Turbulence Lecture 13 ME EN 515 Andrew Ning aning@byu.edu - - PowerPoint PPT Presentation

ME EN 515 Andrew Ning aning@byu.edu

Turbulence

Lecture 13

Turbulent Length Scales

The mixing transition in turbulent flows, Paul E. Dimotakis

Re = 2,500 Re = 10,000

Direct Numerical Simulation (DNS)

https://www.youtube.com/watch?v=4KeaAhVoPIw

Reynolds Averaged Navier Stokes (RANS)

Expected Value

Reynolds Decomposition

2D Incompressible Navier Stokes

∂u ∂x + ∂v ∂y = 0

∂u ∂t + u∂u ∂x + v ∂u ∂y = −1 ρ ∂p ∂x + ν ✓∂2u ∂x2 + ∂2u ∂y2 ◆

(x-dir only)

Time Averaging

∂hui ∂x + ∂hvi ∂y = 0

∂hui ∂t + hui∂hui ∂x + hvi∂hui ∂y = 1 ρ ∂hpi ∂x + ν ✓∂2hui ∂x2 + ∂2hui ∂y2 ◆

• ✓ ∂

∂xhu0u0i + ∂ ∂y hu0v0i ◆

Turbulent Kinetic Energy Equation

∂k ∂t + hUii ∂k ∂xi

+ ∂ ∂xi ✓1 2hu0

iu0 ju0 ji + 1

ρhu0

ip0i 2νhu0 js0 iji

◆

= hu0

iu0 ji∂hUji

∂xi

2νhs0

ijs0 iji

Reynolds Stress Tensor (Boussinesq hypothesis)

Turbulence Models

Vorticity Equation

some vector calculus identities: r ⇥ (rφ) = 0

r ⇥ ( ~ A) = (r ⇥ ~ A) + r ⇥ ~ A

1 2r( ~ A · ~ A) = ( ~ A · r) ~ A + ~ A ⇥ (r ⇥ ~ A)

r ⇥ ( ~ A ⇥ ~ B) = ~ A (r · ~ B) ~ B (r · ~ A) + ( ~ B · r) ~ A ( ~ A · r) ~ B

r · (r ⇥ ~ A) = 0

Turbulent Boundary Layers

Clauser, 1956.

Law of the Wall

y δ

τ τw

1 1 viscous stress Reynolds stress

1 viscous inner layer fractional contribution to stress: Reynolds y+

50 10 20 30 40

Viscous sublayer (laminar inner sublayer)

Log law

Large Eddy Simulation (LES)

https://www.youtube.com/watch?v=iHx_VycHIQc