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物理代写|流体力学代写Fluid Mechanics代考|Circulation, Lift, Kutta Condition

The conformal transformation we discussed previously allows, among others, the generation of asymmetric airfoils with prescribed cambers. These airfoils resemble profiles that are utilized as aircrafts wings, compressors and turbine blade profiles. The significance of the cambered profiles is to generate the necessary force to lift the aircraft, to generate higher total pressure (compressors), and to produce power (turbine). Generation of lift, however, requires the existence of circulation as we briefly discussed in Sect. 6.4. In the context of the potential flow analysis, certain conditions must be fulfilled to bring about a circulation which is a prerequisite for lift generation. Figure $6.25$ exhibits the potential flow around one of those cambered airfoils we designed in the previous section.

The corresponding configuration in the z-plane is the flow around a circle with the circulation $\Gamma$ and an angle of attack $\alpha$. Fig. 6.25a. The complex potential of this configuration is almost the same as in Eq. (6.62) with the exception being that the axis of the dipole flow is turned by the angle $\alpha$. Performing a simple coordinate transformation by substituting in the dipole part of Eq. (6.62) $z=r^{i \theta}$ by $z=r^{i(\theta-\alpha)}$ results in:

$$
F(z)=V_{\infty}\left(z e^{-i \alpha}+\frac{R^2}{z} e^{i \alpha}\right)-i \frac{\Gamma}{2 \pi} \ln \left(\frac{z e^{-i \alpha}}{R}\right)
$$
Assuming a circulation in the clockwise direction, Fig. 6.25a, two stagnation points $S_1$ and $S_2$ are present in the $z$-plane. In the $\zeta$-plane, the transformation of the front stagnation point $S_1$ may be located on the pressure surface (concave side) of the blade, while the rear $S_2$ may be located on the suction side (convex side), Fig. 6.25b. Considering the flow situation at the sharp trailing edge, the fluid particles move from the pressure surface (concave side) of the blade to the suction surface (convex side) with an infinitely large velocity. Increasing the circulation causes both stagnation points to move. For a particular $\Gamma=\Gamma_K$, the Kutta-circulation, the rear stagnation point $S_2$ coincides with the trailing edge. At this point the velocity is zero. Known as the Kutta condition, it specifies that for an airfoil under inviscid flow conditions, to generate enough circulation, the rear stagnation point must coincide with the trailing edge. To satisfy this condition we resort to the complex potential Eq. (6.10) with $F(z)=\Phi+i \Psi$ with the derivative
$$
\frac{d f(z)}{d \zeta}=\frac{d f(z)}{d z} \frac{d z}{d \zeta}=\frac{d f(z) / d z}{d \zeta / d z}=u-i v
$$
Using the Joukowsky transformation function, we find for
$$
\frac{d f(z)}{d \zeta}=\frac{d f(z)}{d z}\left(\frac{z^2}{z^2-a^2}\right) .
$$
For $z \rightarrow \infty$, the expression in the parentheses approaches unity resulting in
$$
\frac{d f(z)}{d \zeta}=\frac{d f(z)}{d z}
$$

物理代写|流体力学代写Fluid Mechanics代考|Generation of Circulation

In the preceding sections we derived the relationship for lift as a function of circulation (Eqs. $6.105$ and 6.143) assuming that a circulation is superposed on the translational flow past the body, without explaining how this circulation has been brought about. The question that needs to be answered is: how can the existence of such a circulation flow be explained? To answer this question we revert to the flow visualization experiments by Prandtl [31] taken from an airfoil subjected to different flow modes. Figure $6.30$ reflect the physical contents of images presented in [31].
We assume that at first the fluid is at rest. Fig. 6.30a, so that the line integral of the velocity along a curve completely surrounding the airfoil is zero, because all velocities are zero. This would correspond to a potential flow situation without circulation immediately after starting Fig. 6.30b. According to Thomson’s theorem, Eq. (6.166), the circulation in a frictionless fluid must remain constant (in this case equal to zero) at all times including the moment when the fluid is suddenly put into a uniform translatory motion with respect to the airfoil. This is apparently in contradiction to the experimental fact that there is a circulation around the airfoil. Considering the infinitely large velocity around the sharp trailing edge in Fig. $6.30 \mathrm{~b}$ of the airfoil, one could suggest that the flow, at the first moment after starting, might be a potential flow without circulation. The presence of the viscosity in the boundary layer, however, causes this large velocity to develop into a surface of discontinuity, Fig. 6.30c. At the sharp trailing edge, the viscosity of the real fluid causes an equalization of the velocity jump, leading to a layer of finite thickness which is occupied by vortices, Fig. 6.30d. This vortical layer, then, is rolled up to a vortex, the so-called starting vortex, Fig. 6.30e, $\mathrm{f}$. This vortex, according to the theorems of Helmholtz (treated in the following section), is always associated with the same particles of fluid, is washed away with the fluid, and is convected downstream as a free vortex. Since this free vortex has a non-zero magnitude, its existence clearly contradicts the Thomson’s theorem. Assuming the validity of the Thomson’s theorem, the process of starting must have generated another vortex with the same magnitude but in the opposite direction so that the sum of their strengths vanishes. In fact, the existence of the free vortex is always associated with the existence of another vortex called bound vortex, Fig. 6.30g. Calculating the circulation around the closed curve $C \equiv A B C D F A, C_B \equiv A B E F A$, and $C_F \equiv B C D F B$, we find $\Gamma=\oint_{(C)} \boldsymbol{V} \cdot d \boldsymbol{C}=\oint_{\left(C_B\right)} \boldsymbol{V} \cdot d \boldsymbol{C}+\oint_{\left(C_F\right)} \boldsymbol{V} \cdot d \boldsymbol{C}=\Gamma_B+\Gamma_F=0$ from which we conclude that $\Gamma_B=-\Gamma_F$. This result is confirmed experimentally verifying the validity of the Thomson’s vortex theorem. The most important feature essential for upholding the Thompson’s theorem is the viscosity effect, without which no vortices can be produced.

In generating the vortex images presented in [6] that we summarized in Fig. 6.30, Prandtl first kept the airfoil in a fixed position that was exposed to a moving fluid. In a second set of experiments, he moved the airfoil relative to undisturbed fluid. The same phenomenon was observed in both cases.

物理代写|流体力学代写Fluid Mechanics代考|ENGG2500

物理代写|流体力学代写流体力学代考|循环,提升,库塔条件


我们前面讨论的保形变换允许,除其他外,生成具有规定弧度的非对称翼型。这些翼型类似于飞机机翼、压气机和涡轮叶片的外形。弧度轮廓的意义在于产生必要的力来提升飞机,产生更高的总压(压气机),并产生动力(涡轮)。然而,升力的产生需要循环的存在,正如我们在6.4节简要讨论的那样。在势流分析的背景下,必须满足某些条件才能产生循环,这是产生升力的先决条件。图$6.25$展示了我们在上一节中设计的弧度翼型周围的潜在流动

相应的配置在z平面是一个圆形的流动,循环为$\Gamma$,攻角为$\alpha$。图6.25a。这种构型的复势几乎与式(6.62)相同,唯一的例外是偶极流的轴被角度$\alpha$所扭转。通过将Eq. (6.62) $z=r^{i \theta}$的偶极子部分替换为$z=r^{i(\theta-\alpha)}$来执行一个简单的坐标变换,结果是:

$$
F(z)=V_{\infty}\left(z e^{-i \alpha}+\frac{R^2}{z} e^{i \alpha}\right)-i \frac{\Gamma}{2 \pi} \ln \left(\frac{z e^{-i \alpha}}{R}\right)
$$假设有一个顺时针方向的循环,如图6.25a所示,有两个驻点 $S_1$ 和 $S_2$ 存在于 $z$-plane。在 $\zeta$-平面,前驻点的变换 $S_1$ 可位于叶片的压力面(凹面),而背面 $S_2$ 可位于吸力侧(凸侧),图6.25b。考虑到锋利后缘处的流动情况,流体颗粒以无限大的速度从叶片的压力面(凹面)向吸力面(凸面)移动。增加循环会使两个停滞点移动。对于一个特定的 $\Gamma=\Gamma_K$库塔循环,后滞点 $S_2$ 与后缘重合。在这一点上,速度为零。被称为库塔条件,它规定了在无粘流动条件下的翼型,要产生足够的环流,后滞点必须与后缘重合。为了满足这个条件,我们利用复势式(6.10) $F(z)=\Phi+i \Psi$ 导数
$$
\frac{d f(z)}{d \zeta}=\frac{d f(z)}{d z} \frac{d z}{d \zeta}=\frac{d f(z) / d z}{d \zeta / d z}=u-i v
$$使用Joukowsky变换函数,我们找到
$$
\frac{d f(z)}{d \zeta}=\frac{d f(z)}{d z}\left(\frac{z^2}{z^2-a^2}\right) .
$$
用于 $z \rightarrow \infty$,括号中的表达式接近unity,结果为
$$
\frac{d f(z)}{d \zeta}=\frac{d f(z)}{d z}
$$

物理代写|流体力学代写流体力学代考|循环的生成


在前几节中,我们推导了作为循环函数的升力关系(方程式。$6.105$和6.143)假设循环叠加在经过人体的平移流上,但没有解释这种循环是如何产生的。需要回答的问题是:如何解释这种循环流动的存在?为了回答这个问题,我们回到Prandtl[31]从机翼受不同流动模式的流动可视化实验。图$6.30$反映了[31]中显示的图像的物理内容。我们假定液体起初处于静止状态。图6.30a,使速度沿完全围绕翼型的曲线的线积分为零,因为所有的速度都为零。这将对应于启动后立即没有循环的潜在流动情况。图6.30b。根据汤姆逊定理,Eq.(6.166),无摩擦流体中的循环在任何时候都必须保持恒定(在这种情况下等于零),包括流体突然相对于翼型进入匀速平动运动的时刻。这显然与实验事实相矛盾,即在翼型周围有一个环流。考虑到翼型图$6.30 \mathrm{~b}$中锋利后缘附近的无限大速度,可以认为在启动后的第一个时刻,流动可能是一个没有循环的势流。然而,边界层中粘度的存在导致这个大速度发展成一个不连续的表面,如图6.30c所示。在锐利的后缘处,真实流体的粘度导致速度跳跃的均衡,导致被涡流占据的有限厚度层,如图6.30d所示。于是,这个涡层被卷成一个涡,即所谓的起始涡,图6.30e, $\mathrm{f}$。根据亥姆霍兹定理(下一节将讨论),这个漩涡总是与相同的流体粒子相关联,被流体冲走,并作为自由漩涡向下游对流。由于这个自由涡旋具有非零量级,它的存在显然与汤姆逊定理相矛盾。假设汤姆逊定理是有效的,开始的过程一定产生了另一个涡旋,其大小相同,但方向相反,因此它们的强度之和消失了。事实上,自由涡的存在总是伴随着另一种涡的存在,即束缚涡,图6.30g。计算闭合曲线$C \equiv A B C D F A, C_B \equiv A B E F A$和$C_F \equiv B C D F B$周围的循环,我们得到$\Gamma=\oint_{(C)} \boldsymbol{V} \cdot d \boldsymbol{C}=\oint_{\left(C_B\right)} \boldsymbol{V} \cdot d \boldsymbol{C}+\oint_{\left(C_F\right)} \boldsymbol{V} \cdot d \boldsymbol{C}=\Gamma_B+\Gamma_F=0$,从中我们得出$\Gamma_B=-\Gamma_F$。实验验证了汤姆逊涡旋定理的有效性。支持汤普逊定理的最重要的特征是粘性效应,没有粘性效应就不能产生涡旋


在生成我们在图6.30中总结的[6]中的涡旋图像时,Prandtl首先将翼型保持在一个固定的位置,暴露在流动的流体中。在第二组实验中,他将翼型相对于未受干扰的流体移动。在两种情况下观察到相同的现象

物理代写|流体力学代写Fluid Mechanics代考

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