A preliminary investigation has been made concerning the effects of polyethylene oxide on the spectral density of turbulent fluctuations in a boundary layer. The Polyox solution was injected into a two-dimensional boundary layer along a flat plate which was towed at 9.5 fps in the Naval Academy's 85-ft tow tank. Measurements of the mean and fluctuating components of the velocity near the wall were made with conical hot film sensors. Polymer injection resulted in an increase in rms level of the fluctuations and in an increase in spectral density below 200 Hz. Above this frequency, the spectral density decreased, the effect being more pronounced at higher frequencies and at higher polymer concentrations. Measurements of the mean and fluctuating components of the wall shear stress were made with flush-mounted hotfilm sensors submerged in the viscous sublayer. Polymer injection resulted in a decrease in mean shear, in rms level, and in spectral density of the fluctuating wall shear stress. The latter effect increased with increasing frequency and all effects increased with higher molecular weight of the additives. All results suggest the polymer shifts the scale of turbulence away from the energy-dissipating small eddies. Nomenclature A = bridge voltage squared at zero flow B = King's formula constant db = decibels (•• -fluctuating voltage CL = fluctuating linearized voltage Ei, = anemometer bridge voltage Eh = time-averaged bridge voltage EL = linearized voltage / = frequency F c (f) = normalized voltage spectral density F,t(f) = normalized velocity spectral density G x (f) = power spectral density of quantity x in a 1-11/ band l7. r (/)A/ = power spectral density of quantity x in a yVoctave band k\ = one-dimensional wave number K = constant m = linearizer exponent n = King's formula exponent Af = temperature difference between sensor and environment TJ. -turbulence intensity n = fluctuating local velocity V = local velocity I? = time-average local velocity r\ : = convection velocity T W = wall shear stress TV = time-averaged wall shear stress r,,/ = fluctuating wall shear stress
The combined production-inventory and capacity expansion problem is modeled as a linear, integer program. The model assumes constant returns to scale in the production function of a firm which must meet, at minimum cost, deterministic demands for a single product over N periods with no backordering. A linear transformation is used to obtain an equivalent form of the model which is then decomposed into fixed cost and variable cost parts. A global optimum is obtained by enumerating on the fixed cost variables and solving transportation sub-problems with the remaining variables. Special demand and cost structures and extensions are discussed, and computational experience presented.
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