Stagnation point von Kármán coefficient.
On the basis of various direct numerical simulations (DNS) of turbulent channel flows the following picture is proposed. (i) At a distance y from either wall, the Taylor microscale lambda is proportional to the average distance l(s) between stagnation points of the fluctuating velocity field, i.e., lambda(y)=B(1)l(s)(y) with B(1) constant, for delta(nu) << y < or approximately equal to delta, where the wall unit delta(nu) is defined as the ratio of kinematic viscosity nu to skin friction velocity u(tau) and delta is the channel's half-width. (ii) The number density n(s) of stagnation points varies with height according to n(s)=(C(s)/delta(nu)(3))y(+)(-1) where y(+) identical with y/delta(nu) and C(s) is constant in the range delta(nu) << y < or approximately equal to delta. (iii) In that same range, the kinetic energy dissipation rate per unit mass, equals 2/3(E(+)((u(tau)(3)/kappa(s)y) where E(+) is the total kinetic energy per unit mass normalized by u(tau)(2) and kappa(s)=B(1)(2)/C(s) is the stagnation point von Kármán coefficient. (iv) In the limit of exceedingly large Reynolds numbers Re(tau) identical with delta/delta(nu), large enough for the Reynolds stress -(uv) to equal u(tau)(2) in the range delta(nu) << y << delta, and assuming that production of turbulent kinetic energy balances dissipation locally in that range and limit, the mean velocity U(+), normalized by u(tau), obeys (d/dy)U(+) approximately equal to 2/3(E(+)/kappa(s)y) in that same range. (v) It follows that the von Kármán coefficient kappa is a meaningful and well-defined coefficient and the log law holds in turbulent channel/pipe flows only if E(+) is independent of y(+) and Re(tau) in that range, in which case kappa approximately kappa(s). (vi) In support of (d/dy)U(+) approximately equal to 2/3(E(+)/kappa(s)y), DNS data of turbulent channel flows which include the highest currently available values of Re(tau) are best fitted by E(+) approximately equal to 2/3(B(4)y(+)(-2/15)) and (d/dy(+))U(+) approximately equal to (B(4)/kappa(s))y(+)(-1-2/15) with B4 independent of y in delta(nu) << y << delta if the significant departure from -(uv) approximately equal to u(tau)(2) at these Re(tau) values is taken into account.
Dallas
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Institute for Mathematical Sciences, Imperial College, London, United Kingdom. vassilios.dallas04@imperial.ac.uk
Vassilicos
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Hewitt
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D003198:Computer Simulation; D008962:Models, Theoretical; D017711:Nonlinear Dynamics; D012212:Rheology / Q000379:methods*