The transport equations for the second-order velocity structure functions 〈(δu)2〉 and 〈(δq)2〉 are used as a scale-by-scale budget to quantify the effect of initial conditions at low Reynolds numbers, typical of grid turbulence. The validity of these equations is first investigated via hot-wire measurements of velocity and transverse vorticity fluctuations. The transport equation for 〈(δq)2〉 is shown to be balanced at all scales, while anisotropy of the large scales leads to a significant imbalance in the equation for 〈(δu)2〉. The effect of using similarity to evaluate the transport equation is rigorously tested. This approach has the desirable benefit of requiring less extensive measurements to calculate the inhomogeneous term of the transport equation. The similarity form of the 〈(δq)2〉 equation produces nearly identical results as those obtained without the similarity assumption. In the case of the 〈(δu)2〉 equation, the similarity method forces a balance at large separation, although the imbalance due to large scale anisotropy remains. The initial conditions of the turbulence at constant R M ≃ 10,400 (28≤ R λ≤ 55) are changed by using three grids of different geometries. Initial conditions affect the shape and magnitude of the second- and third-order structure functions, as well as the anisotropy of the large scales. The effect of initial conditions on the scale-by-scale budget is restricted to the inhomogeneous term of the transport equations, while the dissipation term remains unaffected despite the low R λ. Scales as small as λ are affected by the changes in initial conditions.
Experiments in Fluids – Springer Journals
Published: Aug 2, 2005
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