最近30从www.hoelymoley.com 2023 - 07 - 10 - t12:31:35z //www.hoelymoley.com/feeds/question/20706 https://creativecommons.org/licenses/by-sa/4.0/rdf //www.hoelymoley.com/q/20706 3 blupp //www.hoelymoley.com/users/21677 2021 - 01 - 20 - t21:16:20z 2021 - 01 - 22 - t14:49:56z

< p >我读到浅水方程的推导,我很难理解为什么< span class = " math-container " > $ $ w (\ eta) = \压裂{D \η}{Dt} $ $ < / span >, < span class = " math-container " > w < / span >是美元的垂直速度,< span class = " math-container " > \“埃塔”< / span >是美元(顶部或底部)的垂直边界层和类< span = " math-container " > D / Dt < / span >美元是物质导数。< / p > < p >在大气和海洋流体动力学(第二版。2017年,章3.1.2),意思给了以下解释:< / p > < blockquote > < p >顶部垂直速度的物质导数是一个特定流体元素的位置。但流体在顶部的位置是<跨类= " math-container " > \“埃塔”< / span >,美元因此< span class = " math-container " > $ $ w (\ eta) = \压裂{D \η}{Dt} $ $ < / span > < / p > < /引用> < p >为什么物质导数顶部的垂直速度?< / p > < p > < em >编辑:< / em >添加了一个浅层的图解说明。如你所见,< span class = " math-container " > \“埃塔”< / span >美元不是假定为常数。< a href = " https://i.stack.imgur.com/WyYWR.png " rel = " nofollow noreferrer " > < img src = " https://i.stack.imgur.com/WyYWR.png " alt = "浅层" / > < / > < / p >

//www.hoelymoley.com/questions/20706/-/20709 # 20709 2 BarocliniCplusplus //www.hoelymoley.com/users/704 2021 - 01 - 21 t01:11:35z 2021 - 01 - 21 t01:11:35z

< p >作为复习,让我们参考物质导数的定义:< span class = " math-container " > $ $ \压裂{D} {Dt} = \压裂{\部分}{\部分t} + u \压裂{\部分}{x} \部分+ v \压裂{\部分}{\偏y} + w \压裂{\部分}{\部分z} $ $ < / span > < / p > < p >,如果我们把它应用到<跨类= " math-container " > \“埃塔”< / span >美元,然后我们得到:< span class = " math-container " > $ $ \压裂{D \η}{Dt} = \压裂{\部分\埃塔}{\部分t} + u \压裂{\部分\埃塔}{x} \部分+ v \压裂{\部分\埃塔}{\偏y} + w \压裂{\部分\埃塔}{\部分z} $ $ < / span > < / p > < p >让我们,第二,认为< span class = " math-container " > \“埃塔”< / span >美元是液体的高度。我们可以替换< span class = " math-container”> \埃塔美元< / span >与< span class = " math-container " > z < / span >美元计算液面。因此,我们得到了< span class = " math-container”> $ $ \压裂{Dz} {Dt} | ^{流体}= \压裂{\部分z}{\部分t} | ^{流体}+ u \压裂{\部分z} {x} \部分| ^{流体}+ v \压裂{\部分z}{\偏y} | ^{流体}+ w \压裂{\部分z}{\部分z} | ^{流体}$ $ < / span >由于只有一个欧拉,导数是依赖于< span class = " math-container " > z < / span >美元,它遵循< span class = " math-container " > $ $ \压裂{D \η}{Dt} = \压裂{Dz} {Dt} | ^{流体}= w \压裂{\部分z}{\部分z} | ^{流体}= w | ^{流体}= w (\ eta) $ $ < / span > < / p >

//www.hoelymoley.com/questions/20706/-/20718 # 20718 4 wingtorres //www.hoelymoley.com/users/21649 2021 - 01 - 22 - t09:27:20z 2021 - 01 - 22 - t14:49:56z

< p >这是一个运动自由表面边界条件的声明:不可能有正常的流过它的边界,只有沿着它切向流。同样,正常速度(相对于接口)的自由表面流体的位置是一样的速度。< / p > < p >自由表面的位置定义为< span class = " math-container " > $ $ z = \埃塔(x, y, t) $ $ < / span >然后把物质导数< / p > < p > <跨类= " math-container " > $ $ \压裂{\部分z}{\部分t} + u \压裂{\部分z} {x} \部分+ v \压裂{\部分z}{\偏y} + w \压裂{\部分z}{\部分z} = \压裂{D \η}{Dt} $ $ < / span > < / p > < p >每一项的lh除了< span class = " math-container " > $ \压裂{\部分z}{\部分z} = 1美元< / span >→< span class = " math-container " > $ 0 $ < / span >因为< span class = " math-container " > z < / span >美元是一个独立的坐标(见这< a href = " https://math.stackexchange.com/a/3067984 " > < / >答案供参考),让我们用< / p > < p > <跨类= " math-container " > $ $ w (z = \η)= \压裂{D \η}{Dt} $ $ < / span > < / p >

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