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Concept# Dimensionless quantity

Summary

A dimensionless quantity (also known as a bare quantity, pure quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned.
Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Dimensionless quantities are distinct from quantities that have associated dimensions, such as time (measured in seconds).
The corresponding unit of measurement is one (symbol 1), which is not explicitly shown.
For any system of units, the number one is considered a base unit.
Dimensionless units are special names that serve as units of measurement for expressing other dimensionless quantities. For example, in the SI, radians (rad) and steradians (sr) are dimensionless units for plane angles and solid angles, respectively. For example, optical extent is defined as having units of metres multiplied by steradians.
History
Dimensional

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The presence of aerodynamic vortices is widespread in nature. They can be found at small scales near the wing tip of flying insects or at bigger scale in the form of hurricanes, cyclones or even galaxies. They are identified as coherent regions of high vorticity where the flow is locally dominated by rotation over strain. A better comprehension of vortex dynamics has a great potential to increase aerodynamic performances of moving vehicles, such as drones or autonomous underwater vehicles. An accelerated flat plate, a pitching airfoil or a jet flow ejected from a nozzle give rise to the formation of a primary vortex, followed by the shedding of smaller secondary vortices. We experimentally study the growth, timing and trajectory of primary and secondary vortices generated from a rectangular flat plate that is rotated around its centre location in a quiescent fluid. We systematically vary the rotational speed of the plate to get a chord based Reynolds number \Rey that ranges from 800 to 12000. We identify the critical \Rey for the occurrence of secondary vortices to be at 2500. The timing of the formation of the primary vortex is \Rey independent but is affected by the plate's dimensions. The circulation of the primary vortex increases with the angular position $\alpha$ of the plate, until the plate reaches 30°. Increasing the thickness and decreasing the chord lead to a longer growth of the primary vortex. Therefore, the primary vortex reaches a higher dimensionless limit strength. We define a new dimensionless time $T^*$ based on the thickness of the plate to scale the age of the primary vortex. The primary vortex stops growing when $T^* \approx 10$, regardless of the dimensions of the plate. We consider this value to be the vortex formation number of the primary vortex generated from a rotating rectangular flat plate in a Reynolds number range that goes from 800 to 12000. When $\alpha$ > 30°, the circulation released in the flow is entrained into secondary vortices for $\Rey > 2500$. The circulation of all secondary vortices is approximately 4 to 5 times smaller than the circulation of the primary vortex. We present a modified version of the Kaden spiral that accurately predicts the shear layer evolution and the trajectory of primary and secondary vortices during the entire rotation of the plate.We model the timing dynamics of secondary vortices with a power law equation that depends on two distinct parameter: $\chi$ and $\alpha_{0}$.The parameter $\chi$ indicates the relative increase in the time interval between the release of successive secondary vortices.The parameter $\alpha_{0}$ indicates the angular position at which the primary vortex stops growing and pinches-off from the plate.We also observe that the total circulation released in the flow is proportional to $\alpha^{1/3}$, as predicted by the inviscid theory.The combination of the power law equation with the total circulation computed from inviscid theory predict the strength of primary and secondary vortices, based purely on the plate's geometry and kinematics.The strength prediction is confirmed by experimental measurements.In this thesis we provided a valuable insight into the growth, timing and trajectory of primary and secondary vortices generated by a rotating flat plate. Future work should be directed towards more complex object geometries and kinematics, to confirm the validity of the modified Kaden spiral and explore the influence on the formation number.

HYDROcontest is a challenge open to students from 13 universities from all over the world. The aim of the competition is to design the motorboat that best fulfills the tradeoff between high speed and low energy consumption. In order to optimize the shape of the hydrofoils sustaining the boat, suitable Computational Fluid Dynamics models and parallel simulations were employed. The variational multiscale (VMS) model proposed by Bazilevs et al. (2009) was linearized by means of a Newton method, discretized in time through a fully-implicit scheme, transformed into a dimensionless form and implemented in C++, using the Finite Elements library LifeV.

2014Accurate microscale windfields computations over complex topography is crucial to many particle transport models but remains a challenging task. The objective of this work focuses on the numerical simulations of micro-scale windfields over the steep Gaudergrat ridge, located in the Swiss Alps. These windfields are computed with the objective of driving a snowdrift model, consequently the work concentrates on meteorological situations close to snow storms. As snow transport occurs in the first meters above the surface, this implies a very fine resolution of order tens of meters. Airflow simulations are performed using the meteorological model ARPS (Advanced Regional Prediction System), which is based on a Large Eddy Simulation (LES) formulation of the compressible Navier-Stokes equations. The turbulent airflow features play an important role in the transport of particles. Therefore ARPS turbulence models, the Smagorinsky-Lilly and the 1.5 order Turbulent Kinetic Energy (TKE) closures, have been examined in neutral atmosphere conditions over flat terrain. ARPS mechanical turbulence schemes has hence been tested and the parameters of the Subgrid-Scales (SGS) models have been tuned. ARPS has already been proven suitable for reproducing qualitative features of airflow and over complex alpine terrain with a careful choice of the artificial initialisation and periodic boundary conditions. When lateral periodic boundary conditions are applied for airflow computations over real complex topography, instabilities arise quickly. For a quantitative and stable description of airflow presented in this work, the initialisation and boundary conditions have consequently been improved. In this study, the simulations over the Gaudergrat ridge presented are performed a one-way nesting approach. ARPS is first driven by the outputs of the MeteoSwiss model aLMo which produce initial and time dependent lateral boundary conditions. Then the application of the nesting technique permit to bridge spatial resolutions from 7km (aLMo grid resolution) to 25 m (horizontal resolution in the finer ARPS grid). Such a fine resolution is also required for Large-Eddy Simulations (LES) configuration and it is expected that a large part of the energy is resolved explicitly. The nesting technique has been applied to reproduce two selected days during the Gaudergrat Experiment (Gaudex) with stronger wind, to have conditions as close as possible to winter conditions and when thermal winds are weak. The field measurement campaign, Gaudergrat Experiment (Gaudex), in collaboration with the University of Leeds, was held from June to October 2003 at the Gaudergrat ridge, near Davos, Switzerland. The collected data are used to develop a better understanding of the airflow characteristics and turbulence features as well as to check the model results. The comparison with field data show satisfactory results for the mean flow quantities, whereas the lateral boundary condition forcing suppresses the small scales turbulent motion. A simple method is proposed to spin up turbulent motions in the finer resolution domain. This method is based on the introduction of turbulent perturbations from a precursor simulation onto the mean wind profile at the lateral boundaries. This new configuration facilitate the development of turbulence and resolves explicitly smaller scale motions without altering the mean flow. The spectral analysis of the Gaudex data highlights the fact that the turbulence on the lee side of the Gaudergrat ridge is influenced by local features, whereas at the crest, the effect of the surrounding mountains is recognisable. The statistical analysis of wind speed fluctuations shows that the turbulence in complex terrain is highly intermittent, but can be interpreted as a combination of subsets of isotropic turbulence. In complex terrain, the production of turbulence is not continuous, it is hence difficult to apply the traditional scaling and averaging laws developed for homogeneous horizontal surfaces. The heterogeneous surface conditions are likely to create additional length and time scale to generalise the statistical properties.