# 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|流体力学代写流体力学代考|循环，提升，库塔条件

$$F(z)=V_{\infty}\left(z e^{-i \alpha}+\frac{R^2}{z} e^{i \alpha}\right)-i \frac{\Gamma}{2 \pi} \ln \left(\frac{z e^{-i \alpha}}{R}\right)$$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
$$\frac{d f(z)}{d \zeta}=\frac{d f(z)}{d z}\left(\frac{z^2}{z^2-a^2}\right) .$$

$$\frac{d f(z)}{d \zeta}=\frac{d f(z)}{d z}$$

## 物理代写|流体力学代写流体力学代考|循环的生成

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