ABSTRACT
The starting point of all simulation methods for turbulent flows, whether Direct Numerical Simulations (DNS), Large Eddy Simulations (LES) or RANS methods, are the Navier-Stokes equations, together with a corresponding equation for scalar quantities such as temperature or species concentration. For incompressible flows these equations expressing the conservation laws for mass, momentum, thermal energy/ species concentration read in tensor notation1
Mass conservation: continuity equation
∂ ∂
=
u
(2.1)
Momentum conservation: Navier-Stokes equations
∂ ∂
+ ∂ ∂
= −
∂ ∂
+ ∂
∂ ∂ +
u
t
u u
x p x
u
ρ ν
ρ ρ− ρ
(2.2)
Thermal energy/species concentration conservation:
∂ ∂
+ ∂ ∂
=
∂ ∂ ∂
+ φ φ φ
u
x x Si
(2.3)
where ui is the instantaneous velocity component in the direction xi, p is the instantaneous static pressure, and φ is a scalar quantity which may stand for either temperature T or species concentration C. Sφ is a volumetric source/sink term expressing, for example, heat generation due to chemical or biological reactions or the settling of suspended sediment. ν and Γ are the molecular (kinematic) viscosity and diffusivity (of φ) respectively. Use has been made in the above equations of the Boussinesq approximation so that the
influence of variable density appears only in the buoyancy term, which is the last term on the right hand side of Equation (2.2) involving the reference density ρr and the gravitational acceleration gi in direction xi. Together with an equation of state relating the local density ρ to the local values of T and C, Equations (2.1) to (2.3) form a closed set and are exact equations describing all details of the turbulent motion, including all fluctuations. The Direct Numerical Simulation (DNS) method solves these equations with a suitable numerical technique, introducing no model. As discussed in the Introduction (Chapter 1), such calculations are not feasible in the foreseeable future for practical flows usually having high Reynolds numbers as the computing effort for resolving all scales including the small-scale dissipative motion would be excessive.
