Could anyone explain in simple (but not too simple) words what is the Boussinesq approximation, in the momentum equation? Thank you, Gabriel

Hi, Gabriel,

In the presence of gravity field the momentum equation is:

\rho du/dt=-\nabla p + \nabla^2 u/Re -\rho g j

u is the velocity vector, g is gravity acceleration, j is a unit vector directed upward, \rho is the density, du/dt is a total derivative.

Let, \rho(t,x,y,z)=\rho_0+\rho'(t,x,y,z), \rho_0=const, y looks upward. Then one gets:

(\rho_0+\rho') du/dt=-\nabla (p+\rho_g*y) + \nabla^2 u/Re -\rho' g j

Assume that \rho' is small, but g is large so that \rho'*g=O(1). Neglect small terms. Then

\rho' du/dt=-\nabla (p+\rho_g*y) + \nabla^2 u/Re -\rho' g j

This is Boussinesq approximation. It is much easier to solve since \rho_0 before the time derivative is constant, but is quite accurate, say, for air convection in a room. The reasoning can be extended to other, non-gravity, body forces.

Rgds, Sergei.

The Boussinesq approximation is an approximation carried out to introduce the turbulent viscosity (Reynolds Stress tensor) in a compressible flow. This is done by writting down the regular viscous terms in the equations, as if the density was constant in these terms (putting the spatial derivatives of the density to zero in the viscous terms). One also drops the viscous terms proportional to the pressure (usually in the tensor form, these terms appear with a Kroeneker Delta function). In this approximation the viscous coefficient (say for example kinematic viscousity equivalent) nu is proportional to the speed of the turbulence and the size of the larger Eddies. The turbulence has good chance to be subsonic (otherwse the turbulence dissipates quickly due to shocks) so the turbulent velocity is set to the sound speed, while the scale of the larger Eddies is about the scale of the whole domain and the viscous coefficient becomes: nu=c*L, where c is the sound speed and L is the size of the domain. The Reynolds number of such a flow is of order one, but can be smaller of course.

PG

Hi, Patrick,

Well, it seems that what you say is somewhat unclear.

Reynolds stress tensor is not the same as turbulent viscosity. Reynolds stress tensor can be introduced without any assumptions concerning turbulent viscosity, both for compressible and incompressible flows. It is just (M(\rho u^i u^j)-M(\rho)M(u^i)M(u^j)) e_i e_j by definition, if I remember right. M(..) denotes averaging. One just takes the momentum flux tensor in non-averaged flow, subtracts the momentum flux tensor in averaged flow, and averages the result.

The assumption that Reynolds stress tensor is proportional to the rate-of-strain tensor with a scalar proportionality coefficient is called Boussinesq hypothesis (not approximation). The coefficient is called eddy, or turbulent, viscosity. The hypothesis can be used both for compressible and incompressible flows. It is quite strong hypothesis since if we divide Reynolds stress tensor by rate-of-strain tensor the result, generally, is a tensor of 4th rank (kghm, not sure in my English here). That is, generally, T_{i,j}=A_{i,j}^{k,l} E_{k,l} while Boussinesq hypothesis is T_{i,j}=\mu_{turb} E_{i,j}.

Now, there are many particular models for turbulent, or eddy, viscosity. And, well, I had an impression that you were referring to one of them, specific to compressible flow and also called Boussinesq approximation. Is it so? I never heard about it and is quite interested.

By the way, how many Boussinesq approximations are there?

Rgds, Sergei

Boussinesq (like Euler) was a busy man and got his name attached to lots of equations, approximations, etc. If you are interested in the Boussinesq approximation for buoyancy, you might want to look up an old paper of mine. Gray and Giorgini, 1976. The Validity of the Boussinesq Approximation for Liquids and Gases, Int. J. of Heat and Mass Transfer,vol. 19, pp. 545-551.

To Gabriel,

I would like to quote p35 of the "Hydrodynamic stability"(by Drazin and Reid). I hope it help you.

{The Boussinesq equation

To these equations of motion, Rayleigh(1916) applied the Boussinesq approximation, due independently to Oberbeck(1879) and Boussinesq(1903). The basis of this approximation is that there are flows in which the temperature varies little, and therefore the density varies little, yet the buoyancy drives the motion. Then the variation of density is neglected everywhere except in the buoyancy. On the basis of this approximation for small temperature difference between the bottom and top of the layer of fluid,; density = density@bottom temperature*(1-volumetric expansion coeff.(temp-bottom temperature)) ......}

are you referring to the Boussinesq approximation for bouyant forces or for calculating turbulent stresses. in either case you can find the answer in most fluid mechanics books. try some of the ones Jonas has in the cfd online resources page, there are some online books you can check out