## Modeling drag forces and velocity fluctuation statistics in wall-bounded flows at high Reynolds numbers

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2016-06-22

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Johns Hopkins University

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In this two-part thesis, the long-standing problems of modeling drag forces over smooth and rough surfaces, as well as understanding and modeling velocity fluctuation statistics, are addressed.
In part I, we focus on simulating and modelling rough wall boundary layers. We introduce two techniques to facilitate wall-modeled large-eddy simulations (LES) of turbulent boundary layers: an integral wall modelling technique and a rescaling-recycling inflow generation technique for LES of rough wall turbulent boundary layers. The integral wall model uses the von Karman Polhausen integral method and therefore is algebraic instead of differential. Because of its algebraic nature, the cost of the wall model is Reynolds number independent. Using the integral method, non-equilibrium effects, including flow acceleration, pressure gradient, etc. can be included by solving the vertically integrated momentum equation. The rough wall rescaling-recycling method is a generalization of the flat plate rescaling-recycling inflow generation technique. For the velocity fluctuations at a downstream plane to be recycled for an inflow condition, the downstream velocity signals need to be rescaled according to an inner and an outer length scale. For rough wall boundary layers, the inner layer length scale is imposed by the roughness. To dynamically compute this length scale, we diagnose the dispersive stress and define the inner layer length scale to be the height at which the dispersive stress drops to $10\%$ of its local maximum.
Next, an analytical rough wall model that relates the rough wall topology to its aerodynamic properties is developed and tested by comparing the model predictions with existing experimental and computational measurements. In this analytical rough wall model, the velocity profile within the roughness layer (the layer occupied by the roughness) is modelled as follows: $U=\exp(a(z-h)/h)$, where $U$ is the mean velocity, $z$ is the wall normal coordinate, $h$ is the roughness height and $a$ is the attenuation coefficient; above the roughness, a logarithmic profile $U/u_\tau=1/\kappa\log((z-d)/z_o)$ is used, where $u_\tau$ is the friction velocity, $d$ is the zero-plane displacement, $z_o$ is the effective roughness height and $\kappa$ is the von Karman constants. Initially unknown parameters including $d$ and $z_o$ are determined using fundamental constraints based on velocity continuity, momentum conservation as well as a geometric sheltering model, in which the mutual sheltering among the roughness elements is explicitly accounted for. The model is tested for various rough surfaces, including aligned and staggered arranged cubes, cubes with bi-modal height distribution, cube arrays with Gaussian height distribution, etc. Generally good agreement between the model predictions and LES measurements is found.
In part II of the thesis, new insights for modeling velocity fluctuations in the log region in wall bounded flows at high Reynolds number are presented. First we reformulate the Townsend attached eddy hypothesis and introduce the Hierarchical-Random-Additive-Process formalism. Instead of resorting to a specifically-shaped typical wall eddy, the HRAP formalism represents the space-filling, self-similar, wall-attached eddies as identically, independently distributed random additives. This HRAP formalism is then used to probe the flow physics in the log region. Power-law scaling of the single-point moment generating function of the streamwise velocity fluctuations $\left<\exp(qu)\right>$, where $u$ is the streamwise velocity fluctuation normalized by friction velocity and $q$ is an independent parameter, is predicted. Moreover, a scaling transition in the two-point moment generating function $\left<\exp(qu(x)-qu(x+r))\right>$, where $x$ and $x+r$ are two points separated in the streamwise direction by a distance $r$, is predicted. Those predictions are confirmed using hot-wire measurements from boundary layers at $Re_\tau\sim O(10^4)$, where $Re_\tau$ is the friction Reynolds number, defined based on the boundary layer height and the friction velocity. The measurements are taken from the Melbourne High-Reynolds-Number-Boundary-Layer-Wind-Tunnel.
Next, the HRAP model is used to identify new generalized logarithmic scalings that feature a scaling behavior $\log(\delta/r)$, where $\delta$ is the boundary layer thickness and $r$ is, again, the two-point displacement in the streamwise direction. The same experimental measurements are used to provide empirical support for the new logarithmic laws.

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turbulent boundary layers, drag forces modeling, velocity fluctuation statistics modeling