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|流体力学代写流体力学代考|诱导阻力

$$d D=d L \frac{w_{\text {ind }}}{V_{\infty}} .$$

$$D_{\text {ind }}=\int_{-b / 2}^{b / 2} \frac{w_{\text {ind }}}{V_{\infty}} d L=\int_{-b / 2}^{b / 2} \frac{w_{\text {ind }}}{V_{\infty}} V_{\infty} \Gamma(x) d x=\int_{-b / 2}^{b / 2} w_{\text {ind }} \Gamma(x) d x .$$

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

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