a body. Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref> Forces due to gravity, electric fields and magnetic fields are examples of body forces. Bo ...
s such as
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
. In this equation, density variations are assumed to have a fixed part and another part that has a linear dependence on temperature: :\rho = \rho_0 - \alpha\rho_0(T-T_0), where is the coefficient of
thermal expansion Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions. Temperature is a monotonic function of the average molecular kinetic ...
. The Boussinesq approximation states that the density variation is only important in the buoyancy term. If F = \rho \mathbf is the gravitational body force, the resulting conservation equation is In the equation for heat flow in a temperature gradient, the heat capacity per unit volume, \rho C_p, is assumed constant and the dissipation term is ignored. The resulting equation is where is the rate per unit volume of internal heat production and k is the
thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
. The three numbered equations are the basic convection equations in the Boussinesq approximation.


Advantages

The advantage of the approximation arises because when considering a flow of, say, warm and cold water of density and one needs only to consider a single density : the difference is negligible.
Dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as m ...
shows that, under these circumstances, the only sensible way that acceleration due to gravity should enter into the equations of motion is in the reduced gravity where :g' = g\frac. (Note that the denominator may be either density without affecting the result because the change would be of order .) The most generally used
dimensionless number A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
would be the
Richardson number The Richardson number (Ri) is named after Lewis Fry Richardson (1881–1953). It is the dimensionless number that expresses the ratio of the buoyancy term to the flow shear term: : \mathrm = \frac = \frac \frac where g is gravity, \rho is de ...
and
Rayleigh number In fluid mechanics, the Rayleigh number (, after Lord Rayleigh) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free (or natural) convection. It characterises the fluid's flow regime: a value in a certain ...
. The mathematics of the flow is therefore simpler because the density ratio , a
dimensionless number A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
, does not affect the flow; the Boussinesq approximation states that it may be assumed to be exactly one.


Inversions

One feature of Boussinesq flows is that they look the same when viewed upside-down, provided that the identities of the fluids are reversed. The Boussinesq approximation is ''inaccurate'' when the dimensionless density difference is of order unity. For example, consider an open window in a warm room. The warm air inside is less dense than the cold air outside, which flows into the room and down towards the floor. Now imagine the opposite: a cold room exposed to warm outside air. Here the air flowing in moves up toward the ceiling. If the flow is Boussinesq (and the room is otherwise symmetrical), then viewing the cold room upside down is exactly the same as viewing the warm room right-way-round. This is because the only way density enters the problem is via the reduced gravity which undergoes only a sign change when changing from the warm room flow to the cold room flow. An example of a non-Boussinesq flow is bubbles rising in water. The behaviour of air bubbles rising in water is very different from the behaviour of water falling in air: in the former case rising bubbles tend to form hemispherical shells, while water falling in air splits into raindrops (at small length scales
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...
enters the problem and confuses the issue).


References


Further reading

* * *{{cite book, title=Physical Fluid Dynamics, first=D.J. , last=Tritton , author1-link=David Tritton , year=1988 , isbn=978-0-19-854493-7 , edition=Second , publisher =
Oxford University Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
Fluid dynamics Buoyancy