# 物理代写|空气动力学代写Aerodynamics代考|MAE424

## 物理代写|空气动力学代写Aerodynamics代考|Turbulence Modeling

At high free stream speeds external flows are likely to go through a transition from laminar to turbulence on the airfoil surface close to the leading edge. Depending on the value of the Reynolds number most of the flow on the airfoil becomes turbulent. The Reynolds decomposition technique applied to the Navier-Stokes equations results in new unknowns of the flow field called Reynolds stresses. These new unknowns introduce more unknowns than the existing equations which is called the closure problem of turbulence. In order to close the problem, the Reynolds stresses are empirically modeled in terms of the velocity gradients. All these models aim at finding the suitable value of turbulence viscosity $\mu_{\mathrm{T}}$ applicable for different flow cases. The empirical turbulence models are in general based on the wind tunnel tests and some numerical verification. The simplest models of turbulence are the algebraic models. More complex models are based on differential equations. Although so many models have been introduced, there has not been a satisfactory model developed to reflect the main characteristics of a turbulent flow. Now, we present the well known Baldwin-Lomax model which is used for the numerical solution of attached or separated, incompressible or compressible flows of aerodynamics. This model is a simple algebraic model which assumes the turbulent region to be composed of two different layers. Accordingly the turbulence viscosity reads
$$\mu_T=\left{\begin{array}{l} \left(\mu_T\right)i, \text { for } z \geq z_c \ \left(\mu_T\right)_o, \text { for } z{\mathrm{c}} is the shortest distance where inner and outer viscosity values are equal. The inner viscosity value in terms of the mixing length l and the vorticity \omega reads as$$
\left(\mu_T\right)i=\rho l^2|\omega| R_e \quad \text { and } \quad l=\kappa z\left[1-\exp \left(-z^{+} / A^{+}\right)\right] \quad(2.77 \mathrm{a}, \mathrm{b}) $$Here, \kappa=0.41 is the von Karman constant, \mathrm{A}^{+}=26 damping coefficient and z^{+}=z \sqrt{|\omega| R_e}. The outer viscosity, on the other hand$$ \left(\mu_T\right)_o=K C{c p} F_w F_{k l}(z), \quad F_w=z_{m a k s} F_{m a k s}
$$## 物理代写|空气动力学代写Aerodynamics代考|Initial and Boundary Conditions The study of aerodynamical problems with real gas effects requires solution of a system of partial differential equations which are first order in time and second order in space coordinates. In order to solve Eq. 2.49 to determine the flow field, all dependent variables must be prescribed at time t=0, and for all times t at the boundaries of the computational domain. All the prescribed values must be in accordance with the physics of the problem. As the initial conditions for the unknown values of \mathbf{U} we prescribe the undisturbed flow conditions, i.e., u=1, v=w=0 which represents the impulsive start of the flow. Under these conditions the initial values for the unknown vector in generalized coordinates become$$
\vec{U}(t=0, \xi, \eta, \varsigma)=\left(\begin{array}{c}
\rho^0 \
\rho^0 \
0 \
0 \
\varepsilon^0 \
c_i^0
\end{array}\right)
$\backslash \mathrm{mu}{-} T=\backslash$ left ${$ begin ${$ array $}{}$ istheshortestdistancewhereinnerandouterviscosityvaluesareequal. Theinnerviscosityvalueinterms landthevorticity $y$ 欧米茄readsas $\left(\mu_T\right) i=\rho l^2|\omega| R_e \quad$ and $\quad l=\kappa z\left[1-\exp \left(-z^{+} / A^{+}\right)\right] \quad$ (2.77a, b)Here, $\backslash$ kappa $=0.41$ isthevonKarmanconstant, $\backslash$ mathrm ${A}^{\wedge}{+}=26$ dampingcoefficientand $\mathrm{z}^{\wedge}{+}=\mathrm{Z} \backslash$ sqrt{|\mega| 回 覆}. Theouterviscosity, ontheotherhand $\left(\mu_T\right)_o=K C c p F_w F{k l}(z), \quad F_w=z_{\text {maks }} F_{\text {maks }} \$$## 物理代写|空气动力学代写Aerodynamics代考|初始条件和边界条件 研究具有实际气体效应的空气动力学问题需要求解一个时间一阶、空间坐标二阶的偏微分方程组。为了解 决方程式。 2.49 为了确定流场，必须在时间规定所有因变量 t=0, 并且一直 t 在计算域的边界。所有规定 的值都必须符合问题的物理性质。作为末知值的初始条件 U我们规定了不受干扰的流动条件，即 u=1 ， v=w=0 这代表了流程的冲动开始。在这些条件下，广义坐标中末知向量的初始值变为$$ \vec{U}(t=0, \xi, \eta, \varsigma)=\left(\rho^0 \rho^0 00 \varepsilon^0 c_i^0\right)$$这里，$\rho^0$是密度的初始值，$\varepsilon^0$是能量的初始值，并且$c_i^0$是的初始值$i$种。 至于边界条件：(i) 地表末知数，(ii) 必须提供远场边界条件。 因此: (i) 作为地面无滑移条件:$\mathbf{U}(\mathrm{t}, \xi, \eta, \varsigma=0)=\mathbf{0}$是规定的。(在图。$2.6, \varsigma=0$规定了表面)。在反应流 中，表面的催化性决定了浓度梯度的值， (ii) 在远场：对于$\varsigma_\rho=\varsigma_{\text {maks }} \mathbf{U}\left(\mathrm{t}, \xi, \eta, \zeta=\zeta_{\text {maks }}\right)=\vec{U} \infty$是规定的，并且通量条件在$\xi=\xi$maks 是$\frac{\partial \vec{U}}{\partial \xi}=(\overrightarrow{0})$(iii) 如果存在如图 2.6b 所示的对称条件，我们将垂直于对称的通量规定为$\frac{\partial \vec{U}}{\partial \eta}=(\overrightarrow{0})\$.

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