Question

(1) Consider a turbulent wake far behind a 2D airfoil as shown in Fig. 1. According to a simple model of turbulence, the velocity profile at the far downstream of the wake is given by $u_x(x, y) = V_\infty - Cx^{-1/2} exp(-\alpha y^2 / x)$, where $x$ is the streamwise coordinate, $y$ is the coordinate across the wake, $V_\infty$ is the freestream velocity, and $C$ and $\alpha$ are constants. (a) Please calculate the drag using the following formula based on wake survey (Hint: x is treated as a constant in the integral) (10 points) $D = \rho \int_{-\infty}^{\infty} u_x(x, y)[V_\infty - u_x(x, y)] dy$ (b) Should the drag D depend on x? (5 points) (c) If the drag D depends on x, is there anything wrong? Give your explanation based on your physical intuition. (5 points) Note that you have a special integral called the Gaussian integral. The value of the Gaussian integral can be found in mathematics handbooks or internet.

          (1) Consider a turbulent wake far behind a 2D airfoil as shown in Fig. 1. According to a simple model of turbulence, the velocity profile at the far downstream of the wake is given by
$u_x(x, y) = V_\infty - Cx^{-1/2} exp(-\alpha y^2 / x)$, where $x$ is the streamwise coordinate, $y$ is the coordinate across the wake, $V_\infty$ is the freestream velocity, and $C$ and $\alpha$ are constants.
(a) Please calculate the drag using the following formula based on wake survey (Hint: x is treated as a constant in the integral) (10 points)
$D = \rho \int_{-\infty}^{\infty} u_x(x, y)[V_\infty - u_x(x, y)] dy$
(b) Should the drag D depend on x? (5 points)
(c) If the drag D depends on x, is there anything wrong? Give your explanation based on your physical intuition. (5 points)
Note that you have a special integral called the Gaussian integral. The value of the Gaussian integral can be found in mathematics handbooks or internet.

Added by Loretta C.

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