**5. The LES technique**

The vector expression for the mass conservation equation and the Navier–Stokes equations

( ) <sup>0</sup> ¶ +Ñ× =

*<sup>p</sup> Dt* rr

r

= - +× *ij*

æ ö ¶ ¶ = -- ç ÷ ¢ ¢ ¶ ¶ è ø *j i ij i j j i <sup>u</sup> <sup>u</sup> u u x x*

It can be noted that the mass conservation equation is practically the same for laminar and turbulent flows. However, it is necessary to point out that this equivalence is valid only for the average values with a significant number of samples, meaning that the average of the

Because these stresses vary in their frequencies and sizes as a function of the distance from the walls, turbulent models take into account such variations to solve the closure problem of these

In 1972, Jones and Launder [17] proposed a model with two differential equations in addi‐ tion to the mass and momentum conservation equations. In that model, an equation approxi‐

other one contained the effects of the turbulent kinetic energy dissipation rate "ε". Together, the use of both equations proved to be adequate to obtain good results in turbulent flows whose interest was focused on the study of the influence of average turbulent effects. Both equa‐

> ¶ ¶ ¶¶ ¶ éù æ ö » + +- ê ú ç ÷ ¶ ¶ ¶¶ ¶ ê ú ëû è ø *t ii i t jKj j j i*

n

*Dk k uu u Dt x x x x x*

mated the effect of the turbulent kinetic energy production rate *<sup>k</sup>* <sup>=</sup> <sup>1</sup>

n

s **g**

 t

 r

*V* (1.20)

(1.22)

<sup>2</sup> (*u*' ¯2 + *v*'

(1.23)

ò

¯2 <sup>+</sup> *<sup>w</sup>*' ¯2 ) and the

*<sup>V</sup>* <sup>Ñ</sup> <sup>Ñ</sup> (1.21)

¶*t* r

*D*

tm

are as follows:

332 Numerical Simulation - From Brain Imaging to Turbulent Flows

where:

Laminar Turbulent

extended equations.

**4. RANS models**

tions are as follows:

fluctuating quantities becomes zero.

Large eddy simulation is a technique based on the assumption that the largest eddies in a flow are of sizes relative to the characteristic length of the flow, whereas the smallest ones that form on the walls are isotropic and very similar for different kinds of flow, and therefore being more susceptible to be modeled. Under this consideration, the LES technique solve the largest eddies in an explicit way and the effects of the smallest eddies on largest eddies are modeled using a subgrid scale stress model. The justification for both assumptions is that the largest eddies contain the most part of the kinetic energy in a moving fluid, they transport the majority of conservative properties and they vary the most from flow to flow. Meanwhile, the smallest eddies are considered more "universal" and less important in the total flow; therefore, when they are modeled, it is expected that the error introduced is small. Unlike RANS models, LES technique uses a filtering process instead of an averaging process. It means that the informa‐ tion obtained for the largest scales is instantaneous and does not represent an average information. Instead of considering the influence of the velocity fluctuations equal to zero *u* ′ *ij* ¯ =0 like in the RANS models, it has a value that locally influences the solutions, it means

*u* ′ *ij* ¯ ≠0. In general, the use or the LES technique has produced better results in flow simula‐ tions when RANS models usually fail, like turbulent boundary layers subjected to adverse pressure gradients, the prediction of drag coefficients for immerse bodies in periodic re‐ gimes [21] and other unsteady flows. Governing equations for the LES model in an incom‐ pressible flow are as follows:

$$\frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_i} = \mathbf{0} \tag{1.27}$$

$$\frac{\partial \overline{u}\_{i}}{\partial t} + \frac{\partial \overline{u}\_{i}\overline{u}\_{j}}{\partial \mathbf{x}\_{j}} = -\frac{1}{\rho} \frac{\partial \overline{P}}{\partial \mathbf{x}\_{i}} - \frac{\partial \tau\_{y}}{\partial \mathbf{x}\_{j}} + \frac{\partial}{\partial \mathbf{x}\_{j}} \left| \nu \frac{\partial \overline{u}\_{i}}{\partial \mathbf{x}\_{j}} \right| \tag{1.28}$$

$$(i = 1, 2, 3)$$

In the Eqs. (1.27) and (1.28), *u*¯*<sup>i</sup>* represents the filtered velocity components at different Cartesian positions *xi* , where *P*¯ represents the filtered pressure component, "*<sup>v</sup>*" and "*ρ*" represent the kinematic viscosity and density, respectively. Regarding the local influence of the Reynolds stresses, one of the most used models is the classic Smagorinsky model, which is expressed as follows:

$$
\pi\_{\underline{u}} = \overline{u\_i \underline{u}\_j} - \overline{u}\_i \overline{\underline{u}}\_j \tag{1.29}
$$

$$
\tau\_y - \frac{1}{3}\tau\_{kk}\mathcal{S}\_y = -2\nu\_r\overline{S}\_y = -2\left(C\_s\overline{\Delta}\right)^2||S||\,\overline{S}\_y\tag{1.30}
$$

where the strain tensor of the fluid in each cell is ‖S‖<sup>=</sup> <sup>2</sup>*S*¯ *ijS*¯ *ij*, being *S*¯ *ij* the filtered strain tensor, *vt* the subgrid scale turbulent viscosity, "*Δ*¯"the filter characteristic length. Generally, the size of the filter is proportional to the volume "Ω" of the cell Δ¯ =2(*Ω*) 1/3, and the Smagorinsky constant *Cs* can be varied from 0.05 to 0.1 [22].
