This paper extends the two-level Variational MultiScale (VMS) method for Large-Eddy Simulation, introduced by Hughes et al. [Comput. Visual. Sci., 3:47-59 (2000)], to a threelevel approach that clarifies the role of unresolved scales of motion on the resolved scales.It is shown that the multi-scale framework does not obviate modeling on the large-scales, but that VMS allows for different modeling assumptions to be used on each scale range. In particular, one can choose to neglect the effect of the unresolved scales on the large scales which is reasonable for sufficiently large scale separation.

Surrogate-based optimization methods have become established as effective techniques for engineering design problems through their ability to tame nonsmoothness and reduce computational expense. In recent years, supporting mathematical theory has been developed to provide the foundation of provable convergence for these methods. One of the requirements of this provable convergence theory involves consistency between the surrogate model and the underlying truth model that it approximates. This consistency can be enforced through a variety of correction approaches, and is particularly essential in the case of surrogate-based optimization with model hierarchies. First-order additive and multiplicative corrections currently exist which satisfy consistency in values and gradients between the truth and surrogate models at a single point. This paper demonstrates that first-order consistency can be insufficient to achieve acceptable convergence rates in practice and presents new second-order additive, multiplicative, and combined corrections which can significantly accelerate convergence. These second-order corrections may enforce consistency with either the actual truth model Hessian or its finite difference, quasi-Newton, or Gauss-Newton approximation.

No abstract

Prior studies of wall bounded turbulence control have utilized Direct Numerical Simulation (DNS) which has limited investigations to low Reynolds numbers where viscous effects may play an important role. The current paper utilizes Large Eddy Simulation (LES) with the dynamic subgrid-scale model to explore the influence of viscosity on one popular turbulence control strategy, opposition control, that has been extensively studied using low Reynolds number DNS. Exploiting the efficiency of LES, opposition control is applied to fully developed turbulent flow in a planar channel for turbulent Reynolds numbers in the range Re τ = 100 to 720. As Reynolds number increases, the predicted drag reduction drops from 30% at Re τ = 100 to 19% at Re τ = 720. Furthermore, the ratio of power saved to power input drops by more than a factor of four when Reynolds number increases over this range, indicating that the drag reduction mechanism in opposition control is indeed less effective at higher Reynolds numbers. However, for sufficiently high Reynolds numbers, Re τ > 400, the ratio of power saved to power input becomes constant at a value near 40 indicating that opposition control is a viable turbulence control strategy at high Reynolds numbers.

This paper explores the effects of several wall-based, turbulence control strategies on the structure of the basis functions determined using the proper orthogonal decomposition (POD). This research is motivated by the observation that the POD basis functions are only optimal for the flow for which they were created. Under the action of control, the POD basis may be significantly altered so that the common assumption that effective reduced-order models for predictive control can be constructed from the POD basis of an uncontrolled flow may be suspect. This issue is explored for plane, incompressible, turbulent channel flow at Reynolds number, Reτ=180. Based on well- resolved large eddy simulations, POD bases are constructed for three flows: no control; opposition control, which achieves a 25% drag reduction; and optimal control, which gives a 40% drag reduction. Both controlled flows use wall transpiration as the control mechanism and only differ in the technique used to predict the control. For both controlled flows, the POD basis is altered from that of the no-control flow by the introduction of a localized shear layer near the walls and a nearly impenetrable virtual wall that hinders momentum transfer in the wall-normal direction thereby leading to drag reduction. A major difference between the two controlled flows is that the shear layer and associated virtual wall are located farther away from the physical wall when using optimal compared to opposition control. From this investigation, it is concluded that a no-control POD basis used as a low-dimensional model will not capture the key features of these controlled flows. In particular, it is shown that such an approximation leads to grossly underpredicted Reynolds stresses. These results indicate that a no-control POD basis should be supplemented with features of a controlled flow before using it as a low-dimensional approximation for predictive control.