# Turbulence

### Introduction

Almost all fluid flow which we encounter in daily life is turbulent. Typical examples are flow around (as well as in) cars, aeroplanes and buildings. The boundary layers and the wakes around and after bluff bodies such as cars, aeroplanes and buildings are turbulent. Also the flow and combustion in engines, both in piston engines and gas turbines and combustors, are highly turbulent. Air movements in rooms are turbulent, at least along the walls where wall-jets are formed. Hence, when we compute fluid flow it will most likely be turbulent. In turbulent flow we usually divide the velocities in one time-averaged part ¯vi, which is independent of time (when the mean flow is steady),and one fluctuating part.

There is no definition on turbulent flow, but it has a number of characteristic features

I. 'Irregularity'. Turbulent flow is irregular and chaotic (they may seem random but they are governed by Navier-Stokes equationThe flow consists of aspectrum of different scales (eddy sizes).

II. 'Diffusivity'. In turbulent flow the diffusivity increases. The turbulence increases the exchange of momentum in e.g. boundary layers, and reduces or delays thereby separation at bluff bodies such as cylinders, airfoils and cars.

III. 'Large Reynolds Numbers'. Turbulent flow occurs at high Reynolds number. For example, the transition to turbulent flow in pipes occurs that ReD ≃ 2300, and in boundary layers at Rex ≃ 500 000.

IV. 'Three-Dimensional'. Turbulent flow is always three-dimensional and unsteady. However,when the equations are time averaged, we can treat the flow as two-dimensional (if the geometry is two-dimensional)

VI. 'Continuum'. Even though we have small turbulent scales in the flow they are much larger than the molecular scale and we can treat the flow as a continuum.

The largest scales are of the order of the flow geometry (the boundary layer thickness, for example), with length scale ℓ0 and velocity scale v0. These scales extract kinetic energy from the mean flow which has a time scale comparable to the large scales, i.e.

                        Part of the kinetic energy of the large scales is lost to slightly smaller scales with which


the large scales interact. Through the cascade process, kinetic energy is in this way transferred from the largest scale to the smallest scales. At the smallest scales the frictional forces (viscous stresses) become large and the kinetic energy is transformed (dissipated) into thermal energy. The kinetic energy transferred from eddy-to-eddy (from an eddy to a slightly smaller eddy) is the same per unit time for each eddy size.

The smallest scales where dissipation occurs are called the Kolmogorov scales whose velocity scale is denoted by vη, length scale by ℓη and time scale by τη. We assume that these scales are determined by viscosity, ν, and dissipation, ε. The argument is as follows.

#### viscosity

Since the kinetic energy is destroyed by viscous forces it is natural to assume


that viscosity plays a part in determining these scales; the larger viscosity, the larger scales.

#### dissipation

The amount of energy per unit time that is to be dissipated is ε. The more energy that is to be transformed from kinetic energy to thermal energy, the larger the velocity gradients must be.

## Energy spectrum

As mentioned above, the turbulence fluctuations are composed of a wide range of scales. We can think of them as eddies, It turns out that it is often convenient to use Fourier series to analyze turbulence. In general, any periodic function, g, with a period of 2L (i.e. g(x) = g(x + 2L)), can be expressed as a Fourier series.

## Incompressible flow of Newtonian fluids

The equations are simplified in the case of incompressible flow of a Newtonian fluid. Incompressibility rules out sound propagation or shock waves, so this simplification is not useful if these phenomena are of interest. The incompressible flow assumption typically holds well even with a "compressible" fluid — such as air at room temperature — at low Mach numbers up to about Mach 0.3. With incompressible flow and constant viscosity, the Navier–Stokes equations read[1]

 Navier–Stokes equations (Incompressible flow) ${\displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+\mu \nabla ^{2}\mathbf {v} +\mathbf {f} .}$

in tensor notation:

 Navier–Stokes equations (Incompressible flow) ${\displaystyle \rho \left({\frac {\partial u_{i}}{\partial t}}+u_{j}{\frac {\partial u_{i}}{\partial x_{j}}}\right)=-{\frac {\partial p}{\partial x_{i}}}+\mu {\frac {\partial ^{2}u_{i}}{\partial x_{j}\partial x_{j}}}+f_{i}}$

Here f represents "other" body forces (per unit volume), such as gravity or centrifugal force. The shear stress term ${\displaystyle \nabla \cdot {\boldsymbol {\mathsf {T}}}}$ becomes ${\displaystyle \mu \nabla ^{2}\mathbf {v} }$, where ${\displaystyle \nabla ^{2}}$ is the vector Laplacian.[2]

## Wall region in fully developed channel flow

The region near the wall is very important. Here the velocity gradient is largest as the velocity drops down to zero at the wall over a very short distance. One important quantity is the wall shear stress which is defined as τw = μ ∂¯v1 ∂x2 ____ w From the wall shear stress, we can define a wall friction velocity, uτ , as wall friction τ velocity w = ρu2τ ⇒ uτ = _ τw ρ _1/2

In order to take a closer look at the near-wall region, let us, again, consider fully developed channel flow between two infinite plates,

### Reynolds stresses in fully developed channel flow

The flow is two-dimensional (¯v3 = 0 and ∂/∂x3 = 0). Consider the x2 − x3 plane, Since nothing changes in the x3 direction, the viscous shear stress τ32 = μ _ ∂¯v3 ∂x2 + ∂¯v2 ∂x3 _ = 0 because ¯v3 = ∂¯v2/∂x3 = 0. The turbulent part shear stress, v′ 2v′ 3, can be expressed using the Boussinesq assumption (see Eq. 11.32) −ρv′ 2v′ 3 = μt _ ∂¯v3 ∂x2 + ∂¯v2 ∂x3 _ = 0 and it is also zero since ¯v3 = ∂¯v2/∂x3 = 0. With the same argument, v′ 1v′ 3 = 0. However note that v′2 3 = v2 3 6= 0. The reason is that although the time-averaged flow

### Boundary layer

Up to now we have mainly discussed fully developed channel flow. What is the difference between that flow and a boundary layer flow? First, in a boundary layer flow the convective terms are not zero (or negligible), i.e. the left side of Eq. 6.13 is not zero. The flow in a boundary layer is continuously developing, i.e. its thickness, δ, increases. The flow in a boundary layer is described by. Second, in a boundary layer flow the wall shear stress is not determined by the pressure drop; the convective terms must also be taken into account. Third, the outer part of the boundary layer is highly intermittent, consisting of turbulent/non-turbulentmotion. However, the inner region of a boundary layer (x2/δ < 0.1) is principally the same as for the fully developed channel flow,the linear and the log-law regions are very similar for the two flows. However, in boundary layer flow the log-law is valid only up to approximately x2/δ ≃ 0.1 (compared to approximately x2/δ ≃ 0.3 in channel flow)

## References

1. See Acheson (1990).
2. Batchelor (1967) pp. 21 & 147.