7. Turbulence Turbulent Jet

Instantaneous Mean Turbulent Boundary Layer

U(y)

Instantaneous Mean What is Turbulence? • A “random”, 3-d, time-dependent eddying motion with many scales, superposed on an often drastically simpler mean flow

• A solution of the Navier-Stokes equations

• The natural state at high Reynolds numbers

• An efficient transporter and mixer

• A major source of energy loss

• A significant influence on drag: ‒ increases frictional drag in non-separated flow ‒ delays, or sometimes prevents, boundary-layer separation on curved surfaces, so reducing pressure drag.

● “The last great unsolved problem of classical physics” Momentum Transfer in Laminar and Turbulent Flow

• Laminar: ‒ smooth ‒ no mixing of fluid ‒ momentum transfer by viscous forces

• Turbulent: ‒ chaotic ‒ mixing of fluid ‒ momentum transfer mainly by the net effect of intermingling ρ푈퐿 푈퐿 Regime determined by Reynolds number: Re = = μ ν “Low” Re  laminar; “high” Re  turbulent

“High” or “low” depends on the flow and the choice of 푈 and 퐿 Reynolds’ Decomposition v

mean fluctuation

푢 = 푢ത + 푢′ 푣 = 푣ҧ + 푣′ 푝 = 푝ҧ + 푝′ Alternative Notations

• Overbar for mean, prime for fluctuation: 푢 + 푢′ −ρ푢′푣′

• Upper case for mean, lower case for fluctuation: 푈 + 푢 −ρ푢푣 Reynolds Averaging of a Product

푢 = 푢 + 푢′ 푢′ = 0

cf statistics:

σ 푢2 2 2 ′2 σ2 = 푢′2 = − 푢ത2 푢 = 푢 + 푢 Variance 푁

σ 푢푣 푢푣 = 푢 푣 + 푢′푣′ ρ ≡ 푢′푣′ = − 푢ത푣ҧ Covariance 푁

푢푣 = (푢 + 푢′)(푣 + 푣′) = 푢 푣 + 푢푣′ + 푢′푣 + 푢′푣′

푢푣 = 푢 푣 + 푢푣ഥ′ + 푢ഥ′푣 + 푢′푣′

푢푣 = 푢 푣 + 푢′푣′ Effect of Turbulence on the Mean Flow (i) Mass

vA Mass flux: ρ푣퐴

Average mass flux: ρ푣퐴

The mean velocity satisfies the same continuity equation as the instantaneous velocity Effect of Turbulence on the Mean Flow (ii) Momentum

vA (푥-)momentum flux: (ρ푣퐴)푢 = ρ(푢푣)퐴 u Average momentum flux : (ρ푣퐴)푢 = ρ(푢 푣 + 푢′푣′)퐴 extra term

Net rate of transport of momentum per unit area ρ푢′푣′ from LOWER to UPPER ...

... is equivalent to −ρ푢′푣′ from UPPER to LOWER ...

... has the same dynamic effect as a stress −ρ푢′푣′

The mean velocity satisfies the same momentum equation as the instantaneous velocity, except for additional apparent stresses; e.g. −ρ푢′푣′, −ρ푢′푢′ These are called the Reynolds stresses Total Stress

In simple shear: y 휕푢ത  τ = μ −ρ푢′푣′ 휕푦  ถ turbulent viscous stress stress

τ may be regarded as EITHER: ‒ the force per unit area exerted by upper on lower fluid OR ‒ the rate of transport of momentum (per unit area) from upper to lower fluid

The dynamic effect (average rate of transfer of momentum) is the same Turbulent Transport

B ′ ′ v' 푢 푣 is negative

A τ ≡ −ρ푢′푣′ is positive

U Velocity Fluctuations

Normal stresses: 푢′2, 푣′2, 푤′2 v u Shear stresses: 푣′푤′, 푤′푢′, 푢′푣′

1 Turbulent kinetic energy: 푘 = (푢′2 + 푣′2 + 푤′2) 2

2푘 root − mean − square fluctuation 푢′ 3 Turbulence intensity: = rms = mean velocity 푈 푈 Effect of Turbulence on the Mean Flow (iii) General Scalar

vA Scalar flux: ρ푣퐴 ϕ = ρ(푣ϕ)훢 

Average scalar flux: (ρ푣퐴)ϕ = ρ (푣 ϕ + 푣′ϕ′)퐴

extra term

The mean concentration satisfies the same transport equation as the instantaneous concentration, except for additional turbulent fluxes ρ푣′ϕ′ The Closure Problem

• To close the mean flow equations one must specify the Reynolds stresses −ρ푢′2, −ρ푢′푣′, … and other turbulent fluxes

• A means of doing so is called a turbulence model or turbulence closure

• The simplest model assumes that the turbulent stress can be modelled in a similar manner to the viscous stress. In simple shear: 휕푢 −ρ푢′푣′ = μ 푡 휕푦 This is called an eddy-viscosity model Turbulence Energy: Production and Dissipation

PRODUCTION large eddies by mean flow gradients

ENERGY CASCADE

DISSIPATION small eddies by viscosity Anisotropy

y y

2 constant-stress layer u -uv v2 U u i uj

푢′2 ≠ 푣′2 ≠ 푤′2 Turbulence Transport

• Turbulence is transported by the flow

• Each Reynolds stress satisfies its own scalar-transport equation

• The “source” term is a balance between: ‒ production by mean-velocity gradients ‒ dissipation by viscosity ‒ redistribution by pressure fluctuations

• Turbulence anisotropy is a result of: ‒ anisotropic production by particular velocity gradients ‒ anisotropic production or damping by particular body forces (e.g. buoyancy) ‒ selective damping of wall-normal fluctuations Production of Turbulence Energy

휕푈 휕푈 휕푈 푃 = −2(푢푢 + 푢푣 + 푢푤 ) 11 휕푥 휕푦 휕푧

휕푉 휕푉 휕푉 휕푈 휕푈 휕푈 푃 = −(푢푢 + 푢푣 + 푢푤 ) − (푣푢 + 푣푣 + 푣푤 ) 12 휕푥 휕푦 휕푧 휕푥 휕푦 휕푧

휕푈푗 휕푈푖 푃푖푗 = −(푢푖푢푘 + 푢푗푢푘 ) 휕푥푘 휕푥푘 1 푃(푘) = (푃 + 푃 + 푃 ) 2 11 22 33

휕푈 In simple shear: 푃 = −2푢푣 , 푃 = 푃 = 0 11 휕푦 22 33 휕푈 푃(푘) = −푢푣 휕푦

Energy is preferentially supplied to the streamwise normal stress, 푢푢 PRODUCTION ADVECTION by mean velocity

푢2 푣2 푤2 REDISTRIBUTION by pressure fluctuations

DISSIPATION by viscosity Free Shear Flows

y y y y y y

 y 2 uv u2  y u2 u v2  y 2 v uv

U u u U u u U u u i j i j i j  U  U  U

Mixing layer Wake Jet

• Maximum turbulence where |푈/푦| largest

• 푢푣 has the opposite sign to 푈/푦 and vanishes where this is zero

• Turbulence is anisotropic: 푢2 > 푣2 Wall-Bounded Flows

D U

Pipe flow y y

2 constant-stress layer u -uv v2 U u i uj

Flat-plate boundary layer Parameters in Wall-Bounded Flows

U(y)

Wall shear stress: τ푤

2 Friction velocity: τ푤 = ρ푢τ

푢τ = τ푤/ρ

푦푢τ Non-dimensional height: 푦+ = ν wall units 푈 + } Non-dimensional velocity: 푈 = 푢τ Key Regions of a Turbulent Boundary Layer 휕푈 τ ≈ τ = μ − ρ푢푣 y y 푤 휕푦

Viscous sublayer (푦+ < 5): – viscous stress dominant 휕푈 τ푤 τ푤 = μ 푈 = 푦 휕푦 μ 2 constant-stress layer u -uv v2 U u i uj Log layer (30 < 푦+ < 500): – direct effect of viscosity negligible 휕푈 푢 휕푈 푢τ 푢τ ∝ τ ⇒ = ⇒ 푈 = ln 푦 + constant 휕푦 푦 휕푦 κ푦 κ For smooth walls: 푈 1 푦푢 푈 1 푦푢 = ln τ + 퐵 or = ln 퐸 τ 푢τ κ ν 푢τ κ ν In wall units: 푈 푦푢 (viscous sublayer) 푈+ = 푦+ 푈+ = , 푦+ = τ 푢 ν 1 τ log layer 푈+ = ln 푦+ + 퐵 κ Summary (1)

• Turbulence is a 3-d, time-dependent eddying motion with many scales, causing continuous mixing

• Any instantaneous flow variable may be decomposed as mean + fluctuation

• The process of averaging turbulent variables is called Reynolds averaging and leads to the Reynolds-averaged Navier-Stokes (RANS) equations

• The product of turbulent fluctuations makes a net contribution to the transport of momentum and other quantities via Reynolds stresses and turbulent fluxes Summary (2)

• A turbulence model is a means of specifying turbulent fluxes in order to close the mean-flow equations

• Turbulent energy is: ‒ generated (anisotropically, at large scale) by mean-velocity gradients ‒ redistributed amongst components by pressure fluctuations ‒ dissipated (at small scale) by viscosity

• Turbulence modelling is guided by theory and experiments for fundamental free and wall-bounded flows: in particular, by the log law