Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 108

A first order approach is to relate the Reynolds stresses to the mean rates of defor-
mation, (known as the Boussinesq assumption), similar to the way a laminar shear
stress is related to the velocity gradient. For example:
Laminar shear stress:

Turbulent shear stress:

du d u

τµ = τ µ ρ ≈ ≈− ′′ uu

turb t

dy dx

dv

τ ≈ µ ≈− ′′ ρ vv

turb t

(5.23)

dy
dv dv

τ ≈ µ + ≈− ′′ ρ uv

turb t

dy dy

Equation (5.23) relates the turbulent momentum transport to be proportional to the
mean gradients of velocity. The term µ is sometimes called the eddy viscosity (or
turbulent viscosity) because the mixing effect is based on the motion of small re-
circulating eddies or vortices that transport the fluid particles for momentum ex-
change. The total shear stress is then

du
dy

(5.24)

τµµ =+ ( )

t

The eddy viscosity is not a fluid property but rather is a conceptual term that de-
scribes the Reynolds stresses with mean strain rates (i.e. velocity gradients). It is
dependent on the flow field and determining its value is the objective of RANS
turbulence models. This involves the inclusion of additional equations based on the
convection and diffusion of the flow parameter in question. In the first order method
for relating Reynolds stresses, the additional equations can range from zero to two
which gives rise to its turbulence model classification. This includes:

• Zero equation model: mixing length model.
• One equation model: Spalart-Almaras.
• Two equation models: k-ε models, k-ω models

A second order approach is to directly use the continuity and momentum equa-
tions for the second order moments, which are the Reynolds stresses and turbulent
fluxes instead of the Boussinesq assumption. The motivation is to overcome the
limitations of first order models which assume isotropic turbulence and additional
strains. This results in more additional equations and introduces more unknown
variables. Second order turbulence models include the Reynolds stress models.

5.2.3.1 LES and DNS Approach

During the averaging of the governing equations, some physics may be lost. In the
LES model (Large Eddy Simulation) a filtering function is applied to the equations