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物理代写|流体力学代写Fluid Mechanics代考|Induced Drag Force

So far, we have been dealing with two-dimensional airfoils of infinite span with a bound vortex of constant strength. The superposition of a circulation with a parallel flow generated a lift force, which is the result of the pressure difference between the suction surface (convex surface) and the pressure surface (concave surface). In the case of an airfoil with a finite span, the pressure difference at both tips of the airfoil causes a secondary flow motion.

Figure 6.36a shows the inception of the secondary flow on both tips of a wing. This secondary flow creates tip vortices which induce downward velocities that change the flow pattern of a two-dimensional flow to a three-dimensional one. At the tips, the pressure difference and, thus, the circulation, disappears leading to a circulation distribution that varies from the mid-section of the wing towards both tips, Fig. 6.36b. Immediately behind the trailing edge, a surface separates the flow which has passed over the suction surface from that which passed over the pressure surface. A surface of discontinuity is formed which is occupied by free vortices, Fig. 6.36c and d, as detailed in Sect. 6.6.3. This vortical layer is unstable and rolls itself up to form two discrete vortices with opposite circulation directions, Fig. 6.36e, f. These vortices are responsible for inducing a downward velocity $w_{\text {ind }}$ which is superimposed on the undisturbed velocity $\boldsymbol{V}{\infty}$, changing the effective angle of attack from $\alpha{\infty}$ to $\alpha=\alpha_{\infty}-\varepsilon$ and the resultant velocity to $\boldsymbol{V}R$, as shown in Fig. 6.37. According to the Kutta-Joukowsky theorem, in an inviscid flow field, the lift force is perpendicular to the plane spanned by the velocity and the circulation vectors. Considering an infinitesimal lift force brought about by an infinitesimal wing span $d x$ as $d I .=\rho V{\infty} \Gamma d x$, the infinitesimal induced drag is calculated from $d \boldsymbol{D}=d \boldsymbol{I} \cdot \tan \varepsilon$ with $\varepsilon$ as the induced angle. Since $V_{\infty}>>w_{\text {ind }}$, we may approximate $\tan \varepsilon=\varepsilon$, which leads to
$$
d D=d L \frac{w_{\text {ind }}}{V_{\infty}} .
$$
Integrating Eq. (6.207) gives
$$
D_{\text {ind }}=\int_{-b / 2}^{b / 2} \frac{w_{\text {ind }}}{V_{\infty}} d L=\int_{-b / 2}^{b / 2} \frac{w_{\text {ind }}}{V_{\infty}} V_{\infty} \Gamma(x) d x=\int_{-b / 2}^{b / 2} w_{\text {ind }} \Gamma(x) d x .
$$

物理代写|流体力学代写Fluid Mechanics代考|Steady Viscous Flow Through a Curved Channel

Solving the Navier-Stokes equation, we investigate the influence of curvature and pressure gradient on the flow temperature and velocity distribution. The flows within curved channels under adverse, zero, and favorable pressure gradients are encountered in numerous practical devices such as compressor and turbine blades, diffusers and nozzles. Within these devices the distribution of flow quantities such as the temperature and velocity and consequently the flow behavior are affected primarily by the curvature and pressure gradient. To calculate the above quantities, conservation laws of fluid mechanics and thermodynamics are applied. For an incompressible Newtonian fluid, the Navier-Stokes equation describes the flow motion completely. This equation has exact solutions for only a few special cases. For the major part of practical problems encountered in applied fluid mechanics, however, it is hardly possible to find any exact solutions. This deficiency is in part due to the complexity of the individual flow field and its geometry under consideration. Despite this fact, the existence of exact solutions of fluid mechanics problems including the velocity and temperature distribution within viscous flows are of particular interest to the computational fluid dynamics (CFD) community dealing with development of CFD-codes. A comprehensive code assessment and validation requires both the experimental verification and theoretical confirmation. For the latter case, a comparison with existing exact solutions exhibits an appropriate procedure to demonstrate the code capability. For symmetric flows through channels with positive and negative pressure gradients exact solutions are found by Jeffery [40]. For asymmetric curved channels with convex and concave walls, exact solutions of the Navier-Stokes equation are found by Schobeiri $[42,43]$, where the influence of the wall curvature on the velocity distribution is discussed. Furthermore, a class of approximate solutions of Navier-Stokes is presented in [44].

This section treats the influence of curvature and pressure gradient on temperature and velocity distributions by solving the energy and momentum equations. Under the assumption that the flow is two dimensional, steady, incompressible, and has constant viscosity, the conservation laws of fluid mechanics and thermodynamics are transformed into a curvilinear coordinate system. The system describes the twodimensional, asymmetrically curved channels with convex and concave walls. As a result, exact solutions for the equation of energy as well as the Navier-Stokes equation are found.

物理代写|流体力学代写Fluid Mechanics代考|ENGR30002

物理代写|流体力学代写流体力学代考|诱导阻力


到目前为止,我们一直在处理具有恒强度边界涡的无限跨度二维翼型。循环与平行流的叠加产生升力,这是吸力面(凸面)和压力面(凹面)之间的压力差的结果。在有限跨度翼型的情况下,翼型两端的压差导致二次流运动


图6.36a显示了翼尖两侧二次流的起始。这种二次流产生了叶尖涡,引起了向下的速度,从而改变了二维流的流动模式为三维流。在翼尖处,压差和循环消失,导致从翼中部到翼尖的循环分布变化,图6.36b。紧靠后缘的后面,有一个表面将通过吸力面与通过压力面的流动分开。形成一个不连续面,由自由涡占据,图6.36c和d,详见6.6.3节。这个涡层是不稳定的,它自身卷起形成两个循环方向相反的离散涡,图6.36e, f。这些涡负责诱导一个向下的速度$w_{\text {ind }}$叠加在未受干扰的速度$\boldsymbol{V}{\infty}$上,使有效迎角由$\alpha{\infty}$变为$\alpha=\alpha_{\infty}-\varepsilon$,使合成的速度变为$\boldsymbol{V}R$,如图6.37所示。根据库塔-朱可夫斯基定理,在无粘流场中,升力垂直于速度向量和循环向量张成的平面。考虑无穷小翼展$d x$为$d I .=\rho V{\infty} \Gamma d x$所带来的无穷小升力,以$\varepsilon$为诱导角,从$d \boldsymbol{D}=d \boldsymbol{I} \cdot \tan \varepsilon$计算无穷小诱导阻力。由于$V_{\infty}>>w_{\text {ind }}$,我们可以近似$\tan \varepsilon=\varepsilon$,从而得到
$$
d D=d L \frac{w_{\text {ind }}}{V_{\infty}} .
$$
积分式(6.207)给出
$$
D_{\text {ind }}=\int_{-b / 2}^{b / 2} \frac{w_{\text {ind }}}{V_{\infty}} d L=\int_{-b / 2}^{b / 2} \frac{w_{\text {ind }}}{V_{\infty}} V_{\infty} \Gamma(x) d x=\int_{-b / 2}^{b / 2} w_{\text {ind }} \Gamma(x) d x .
$$

物理代写|流体力学代写流体力学代考|通过弯曲通道的稳定粘性流


求解Navier-Stokes方程,研究了曲率和压力梯度对流动温度和速度分布的影响。在许多实际装置中,如压气机和涡轮叶片、扩散器和喷嘴等,都遇到了不利、零和有利压力梯度下弯曲通道内的流动。在这些装置中,流动量的分布,如温度和速度,从而流动行为主要受曲率和压力梯度的影响。为了计算上述量,应用了流体力学和热力学的守恒定律。对于不可压缩的牛顿流体,Navier-Stokes方程完全描述了流动运动。这个方程只有在少数特殊情况下才有精确解。然而,对于应用流体力学中遇到的大部分实际问题,几乎不可能找到任何确切的解决方案。这种不足部分是由于单个流场及其几何结构的复杂性。尽管如此,流体力学问题(包括粘性流动中的速度和温度分布)的精确解的存在,对计算流体动力学(CFD)社区处理CFD代码的开发特别有兴趣。一个全面的代码评估和验证需要实验验证和理论确认。对于后一种情况,与现有精确解的比较显示了演示代码功能的适当过程。对于具有正负压梯度通道的对称流动,Jeffery[40]给出了精确解。对于凹凸壁面的非对称弯曲通道,Schobeiri $[42,43]$给出了Navier-Stokes方程的精确解,讨论了壁面曲率对速度分布的影响。进一步,在[44]. . . >中给出了一类Navier-Stokes近似解


本节通过求解能量和动量方程来讨论曲率和压力梯度对温度和速度分布的影响。在流体为二维稳态、不可压缩、黏度恒定的假设下,将流体力学和热力学的守恒定律转化为曲线坐标系。该系统描述了具有凸壁和凹壁的二维非对称弯曲通道。结果得到了能量方程和Navier-Stokes方程的精确解

物理代写|流体力学代写Fluid Mechanics代考

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