Ludwig Prandtl | EPFL Graph Search

Ludwig Prandtl (4 February 1875 – 15 August 1953) was a German fluid dynamicist, physicist and aerospace scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used for underlying the science of aerodynamics, which have come to form the basis of the applied science of aeronautical engineering. In the 1920s, he developed the mathematical basis for the fundamental principles of subsonic aerodynamics in particular; and in general up to and including transonic velocities. His studies identified the boundary layer, thin-airfoils, and lifting-line theories. The Prandtl number was named after him. Early years Prandtl was born in Freising, near Munich, on 4 February 1875. His mother suffered from a lengthy illness and, as a result, Ludwig spent more time with his father, a professor of engineering. His father also encouraged him to observe nature and think about his observations. Prandtl entered the Technisc

Theoretical and numerical studies of atmospheric boundary-layer flows over complex terrain

Marco Giovanni Giometto

Atmospheric boundary-layer (ABL) flows over complex terrain have been the focus of active research, given their impact on weather and climate variability. Surface complexity is understood in a broad sense and includes variation in roughness properties, inclination of the underlying surface, presence of heterogeneous forcing mechanisms (e.g., buoyancy, humidity), to name but a few. Most assumptions of classical boundary-layer similarity theory do not hold under such conditions, complicating matters from both a measurement and modeling perspective. Here, a combination of analytical and numerical approaches are used to address two among the most relevant problems: turbulent slope flows, and ABL flows over multi-scale rough surfaces. The first part of the thesis focuses on slope flows: the building blocks of local weather in mountainous regions. To understand the system conceptually, a closed-form analytic solution to the Prandtl slope flow model is first derived, prescribing transfer coefficients in accordance to the O'Brien K-theory model. Profiles are characterized by stark variations in both phase and amplitude of extrema compared to the classic constant-K and a more recent solution, valid within the Wentzel-Kramers-Brillouin theory, shedding new light on this long-standing geophysical problem. In addition, direct numerical simulation is used to study the turbulent structure of anabatic and katabatic flows, and to describe the sensitivity of the solution to variations in the parameter space, within the conceptual framework of the Prandtl model. Variations in the sloping angle from the vertical wall setup are shown to induce a progressive departure of averaged profiles between the two flow regimes, ultimately resulting in stark differences at gentle sloping angles. The thermodynamical mechanisms responsible for sustaining mean and turbulent kinetic energy are used to further distinguish between flow regimes, and to propose a qualitative partition of the boundary layer in slope flows. The DNS setup is additionally adopted to identify coherent structures in katabatic flows over steep slopes. Coherent motions are responsible for the maintenance of turbulence in the ABL, hence their characterization is of fundamental importance toward a better understanding of boundary-layer dynamics. Packets of hairpins are found to connect in the streamwise direction to form large-scale motions (LSMs). For the lower sloping angles that are considered, it is then shown how LSMs further align to form very-large-scale motions (VLSMs). LSMs and VLSMs are found to be dominant contributors to streamwise momentum variance and turbulent momentum transfer in the above-jet regions. Next, drag properties of fractal-like sea ice surface morphologies are examined within the large-eddy simulation framework. The effects of large-scale surface features on wind flow are accounted for by an immersed boundary method. Conversely, the drag forces caused by subgrid-scale features are modeled through a novel dynamic roughness approach, in which the hydrodynamic roughness length parameter is determined using the first-principles based constraint that the total momentum flux (drag) must be independent of the grid-filter scale. This approach leads to accurate flow predictions, and provides an estimate of the otherwise unknown roughness parameter for sea ice surfaces, of use in climate, weather prediction and scalar transport models to evaluate the hydrodynamic roughness length.

EPFL2016

An Intercomparison of LES Subgrid-Scale Models for Simulations of Slope Flows

Jiannong Fang, Marco Giovanni Giometto, Marc Parlange

We propose an inter-comparison of Smagorinsky type subgrid-scale models for the simulation of high Reynolds number katabatic flows, with the purpose of testing the validity of some of the model's assumptions for the considered application. Katabatic flows arise along sloping surfaces under stably stratified conditions and are a direct consequence of surface cooling. Understanding their structure is of great interest in meteorology because of the broad band of areas and scales that they cover, influencing from local valleys microclimate (e.g. over Salt Lake and Phoenix valleys) to synoptic scale motions (e.g. over Antarctica). Large-eddy simulation (LES), on the other hand, represents an important tool to study high-Reynolds number flows and has undergone significant developments and validations in the past years, confirming its reliability when adopted to simulate a variety of atmospheric (and industrial) flows. The stable stratification of the environment and the complex dynamics that arise close to the surface in katabatic winds – leading eventually to the formation of the so-called low level jet – represent a tough test case for closure models, questioning their applicability over a broad range of filter (grid) scales and motivating the proposed work. The study is performed in the idealized framework of the Prandtl model, the set of normalized, filtered Boussinesq equations is solved on a regular domain, adopting an in-house mixed pseudo-spectral / finite difference code. The original static Smagorinsky model (SMAG), the standard dynamic model with planar averaging of the coefficient (PASI) and the more recent scale-dependent Smagorinsky model with Lagrangian averaging of the coefficients (LASD) are adopted to independently parametrize the subgrid-scale terms for kinematic momentum and energy, which arise from the filtering operation. Results focus on the statistically steady state solution of the problem and show how mean profiles are consistent across subgrid-scale models and most of the discretizations, whereas second order statistics show strong dependence on the spatial resolution, which is partially alleviated once the scale invariance assumption of the model coefficient is relaxed. For the considered resolutions we show that scale-invariance of the model coefficient does in fact not hold – on average – in proximity of the low level jet and close to the wall. At these location of strong shear production the support of the LES filter is approaching the integral scale and the LASD model is therefore expected to provide a more accurate estimation of the Smagorinsky coefficient. Considerations regarding the dissipation rates of the different subgrid-scale models and on mixing efficiency of resolved and modeled turbulence will be discussed.

2014

Katabatic Flow: A Closed-Form Solution with Spatially-Varying Eddy Diffusivities

Jiannong Fang, Marco Giovanni Giometto, Peter Monkewitz, Marc Parlange

The Nieuwstadt closed-form solution for the stationary Ekman layer is generalized for katabatic flows within the conceptual framework of the Prandtl model. The proposed solution is valid for spatially-varying eddy viscosity and diffusivity (O'Brien type) and constant Prandtl number (Pr). Variations in the velocity and buoyancy profiles are discussed as a function of the dimensionless model parameters and , where is the hydrodynamic roughness length, is the Brunt-Vaisala frequency, is the surface sloping angle, is the imposed surface buoyancy, and is a reference velocity scale used to define eddy diffusivities. Velocity and buoyancy profiles show significant variations in both phase and amplitude of extrema with respect to the classic constant model and with respect to a recent approximate analytic solution based on the Wentzel-Kramers-Brillouin theory. Near-wall regions are characterized by relatively stronger surface momentum and buoyancy gradients, whose magnitude is proportional to and to . In addition, slope-parallel momentum and buoyancy fluxes are reduced, the low-level jet is further displaced toward the wall, and its peak velocity depends on both and .