DE LA GESTIÓN DE INVENTARIOS

When a fluid travels over a solid surface, a near-wall layer develops where vis- cous forces become important; this layer is known as a boundary layer. Boundary layers develop due to the velocity no-slip condition on solid surfaces. When fluid passes over a flat plate for instance, a boundary layer forms at the front edge of the plate and then develops as it moves downstream. The boundary layer in the region close to the front of the plate is laminar, as the local Reynolds number is low. Downstream, the boundary layer will undergo transition to turbulence as local Reynolds number increases. A schematic of this process is shown in Fig- ure 1.3. Channel flow, due to its periodic boundary conditions, develops only in time.

A turbulent boundary layer contains an inner layer and an outer layer. Within the inner layer viscous effects dominate and the the rate of turbulent kinetic en- ergy production exceeds dissipation. Turbulent kinetic energy can be produced by fluid shear, friction or buoyancy [64]. Therefore, the eddies created by wall- bounded vortices and the fluid shear caused by the no-slip condition at the wall all produce turbulent kinetic energy in the inner layer. In the outer layer inertial effects dominate and turbulent dissipation exceeds production. In between the

CHAPTER 1. INTRODUCTION 6

Figure 1.4: Turbulent boundary layer mean-velocity profile, with different lay-

ers/sublayers annotated [34].

two layers is the log-law region, where turbulent energy production equals dissi- pation. Figure 1.4 shows the mean-velocity profile of a typical turbulent boundary layer. As can be seen, the inner layer is comprised of a viscous sublayer, a buffer layer and a log-law region. The mean-velocity profile in the viscous sublayer is linear. The log-law region is so-called because the mean-velocity profile in this region obeys the semi-empirical model: U+ = 1_k log y++ C, where k is the Von Karman constant (thought to be universal at k_{≈ 0.4) and C is a constant specific} to a boundary layer. The log-law region is where both viscous effects and inertial effects play an important role. In channel flow, the outer layers from the upper and lower walls meet in the centre of the channel and interact. However, both walls still have their own inner layer velocity profiles as in Figure 1.4, although their shape may be influenced by the interaction of the outer layers. Control via wall-based actuation will have the greatest effect on the region of the flow closest to the wall, i.e. the inner layer.

Figure 1.5 shows mean velocity profiles for laminar and turbulent channel flows, where the channel walls are located at y = _{±1. The mean velocity profile for} laminar channel flow is an exact solution to the Navier-Stokes equations, and is parabolic. The skin-friction drag at each wall is proportional to the wall-normal gradient of the spatial-mean velocity profile which for a non-dimensionalised chan- nel flow is defined as:

¯ D±1 := 1 Re ∂_hUi ∂y     y=±1 . (1.8)

Figure 1.5 shows that the gradient of the turbulent mean velocity profile at the walls is much larger than for the laminar flow. Hence why turbulent flows have higher skin-friction drag. Therefore, for constant Reynolds number flow, the only way to reduce skin-friction drag is by reducing the gradient of the mean velocity profile, either directly or (usually) indirectly.

1.3. COHERENT STRUCTURE AND MOTION 7
U 0 0.2 0.4 0.6 0.8 1 y -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Figure 1.5: Plots of mean velocity profiles for laminar (_{−−) and Re = 2230 turbulent} (_{−) channel flow.}

The skin-friction coefficient for a turbulent channel flow is defined as [37]: Cf := τ_w∗ 1 2ρ ∗_U∗ b 2, (1.9)

where U∗_b is bulk velocity which is defined as:

U∗_b := 1 2h∗ Z h∗ −h∗hU ∗_{i dy}∗ . (1.10)

Fukagata et al. [37] derived an equation, commonly known as the FIK identity, which expresses the skin-friction coefficient as a sum of the laminar and turbulent contributions to skin-friction drag. The FIK identity for a fully-developed channel flow is given as [37]: Cf = 12 Reb |{z} Laminar + 12 Z 1 −1 (_−y)(−hu0_v0_i)dy | {z } Turbulent , (1.11)

where Reb is bulk Reynolds number defined as:

Reb :=

U∗_bh∗

ν∗ , (1.12)

and _hu0_v0_{i is a Reynolds stress, where u}0 _{:= U}∗_{− U}∗
_/U∗

b and v0 := V∗/U∗b. As

a constant mass flow rate is assumed, bulk velocity non-dimensionalised by U∗_cl will be Ub = 2/3. From (1.11) we see that the laminar contribution to skin-

CHAPTER 1. INTRODUCTION 8

turbulent contribution is a weighted integral of the Reynolds stress term_−hu0_v0_i.

The weightings are such that the turbulent drag contribution is largest nearest the walls. The FIK identity shall be used in later chapters to analyse the effects of wall-based controllers on turbulent channel flow.