# 物理代写|空气动力学代写Aerodynamics代考|ENGR087

## 物理代写|空气动力学代写Aerodynamics代考|Thin Shear Layer Navier-Stokes Equations

In the open form of Navier-Stokes Eqs. (2.53) , we observe the existence of second derivatives for the velocity and the temperature. This implies that the Navier-Stokes equations are second order partial differential equations. When the freestream speed is high, the Reynolds number is high. This makes the gradients of the flow parameters to be high normal to the surface as compared to the gradients parallel to the surface. Therefore, we can neglect the effect of the viscous terms which are parallel to the flow surface and simplify Eqs. 2.53. Let us now, perform some order of magnitude analysis for the simplification process on a simple wing surface immersed in a high free stream speed given in Fig. 2.7.

Since we consider the air flowing over the wing as a real gas, the boundary conditions on the surface will be (i) no slip condition and (ii) the wall temperature specification. According to Fig. 2.7, the wing surface is almost parallel to xy plane where the molecular diffusion parallel to the $\mathrm{xy}$ plane is negligible compared to the diffusion taking place normal to the surface. This is because of high free stream speed transporting the properties in the parallel direction much faster than the molecular diffusion. On the other hand, because of no slip condition, the gradients which are normal to the surface are much higher than the gradients parallel to the surface. The order of magnitude analysis performed on the terms of Eq. $2.53$ gives
$$\frac{1}{R_e} \frac{\partial \hat{\mu}}{\partial \hat{z}}\left(\frac{\partial}{\partial \hat{z}}\right) \gg \frac{1}{R_e} \frac{\partial \hat{\mu}}{\partial \hat{x}}\left(\frac{\partial}{\partial \hat{x}}\right), \ldots, \frac{1}{R_e} \frac{\partial \hat{\mu}}{\partial \hat{y}}\left(\frac{\partial}{\partial \hat{y}}\right) .$$
The approximate form of the equations result in modeling an external real gas flow which takes place in a thin shear layer around the wing surface. Therefore, the first approximate form of Eq. $2.53$ is called ‘Thin Shear Layer Navier-Stokes Equations’ which are to be introduced next

## 物理代写|空气动力学代写Aerodynamics代考|Boundary Layer Equations

In the attached or slightly detached external flow cases, we can obtain the surface pressure distribution using the methods described in Sect. 2-A and further simplify set of Eqs. $2.49$ and 2.54. In these simplifications we again resort to the order of magnitude analysis. Assuming again that the viscous effects are only in the vicinity of the surface of the body, we can consider the gradients and the diffusion normal to the surface we obtain
Continuity: $\frac{\partial \rho}{\partial t}+\frac{\partial \rho u}{\partial x}+\frac{\partial \rho w}{\partial z}=0$
Continuity of
The species: $\rho \frac{\partial c_i}{\partial t}+\rho u \frac{\partial c_i}{\partial x}+\rho w \frac{\partial c_i}{\partial z}=\frac{\partial}{\partial z}\left(\rho D_{12} \frac{\partial c_i}{\partial z}\right)+\dot{w}i$ $\mathrm{x}-$ momentum: $\rho \frac{\partial u}{\partial t}+\rho u \frac{\partial u}{\partial x}+\rho w \frac{\partial u}{\partial z}=-\frac{\partial p}{\partial x}+\frac{\partial}{\partial z}\left(\mu \frac{\partial u}{\partial z}\right)$ $\mathrm{z}-$ momentum $: \frac{\partial p}{\partial z}=0$ Energy: $$\rho \frac{\partial h}{\partial t}+\rho u \frac{\partial h}{\partial x}+\rho w \frac{\partial h}{\partial z}=\frac{\partial p}{\partial t}+u \frac{\partial p}{\partial x}+\mu\left(\frac{\partial u}{\partial z}\right)^2+\frac{\partial}{\partial z}\left(k \frac{\partial T}{\partial z}\right)+\frac{\partial}{\partial z}\left(\rho D{12} \sum_i h_i \frac{\partial c_i}{\partial z}\right)$$
Here, $x$ is the direction parallel to the surface, $z$ is the normal direction and $\mathrm{h}_i$ in Eq. $2.62$ is the enthalpy of species $i$.

The real gas effect in an external flow can be measured with the change caused in the stagnation enthalpy. If we neglect the effect of vertical velocity component, the stagnation enthalpy of the boundary layer flow reads: $h_{\mathrm{o}}=h+u^2 / 2$. The normal gradient of the stagnation enthalpy at a point then reads

## 物理代写|空气动力学代写空气动力学代考|薄剪切层Navier-Stokes方程

$$\frac{1}{R_e} \frac{\partial \hat{\mu}}{\partial \hat{z}}\left(\frac{\partial}{\partial \hat{z}}\right) \gg \frac{1}{R_e} \frac{\partial \hat{\mu}}{\partial \hat{x}}\left(\frac{\partial}{\partial \hat{x}}\right), \ldots, \frac{1}{R_e} \frac{\partial \hat{\mu}}{\partial \hat{y}}\left(\frac{\partial}{\partial \hat{y}}\right) .$$

## 物理代写|空气动力学代写空气动力学代考|边界层方程

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