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
THANK Y0U