An Introduction to Computational Fluid Dynamics Versteeg

48 Turbulenc. .nd Its modelling

f1&.3.7 Mergmg ofturbulent pol' andtransition to twbulcnce Ina naruraJ flat plateboundary layer

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flow, the intcrmlucm formation of turbulent slugs takes place at Reynolds numbersaround 2000 gi\ ing rise to alternate turbulent and laminar regions along the lengthof the pipe. At Reynolds numbers above 2300 the turbulent slugs link up and theentire pipe is filled" ith turbulent flow.

Final comments

It is clear from the above descriptions of transition in jets, flat plate boundary layersand pipe flows that there are a number of common features in the transitionprocesses: (i) the amplification of initially small disturbances, (ii) the developmentof area') with concentrated rotational structures. (Iii) the formation of intense smallscale motions and finally (iv) the growth and merging of these areas of small scalemotions uno fully turbulent fio" s.

The transition to turbulence is strongly uffecied by factors such as pressuregradient, disturbance levels, wall roughness and heat transfer. The discussions onlyapply to subsonic incompressible flows, The appearance of significant compress-ibility effects in flow) at Mach numbers above about 0.7 greatly complicates thestability theory.

It should be noted that although a great deal has been learnt from simple flowsthere is no comprehensive theory of transition. Recent advances in supercomputertechnology have made it possible to simulate the events leading up to transition,including turbulent spot formation, by solving the complete. time-dependent Navier-Stokes equations at modest Reynolds numbers for a number of very SImplegeometries. Kleiser and Zang (1991) give a review of the state of the art whichhighlights very favourable agreement between experiments and (extremelyexpensive) computations.

For engineering purposes the major case where the transition process influences asizeable fraction of the flow IS that of external \\'311 boundar) layer flows atintermediate Reynolds numbers. This occurs in certain rurbomachines, helicopter

56 TurtMJlence and Its modelling

F12.3.9 V~uah lion of a jet flow