Stochastic flow approach to model the mean velocity profile of wall-bounded flows

There is no satisfactory model to explain the mean velocity profile of the whole turbulent layer in canonical wall-bounded flows. In this paper, a mean velocity profile expression is proposed for wall-bounded turbulent flows based on a recently proposed stochastic representation of fluid flows dynamics. This original approach, called modeling under location uncertainty, introduces in a rigorous way a subgrid term generalizing the eddy-viscosity assumption and an eddy-induced advection term resulting from turbulence inhomogeneity. This latter term gives rise to a theoretically well-grounded model for the transitional zone between the viscous sublayer and the turbulent sublayer. An expression of the small-scale velocity component is also provided in the viscous zone. Numerical assessments of the results are provided for turbulent boundary layer flows, pipe flows and channel flows at various Reynolds numbers.

Figure 1

Velocity profiles for the turbulent boundary layer at ${\mathrm{Re}}_{\tau }=1306$. The green curve corresponds to the reference data. The blue dot lines show the classic laws (linear then logarithmic), and the red dots correspond to the profile of the model under location uncertainty.

Figure 2

Velocity profiles for the turbulent boundary layer at ${\mathrm{Re}}_{\tau }=1437$. The green curve corresponds to the reference data. The blue dot lines show the classic laws (linear then logarithmic), and the red dots correspond, to the profile of the model under location uncertainty.

Figure 3

Velocity profiles for the turbulent boundary layer at ${\mathrm{Re}}_{\tau }=1709$. The green curve corresponds to the reference data. The blue dot lines show the classic laws (linear then logarithmic), and the red dots correspond to the profile of the model under location uncertainty.

Figure 4

Velocity profiles for the turbulent boundary layer at ${\mathrm{Re}}_{\tau }=1989$. The green curve corresponds to the reference data. The blue dot lines show the classic laws (linear then logarithmic), and the red dots correspond to the profile of the model under location uncertainty.

Figure 5

Velocity profiles for the pipe flow at ${\mathrm{Re}}_{\tau }=180$. The green curve corresponds to the reference data. The blue dot lines show the classic laws (linear then logarithmic), and the red dots correspond to the profile of the model under location uncertainty.

Figure 6

Velocity profiles for the pipe flow at ${\mathrm{Re}}_{\tau }=360$. The green curve corresponds to the reference data. The blue dot lines show the classic laws (linear then logarithmic), and the red dots correspond to the profile of the model under location uncertainty.

Figure 7

Velocity profiles for the pipe flow at ${\mathrm{Re}}_{\tau }=550$. The green curve corresponds to the reference data. The blue dot lines show the classic laws (linear then logarithmic), and the red dots correspond to the profile of the model under location uncertainty.

Figure 8

Velocity profiles for the pipe flow at ${\mathrm{Re}}_{\tau }=1000$. The green curve corresponds to the reference data. The blue dot lines show the classic laws (linear then logarithmic), and the red dots correspond to the profile of the model under location uncertainty.

Figure 9

Mean velocity profiles for the channel flow at ${\mathrm{Re}}_{\tau }\approx 5200$ [32]. The green curve corresponds to the reference data. The blue dot lines show the classic laws (linear then logarithmic), and the red dots correspond to the profile of the model under location uncertainty.