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物理代写|流体力学代写Fluid Mechanics代考|Helmholtz Theorems

Research on vortex flow has been initiated by the fundamental paper [36] of H.L.F. Helmholtz (1821-1894), a physicist and a professor of physiology and anatomy at the University of Königsberg, Bonn, Heidelberg and Berlin. In his paper, Helmholtz established his three theorems of vortex motion. Assuming incompressible frictionless fluids subjected to flow forces defined by a potential, Helmholtz [36] published a paper about the vortex motions in which he stated his vortex theorems. These theorems are translated from German and appear in an excellent textbook by Prandtl and Tietiens [31]. They reflect the quintessence of the vortex flow motion treated in Helmholtz original work. Before starting with the discussion of Helmholtz theorems, it is helpful to become familiar with the nomenclature sketched in Fig. 6.31.

A vortex line, Fig. 6.31a is a line tangent to the rotation vector $\nabla \times V$. The vortex lines may form a vortex tube, Fig. 6.3lb. Reducing the cross sections of a vortex tube to an infinitely small size, we obtain a vortex filament. Thus, a vortex filament is essentially a vortex tube with an infinitely small cross section but a finite value of circulation. This particular configuration allows one to apply the Stoke’s theorem Eq. (6.147) without integrating the rotation vector
$$
\Gamma=(\nabla \times V) \cdot n d S .
$$

Since the unit vector $\boldsymbol{n}$ is parallel to the rotation vector $\nabla \times \boldsymbol{V}$, we may re-arrange Eq. $(6.167)$
$$
\nabla \times V=n \frac{\Gamma}{d S} .
$$
Since $d s$ is, per definition, infinitely small and $\Gamma$ has a finite value, the rotation vector $\nabla \times V$ must be infinitely large, indicating that the vortex filaments represent a singularity. This and many other types of singularities are used for dealing with more complicated issues particularly in aerodynamics.

Ignoring the friction forces and assuming that there exists a potential acting on fluid particles, Helmholtz formulated in his original paper [36], three theorems: The first theorem states that no fluid particle can have a rotation if it did not originally rotate. This theorem reflects the physical content of the differential part of Eq. (6.166). The second theorem states that the fluid particles, which at any time are part of a vortex line, always belong to that same vortex line. This theorem is the consequence of the integral part of Eq. (6.166), stating that the circulation remains constant. The third theorem states that the product of the cross section area and angular velocity of an infinitely thin vortex filament remains constant over the whole length of the filament and keeps the same value even when the vortex moves. It further states that the vortex filament must, therefore, be either closed curves or end on the boundaries of the fluid.

物理代写|流体力学代写Fluid Mechanics代考|Induced Velocity Field, Law of Bio-Savart

We consider now an isolated vortex filament with the strength $\Gamma$ imbedded in an inviscid irrotational flow environment, as shown in Fig. 6.33.

In a distance $r$ from the point $A$, a differential element $d \xi$ of the vortex filament at the point $B$, induces a differential velocity vector field $d \boldsymbol{V}$. The velocity vector is perpendicular to the plane spanned by the normal unit vector $\boldsymbol{n}$ and the unit vector $\boldsymbol{e}$ in the $\boldsymbol{r}$-direction. The unit vector $\boldsymbol{n}$ is perpendicular to the infinitesimal cross section $d S$, whereas the unit vector $\boldsymbol{e}$ points from the center point of the element $A$ to the position $B$, where the velocity $d V$ is being induced. The relationship describing the velocity field is analogous to the one discovered by Bio and Savart through electrodynamic experiments. It describes the magnetic field induced by a current through a conducting wire. In an aerodynamic context, the conducting wire corresponds to the vortex filament, its current corresponds to the vortex strength $\Gamma$ and the induced magnetic field corresponds to the induced velocity field.

To present the derivation, first we provide the mathematical tool essential to arriving at the Bio-Savart law. Let us decompose an arbitrary vector point function $V$ into an irrotational part that can be expressed in terms of the gradient of a potential and a rotational or solenoidal part
$$
\boldsymbol{V}=\nabla \Phi+\nabla \times \boldsymbol{U}
$$
with $\Phi$ as a scalar potential and $\nabla \times \boldsymbol{U}$ as the solenoidal part of Eq. (6.186). Taking the curl of Eq. (6.186) gives
$$
\nabla \times \boldsymbol{V}=\nabla \times(\nabla \Phi)+\nabla \times(\nabla \times \boldsymbol{U})=\nabla \times(\nabla \times \boldsymbol{U})
$$
The first term on the right-hand side of Eq. (6.187) is the curl of the gradient of the scalar field $\Phi$ that identically vanishes. The divergence of Eq. (6.186) delivers
$$
\nabla \cdot \boldsymbol{V}=\Delta \Phi+\nabla \cdot(\nabla \times \boldsymbol{U})=\nabla^2 \Phi=\Delta \Phi
$$ with $\nabla \cdot \nabla \Phi=\Delta \Phi$ and $\nabla \cdot(\nabla \times \boldsymbol{U}) \equiv 0$. Equation $(6.188)$ is an inhomogeneous partial differential equation called Poisson’s equation. What makes the Poisson’s equation (6.188) a special case where $\Delta \Phi=0$, is the Laplace equation we treated in the preceding sections. Using the vector identity for $\nabla \times(\nabla \times \boldsymbol{U})=\nabla(\nabla \cdot \boldsymbol{U})-$ $\Delta \boldsymbol{U}, \mathrm{Eq} .(6.187)$ reads
$$
\nabla \times \boldsymbol{V}=\nabla(\nabla \cdot \boldsymbol{U})-\Delta \boldsymbol{U}
$$

物理代写|流体力学代写Fluid Mechanics代考|ZEIT2503

物理代写|流体力学代写流体力学代考|Helmholtz定理

. . Helmholtz定理


关于旋涡流动的研究是由H.L.F. Helmholtz(1821-1894)的基础论文[36]发起的。Helmholtz是一位物理学家,也是Königsberg、波恩、海德堡和柏林大学的生理学和解剖学教授。在他的论文中,亥姆霍兹建立了涡旋运动的三个定理。假设不可压缩无摩擦流体受到由势定义的流动力,Helmholtz[36]发表了一篇关于旋涡运动的论文,在其中他阐述了他的旋涡定理。这些定理是从德语翻译过来的,出现在Prandtl和Tietiens的一本优秀教科书[31]中。它们反映了亥姆霍兹原著中所处理的旋涡流动运动的精髓。在开始讨论亥姆霍兹定理之前,熟悉图6.31中描绘的命名法是有帮助的


图6.31a是一条与旋转矢量$\nabla \times V$相切的线。旋涡线可以形成旋涡管,如图6.3lb。将涡旋管的横截面缩小到无限小的尺寸,就得到了涡旋灯丝。因此,涡旋灯丝本质上是一个具有无限小截面但循环有限值的涡旋管。这种特殊的配置允许应用斯托克定理Eq.(6.147),而无需对旋转向量
$$
\Gamma=(\nabla \times V) \cdot n d S .
$$ 积分


由于单位向量$\boldsymbol{n}$平行于旋转向量$\nabla \times \boldsymbol{V}$,我们可以重新排列方程式$(6.167)$
$$
\nabla \times V=n \frac{\Gamma}{d S} .
$$
由于$d s$根据定义是无限小的,而$\Gamma$有一个有限的值,旋转向量$\nabla \times V$必须是无限大的,这表明涡旋丝代表一个奇点。这种奇点和许多其他类型的奇点被用来处理更复杂的问题,特别是在空气动力学中


忽略摩擦力,假设存在作用在流体粒子上的势能,Helmholtz在他的原始论文[36]中提出了三个定理:第一个定理指出,如果流体粒子本来没有旋转,那么它就不可能旋转。这个定理反映了式(6.166)中微分部分的物理内容。第二个定理指出,流体粒子,在任何时候都是一条涡线的一部分,总是属于同一条涡线。该定理是式(6.166)积分部分的推导,说明循环保持不变。第三个定理指出,无限细的旋涡长丝的横截面积和角速度的乘积在长丝的整个长度上保持恒定,并且即使在旋涡移动时也保持相同的值。因此,涡丝必须是闭合曲线或终止于流体边界上

物理代写|流体力学代写流体力学代考|诱导速度场,生物-萨伐尔定律


我们现在考虑一个强度为$\Gamma$的孤立涡丝嵌入无粘无旋流动环境中,如图6.33所示。


在距离$A$点$r$处,位于$B$点的涡旋丝的微分元$d \xi$,诱发了一个微分速度向量场$d \boldsymbol{V}$。速度向量垂直于由单位向量$\boldsymbol{n}$和单位向量$\boldsymbol{e}$在$\boldsymbol{r}$方向张成的平面。单位向量$\boldsymbol{n}$垂直于无穷小的截面$d S$,而单位向量$\boldsymbol{e}$从元素$A$的中点指向位置$B$,在那里速度$d V$被诱导。描述速度场的关系类似于Bio和Savart通过电动力学实验发现的关系。它描述通过导线的电流所引起的磁场。在气动环境下,导线对应旋涡丝,其电流对应旋涡强度$\Gamma$,感应磁场对应感应速度场


为了给出推导过程,我们首先提供得到比奥-萨伐尔定律所必需的数学工具。让我们把一个任意向量点函数$V$分解成一个无转动部分,这个无转动部分可以用势和旋转部分或螺线部分
$$
\boldsymbol{V}=\nabla \Phi+\nabla \times \boldsymbol{U}
$$
的梯度表示,其中$\Phi$作为标量势,$\nabla \times \boldsymbol{U}$作为式(6.186)的螺线部分。取式(6.186)的旋度,得到
$$
\nabla \times \boldsymbol{V}=\nabla \times(\nabla \Phi)+\nabla \times(\nabla \times \boldsymbol{U})=\nabla \times(\nabla \times \boldsymbol{U})
$$
式(6.187)右边的第一项是相同消失的标量场$\Phi$的梯度的旋度。Eq.(6.186)的发散产生
$$
\nabla \cdot \boldsymbol{V}=\Delta \Phi+\nabla \cdot(\nabla \times \boldsymbol{U})=\nabla^2 \Phi=\Delta \Phi
$$与$\nabla \cdot \nabla \Phi=\Delta \Phi$和$\nabla \cdot(\nabla \times \boldsymbol{U}) \equiv 0$。方程$(6.188)$是一个叫做泊松方程的非齐次偏微分方程。使泊松方程(6.188)成为$\Delta \Phi=0$的特例的是我们在前面几节中处理过的拉普拉斯方程。使用向量标识$\nabla \times(\nabla \times \boldsymbol{U})=\nabla(\nabla \cdot \boldsymbol{U})-$$\Delta \boldsymbol{U}, \mathrm{Eq} .(6.187)$读取
$$
\nabla \times \boldsymbol{V}=\nabla(\nabla \cdot \boldsymbol{U})-\Delta \boldsymbol{U}
$$

物理代写|流体力学代写Fluid Mechanics代考

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