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THE BOUSSINESQ EDDY-VISCOSITY ASSUMPTION

As discussed in the previous section, the RANS equation is unclosed. The Reynolds stresses introduced six new unknowns into the problem. Several methods could be used to estimate this tensor:

Adding more equations to the problem: It is possible to consider various moments of the Navier-Stokes equation. A moment equation is simply the averaged NS equations multiplied by a fluctuating component, such as u_i’. However, for each additional equation, additional unknowns are also being generated. The closure problem remains unsolved.
Modeling the Reynolds stresses: Boussinesq (1877) postulated that the transfer of momentum due to turbulent eddies could be modeled with an eddy viscosity [1]. This is in analogy with Newton’s hypothesis, which assumes that the viscous stresses τ_{vis _i, j} are proportional to the strain rate tensor S_i, j. The molecular viscosity μ simply represents the coefficient of proportionality.
(1) τ_{vis _i, j} = 2μS_i, j = μ⎛⎝(∂u_i)/(∂x_j) + (∂u_j)/(∂x_i)⎞⎠
For two-dimensional cases such as the channel flow, Eq.(1↑) reduces to the well known
(2) τ_{vis _x, y} = μ(∂u)/(∂y)
where x is the stream-wise direction and y the distance from the wall. u is the velocity in the x-direction. It is important to realize that Eq.(1↑) is an hypothesis. This linear assumption provides very good results for a wide range of fluids, including air in standard conditions. However, some fluids behave differently. For example, blood is a fluid whose viscous stresses are not proportional to the strain rates. These are called non-Newtonian fluids. In the turbulent case, Boussinesq postulated that the Reynolds stresses would behave in a manner similar to the viscous stresses. The turbulent stresses can therefore be written
(3) τ_i, j = ρu_i’u_j’ = 2μ_tS_i, j − (2)/(3)ρkδ_ij
where μ_t is called the eddy viscosity, S_i, j is the mean strain rate tensor and k = (1)/(2)u_i’u_i’ is the turbulent kinetic energy. The symbol δ_ij is called the Kronecker delta. This function is 1 if i = j and 0 otherwise. The reader probably noticed the apparition of a new term in Eq.(3↑). In order to understand the reason why this term was introduced, consider the trace of the turbulent stress tensor without this last term.
(4) τ_i, i = ρu_i’u_i’ = 2ρk = 2μ_t(∂u_i)/(∂x_i) = 0
for an incompressible flow (from the continuity equation). Hence, the last term in Eq.(3↑) is included to ensure that the turbulent kinetic energy k is not zero everywhere in the flow field.

Issues related to the Boussinesq assumption

The Boussinesq approximation is widely used in the computational fluid dynamics (CFD) world. Most of the codes currently used in industry are based on this hypothesis. It is therefore important to recognize the limits of this approach. For example, consider a fully developed turbulent channel flow. x is the stream-wise direction and y the distance from the wall. The velocities in the x, y and z directions are respectively u, v and w. In the fully developed region,

(5) (∂U)/(∂x) = (∂V)/(∂y) = (∂W)/(∂z) = 0

so that Eq.(3↑) becomes

(6) ⎧⎪⎨⎪⎩ u’² − 2 ⁄ 3k = ν_t[0] v’² − 2 ⁄ 3k = ν_t[0] w’² − 2 ⁄ 3k = ν_t[0]

Boussinesq assumption predicts u’² = v’² = w’², but experiments (see Fig. 1↓) clearly show that u’² > w’² > v’² [2]. Hence, it appears that a linear eddy viscosity hypothesis fails to predict reliably the anisotropy of turbulence. In order to overcome this problem, a non linear model should be used.