Fluid turbulence

TRANSITION TO TURBULENCE

S.HARIRAM AE08S022

* Turbulent flow often originates as an instability of laminar flows. * Experimental results suggest that the transition is related to presence of a disturbance and its amplification. * The instabilities at least during the initial stage of their development may be understood using linear stability analysis. * For free shear flows, the primary instabilities are inviscid. * For boundary layer flows, it depends critically upon viscosity.

Osborne Reynolds experiment * A fine line of dye is injected upstream at the centre of the pipe * Flow velocities are increased by small stages Observations

Linear Instability theory * It explains the mechanism of transition from laminar to turbulent state by 1.superimposing the disturbance on the main flow.

2.examining the decay or amplification of the disturbance. * If the perturbations die away with time , the basic flow is said to be stable. * If the perturbations grow in time, the basic flow is unstable. (i.e., the transition to turbulent flow is possible)

* Only first order (linear) terms are considered.

Start with 2-D incompressible, unsteady flow: Navier-Stokes equations:

Continuity

X -momentum

Y -momentum

substituting u = U + u'; v = v'; p = P + p' .....

For 2-D flow, the Stream-function can be defined as:

wave number

angular frequency damping

substituting this in NSEs

degree of

Orr Sommerfeld equation – in linear fourth order ODE form * an eigen value problem which

* right combination of wave number , c and Re gives non trivial solutions

Rayleigh's criterion : “ The existence of inflexion point in the velocity profile implies the necessary condition for instability “

Squire's theorem : “ Two dimensional disturbances become unstable earlier than the three dimensional disturbances ”

Comparison with experiments

Flow type Stability analysis Re.crit Poiseuille (pipe) stable for all Re Poiseuille (plane) 7696 Plane Couette stable for all Re Free shear flow 0 Boundary layer flat plate (Blasius profile) 1.12*10^5

Experiments Re.crit 2000 1350 370 4 3.5-6*10^5

Factors affecting Transition

* Free sream turbulence

* Pressure gradient

* Surface roughness

* Density gradient

* Surface curvature

* Surface cooling

Free shear flows - Mechanism

Boundary layer transition-mechanism

secondary 3D instability

flo w

di re

ct io n

x

turbulent boundary layer

outflow

turbulent spots

Branch II

exponential growth of 2D TS waves Branch I

flat plate

inflow

Λ −vortices

hairpin vortices

SUMMARY Laminar Instability Inviscid

Viscous

Coherent vortices

Unstable TS waves

Merging of coherent vortices

Hair-pin vortices

2-D Turbulence

Turbulent spots

Turbulent

Transition

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