狂舞飛圈-簡單飛機的飛行動力研究
本實驗主要是探究雙圈圈簡單飛機的飛行原理，歸納圈圈結構對飛行距離、升力的影響，以及氣流流經機體時發生的作用。研究結果如下：一、實際發射，歸納影響滑行距離的變因。1. 前後圈直徑比值約為0.8 時滑行距離為最大。2. 前後圈寬度比值越接近1 時，滑行距離越遠，但影響不大。3. 圈圈間隔在21cm 時，滑行距離最大。二、設置風洞，模擬飛機飛行，測量升力1. 圈圈寬度越大，升力越大。2. 升力最大值出現在圈圈仰角25 度左右，風速越快，升力越大。3. 鋁片仰角在20°時升力最大，升力與角度的關係式為 F = 5×10?7θ4 + 4×10?5θ3 ? 0.0083θ2 + 0.2615θ + 0.13744. 風速越快，升力越大，在仰角20°時升力與風速的關係大約為F = 0.4579V2 - 0.9231V +1.4772 。5. 鋁片面寬每增加1cm，升力也增加0.1513gw。前後長每增加1cm，升力即增加0.1263gw。三、設置蒸汽氣流，觀察簡單飛機的氣流場1. 蒸汽流通過圈圈時，會發生附壁現象，而且簡單飛機使氣流往下偏折，飛機得到升力。四、理論演繹︰1. 以康達效應的理論推算出升力，與實際測量得的升力約相等，驗證升力確實由康達理論造成。2. 墊高簡單飛機前圈，使得軸線提高，確實影響了飛行距離，墊高1cm 以內，飛行距離均增加了，以實際的改進證實升力確實是康達效應。This experiment mainly discusses the flying principle of the simple plane which is made up of a straw with two paper circles, one bigger than the other, stuck on both two ends of it. We first launched the simple plane actually and concluded the factors which influenced the sliding distance of the plane, including the distance between two circles, diameter and width of the two circles. Second, we set up a simple wind-tunnel and simulated the flight, in order to measure the strength of lift. Third, we set up the steam air flow and observed the change of the air current in the steam flow while flowing through the plane. The Phenomenon of Wall Enclosing happened and made the flows downward, and the plane gained the lift at the same time. Finally, we deduced that there are two sources of lift and Benoulli's law is not suitable for it. The Coanda Effect can be applied to figure out 54 percent of lift. And the current, blocked by the plane, also offers some lift. In order to prove that the Coanda Effect does effect, we padded the first circle to enlarge the angle of elevation of the axis of the two circles. It really affected the sliding distance of the plane. While the first circle is padded up within 1 cm, the sliding distance of the plane increases. Practical improvement proves that Coanda Effect accounts for the lift.
翻轉「膜」力
The starting point of this experiment is to study the structure of soap-film. By changing the height of the triangular prisms, cuboids and pentagonal prisms, I observed the patterns set by the soap within the frameworks. It is surprised that when the proportion of prism is in a specific range, the phase in the middle of the structure will overturn 90 degree and then transmitted into another kind of balance pattern. I named this process “phase transition”. According to the experiment ,we can conclude the change of film patterns within variable prisms are all applied to this regular cycle:：
We know the soap films are forever attempting to minimize their energy. It stands to reason that surface tension tend to set up the film in its minimal surface. From the point of Mathematic, each structure should have only one single balance pattern, which is set up on the base of Fermat point and this pattern should stand to the minimize of it’s energy. However, we discovered that in some specific cases, one structure can allowed two kinds of balance films-patterns to exist. In these cases, any small vibration can cause the happening of “phase transition”. To sum up, I presume some structures have two different types of balance film-patterns: one of which stands to the local minimum (in this condition the pattern’s surface area isn’t the smallest); the other stands to the absolute minimum (in this condition the pattern’s surface area is the smallest).
There is an energy valley separate local minimum from absolute minimum. The second pattern (local minimum) will appear when the structure is blocked from attaining its absolute minimum, but surface intention is not powerful enough to support the film jumping over the energy valley. In this condition, if we works on the structure (such as blowing), which would provide the film of energy to cross the valley, and then phase transition take place. Vice versa, we can also force the film to jump from absolute minimum to local minimum and phase transition will occur as well. In a word, phase transition can happen in each two way, which connects the two types of balance pattern.
This report lays stress to find out the condition of phase transition. We also analyze the structure of soap-film by its included angles and surface area in hope to go deep into the science of soap-film.
我們實驗的出發點在於研究泡膜的立體結構。藉由改變正立方柱的高，觀察其平衡薄膜形式，意外的發現當正立方柱的邊長比在某個範圍時，泡膜結構中央會瞬間90 度翻轉，形成另一種平衡型式，我們將這個過程命名為面轉變(Phase Transition)。為了進一步了解面轉變發生的相關因素，我們設計了一連串的實驗，針對正三角柱、正四角柱、正五角柱、正六角柱發生面轉變的時機和條件分析討論。此外，我們還分析了泡膜結構中膜與膜夾角的特性、最小表面積和表面能之間的相關性，對於泡膜的立體結構做了一系列深入的探討。
漩渦也有形
流體旋轉時，外圍及底部流體，因槽壁及槽底摩擦力的影響，流速較慢，相對的壓力也較大，導致外圍的水流會轉入中心。發現本實驗的渦流為強迫與自由漩渦組成。實驗中，探討f(轉動器的頻率)、H(總水深)、y(?入深度)、R(轉盤半徑)四者與角形數間的關係。若y、R 愈大、H 越小，隨著f 的增大，可觀察到的形狀邊數越多；反之，若y、R 愈小、H 越大，則f 愈高，所形成的圖形半徑愈大，易超過轉盤，不易觀察。依白努利方程式，外層水流的流速較慢，而內層水流的流速較快，故外層壓力大而內層壓力小，水會由外往內流，而此渦動流於轉動液面產生的剪力，可能為產生N 邊形漩渦的主要原因之一。流體旋轉系統中，因轉動而產生流體離心力與內外層壓力差交互作用下，於某特定相關的因素條件下，形成特定角形數漩渦，是本實驗的重要發現。When fluids are in rotation, fictitious force given by the container brings about the relative decrease of speed of the bottom and outer layer of water, which causes its pressure to increase, and water to spin inward, resulting in a vortex motion with N-corner polygons formed at the surface of the rotating plate. During this experiment, we discover that the vortices consisted of free and forced vortex and the polygons vary as control parameters f(rotation frequency), H(height of fluid), y(depth of the plate), and R(radius of the plate) change. The larger y and R are,the smaller H is, the more corners show up as f increases. On the contrary, the smaller y and R are,the larger H is, few polygons are identified since the rotating radius of polygons are larger than the plate. According to Bernoulli’s principle, smaller velocity of the outer-layer water causes water pressure to increase and water to spin inward. During this process, shear force is developed at the surface of the rotating fluid, which we believe is the main cause of N-corner polygons. In a rotating system, the interaction of centrifugal force and differential pressure causing a certain Ncorner polygon to be formed under different controlled parameters is our main discovery.
培地茅根系碎形維度及抗拉力
本研究首先確認培地茅根系具有碎形之基本特性，再進一步以方格覆蓋法計算之碎形維度來分析培地茅根系在不同時間及環境因素下的生長。主要探討碎形維度與抓地力之關係，並設計以實際根系模型來加以模擬，並發展出一可描述抓地力與碎形維度及深度關係的方程式。我們的結論為：（1） 經由方格覆蓋法之計算，培地茅此種植物，不管是整個根系或單枝根，均具有碎形基本特性，適合進一步實驗研究。（2） 碎形維度會隨著培地茅生長時間增長而增加，並且在自然光照及30℃左右會有較大值，而種植於土壤中根系發展較廣，其碎形維度比種植於沙耕中來的高。（3） 實驗結果顯示，抓地力受碎形維度及根系深度兩因素影響，而培地茅根系對土壤有較強的抓地力，推測是因為兩者根系皆又深又長，土中培地茅根碎形維度較大，接觸面積較廣，而又進一步以矽膠模型做實驗驗證。（4） 矽膠模型之目的在於減少難控制之自然變因，實驗之前，測量了根系模型與洋菜凍之基本性質，實驗結果顯示抓地力與碎形維度及根系深度皆呈正向關係，可用數學方程式加以描述。This project is mainly a research into the fractal dimension of the vetiver root system. First, we confirm the vetiver root system has the basic fractal structure by checking its self-similarity, then using box-counting method to calculate fractal dimension. We begin with a fundamental investigation into the relation between different time and environmental factors and fractal dimension. Then we move to our main point: the relation between fractal dimension and its pull-out resistance. In the next step, we make a fundamental silicon model, simulating the vetiver root system, to continue our experiments. In the end, we develop a formula that can describe the relation between its pull-out resistance, roots depth and fractal dimension. Here are our conclusions: （1） After using box-counting method to calculate fractal dimension, we discover that not only the whole vetiver root system but also a single vetiver root has the basic fractal structure. （2） Fractal dimension increases when time goes on. Also the value of fractal dimension is larger in natural sunlight and the temperature at about 30℃.The vetiver root system grows more widely in soil than those in sand. That’s why it has larger fractal dimension. （3） Data shows that its pull-out resistance is influenced by both fractal dimension and the depth of the roots. The vetiver roots, in the meantime, show greater pull-out resistance than some other plants. Thus we draw the assumption that the vetiver root system grows deep and wide, and in natural soil its fractural dimension is greater and reaches greater area. Therefore, a silicon model is constructed to further confirm the findings of the experiment.（4） The design of the silicon model is to reduce the uncontrollable variables in nature. Before starting the experiment, we measured some basic characteristics of the silicon model, including density and angle of repose. Furthermore, the experiment demonstrates that pull-out resistance and fractural dimension have a commensurate mutual relation: the stronger the pull-out resistance, the wider the fractural dimension and the deeper the root system. Thus we derive a math formula to describe this relation.