Abstract:
It has been known for sometime that turbulence is due to the ahaatic motion ofa large number of small eddies. By neglecting the non-lineax terms in the Navier-Stokes equations, the authors obtained recently a class of solutions which representthe velocity distxibutians of spherical vortices moving in an incompressible viscousfluid. It is obvious that, after the linearization of the Navier-Stokes equations of mn-Lion, velocitp distributions obtained. by differentiating the spherical vortex solutionpartially with respect to the coordinates along any direction in space for any numberof times also satisfp the equations of motion.We investigate the homagenEaus isotropic turlulerit motion of a liquid in its finalperiod of decay, when the non-linear terms in the Navier-Stokes equations can beneglected-The type ofvortices which are assumed to give rise to turbulence, is deterrnined by the following twoccanditians:In the first place, the vorticity distributionof the eddy should be concentrated in the vicinity of the geometrical center of thevortex instead of being spread over a largre volume; secondly, the total angular mamentom of the vertex should he constant instead of decreasing ar increasing with time. The first cnnditian requires that only the spherical vortex solution with theparameter
k =5/4 and its partial dexivatives shomld be chosen. only in this specialcase, amongall the spherical voxtices, has its voxticity concentrated inthe regionaround its center. Solutions derived by partially differentiating its velocity distribition with respect to the ooaxdinates also have this property when compared with thesolutions obtained from those of the spherical vortices with other values of theparameter
k by similar process.The second condition determines that partial cliff erentiation of the solution of the spherical vortex
k=5/4 with respect to the coordinates only once is sufflclent. Partial differentiation with reRpect to the coordinate along the axis of symmetry of the velocity distribution yields the varticity η, the stream function ψ and the oharacteristio velocity
U given. in the paper [(2.4) and (2.5)]. In the stream function ψ, we have introduced aftez the gaztial differentiation the part due to a uniform flow along the axia of symmetry. Without this part, the solution represents the motion of a standing axially symmetrical vortex. Partial differentiation with respent to the coordinates along any other dizeation in space, leads to the solution of asymmetrical vortices in general[(2.12) and (2.13)]. According to Synge and Lin, the double velocity correlation between two points in the fluid is the product of the velocity components
u#em/em#′
u#em/em#″ averaged ovex all the directions of orientation of the axis of symmetry of the axially symmetrical eddy, and over all the positions of the vortex in the whole fluid (3.2). If the varteg Gcera asymmetriaal, another average over all the independent directions of differenti.a- dons should be added (3.3). Tn the actual computation, for the sake of convenience however, the double scalar vortiaitp anrrelation zs obtained first, and the double velocity correlation is found by solving the differential relations between them. Based upon either the axially symmetrical or aspmmetxiaal vortex as the element of turbulence, the calculatien leads to the same Millionshikov solution (4.4) of the Karman-Howarth equation, in vPhiah the ternus involving the triple velocity corre- Lations are neglected. Tt is well-known that both the velocity correlation and the law of turbulent energy decay of this solution agree with. Batahalor and Townsend's experiments. The present paper paints out finallp that the structure of turbulence, as revesled by the double velocity cnrrel.ation function, depends upon the vortiaity struofurs of its component eddies. h'urthermare, since their velocity distribution of a furbulent fluid in its final. period of decay ie now known, triple and higher order velocity correlations or correlation functions of oilier kinds can tie oalaulaLed. These theoretical predictions can be subjected to the direct experimental. verification.
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