## A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow.(English)Zbl 1145.76393

Summary: Fully developed incompressible turbulent pipe flow at bulk-velocity- and pipe-diameter-based Reynolds number $$Re_{D}=44000$$ was simulated with second-order finite-difference methods on 630 million grid points. The corresponding Kármán number $$R^{+}$$, based on pipe radius $$R$$, is 1142, and the computational domain length is $$15R$$. The computed mean flow statistics agree well with Princeton Superpipe data at $$Re_{D}=41727$$ and at $$Re_{D}=74000$$. Second-order turbulence statistics show good agreement with experimental data at $$Re_{D}=38000$$. Near the wall the gradient of with respect to $$\ln(1 - r)^{+}$$ varies with radius except for a narrow region, $$70 < (1 - r)^{+} < 120$$, within which the gradient is approximately 0.149. The gradient of with respect to $$\ln{(1 - r)^{+}+a^{+}}$$ at the present relatively low Reynolds number of $$Re_{D}=44000$$ is not consistent with the proposition that the mean axial velocity is logarithmic with respect to the sum of the wall distance $$(1 - r)^{+}$$ and an additive constant $$a^{+}$$ within a mesolayer below 300 wall units. For the standard case of $$a^{+}=0$$ within the narrow region from $$(1 - r)^{+}=50$$ to 90, the gradient of with respect to $$\ln{(1 - r)^{+}+a^{+}}$$ is approximately 2.35. Computational results at the lower Reynolds number $$Re_{D}=5300$$ also agree well with existing data. The gradient of with respect to $$1 - r$$ at $$Re_{D}=44000$$ is approximately equal to that at $$Re_{D}=5300$$ for the region of $$1 - r > 0.4$$. For $$5300 < Re_{D} < 44000$$, bulk-velocity-normalized mean velocity defect profiles from the present DNS and from previous experiments collapse within the same radial range of $$1 - r > 0.4$$. A rationale based on the curvature of mean velocity gradient profile is proposed to understand the perplexing existence of logarithmic mean velocity profile in very-low-Reynolds-number pipe flows. Beyond $$Re_{D}=44000$$, axial turbulence intensity varies linearly with radius within the range of $$0.15 < 1 - r < 0.7$$. Flow visualizations and two-point correlations reveal large-scale structures with comparable near-wall azimuthal dimensions at $$Re_{D}=44000$$ and 5300 when measured in wall units. When normalized in outer units, streamwise coherence and azimuthal dimension of the large-scale structures in the pipe core away from the wall are also comparable at these two Reynolds numbers.

### MSC:

 76F65 Direct numerical and large eddy simulation of turbulence 76M20 Finite difference methods applied to problems in fluid mechanics
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### References:

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