輔導教授：楊宗哲
指導老師：李文堂
學生：呂軒豪

1-1

When a fluid jet falling vertically strikes a horizontal plate, fluid is expelled radially, and the layer generally thins until reaching a critical radius at which the layer depth increase abruptly. This phenomenon is called the Circular Hydraulic Jump .

1-2
 Predictions for the jump radius based on inviscid theory were presented by Lord
Rayleigh(1914).
 The dominant influence of fluid viscosity on the jump radius was elucidated by Watson(1964).
 Ellegaard(1998)identified that a striking in stability may transform the circular hydraulic jump into regular polygons.

1-3
 We find when a fluid jet strikes to a container, at the moment when the flow over the container’s boundary the circular hydraulic jump transform into rotating polygons, this is referred to as
Rotating Hydraulic Jump.
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影片 ( 慢放 )

Background
2-1
 Rayleigh regarded hydraulic jump as a discontinuity (shock). Close to the jet the fluid layer is thin and the motion is rapid, further away it is an order of magnitude thicker and moves correspondingly slower.

2-2
 Rayleigh’s shock conditions imply that the fluid before and after jump are respectively “supercritical” and
“subcritical” , which means the average velocity is respectively larger and smaller than the small amplitude wave gh
.

2-3
 When a jet of viscous ethylene glycol strikes a container, a circular hydraulic jump is formed.
 As height of h ext is increased, vertical rollers are formed surrounding the jump.
 The roller is formed owing to velocity gradient of the fluid layer.
 The vertical structure of flow now plays a crucial role, it produces multiple vortices around the jump.
 The vortex produces a horizontal pressure gradient
 p


2
R
2
2

: angular velocity of the roller. ;
R= h ext
/2 ,

=density of the fluid .

 液體旋轉示意圖：
上層液體向外流
下層液體向外流且受到較大的黏滯阻力
2-4

影片
1
2/3 秒後
2/3 秒後
影片 2
2-5

2-6
 控制濃度固定(及黏滯係數固定)、流量固定，
改變液深 h ext ，探討邊數和 h ext 關係。
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 控制流量固定、液深固定，改變溶液濃度，探
討邊數和黏滯係數的關係。
 控制液深固定、濃度固定，探討邊數和流量的
關係。

 We measure the Reynold number of
Rotating Hydraulic Jump.
 We assume that
N
 k

N: the polygon number.
a

Q b h c ext
: kinematical viscosity of fluid.
Q: flow rate.
We do experiment to find a,b,c. And know the dependence of number of polygon.
2-7

 用可以改變高度的容器，重做深度對邊數的
實驗，在每一個穩定的多邊形旁放入膠片量
出 v ，求出 Vortex 之
 p ，算出 Vortex 大小
對邊數的關係。
 架高盤子(透明盤)，液體中加入鋁粉(不起
化學變化)由底端拍出較清晰的 Vortex 。
 數據分析整理。

 彭黃勝、范治明、蔡國棟和李志強：水牆，中華民國
中小學科學展覽第二十一屆至三十屆優勝作品專輯
國立台灣科學教育館編印，頁301-307
 周雨剛等四人：利用因次分析法研究圓形水躍的變因，
中華民國第四十六屆中小學科學展覽會作品說明書。
 Clive Elligard, “Creating corners in kitchen sinks”,
Nature, Vol.392, P767-768, 1998.
 Thomas R. N. Jansson, “Polygons on a rotating fluid surface”, Physics Review Letters,174502, 2006(May)