0.8 Multidimensional laminar flow  (Page 7/8)

Recall that the convection terms are the convective derivative of the $x$ component of momentum. Thus the equation of motion can be expressed as follows.

This equation can be used to describe the diffusion of momentum along a streamline which originated at $x=0$ at $t=0$ . If the transverse velocity is zero, each streamline would be at a constant value of $y$ . With steady flow, there is a one-to-one mapping between $x$ and $t$ along each streamline. However, this mapping is not the same for all streamlines because different streamlines have different velocities. Beyond the boundary layer, the velocity is the free stream velocity and at the surface of the plate, $y=0$ , the velocity is zero. We will make the assumption that the mapping for the entire boundary layer can be approximated by using the average of the free stream velocity and the velocity at the plate.

Also, we assume for the first approximation that the transverse velocity is zero such that each streamline is at a constant value of $y$ . The diffusion equation now transforms into a parabolic PDE in $x$ and $y$ .

This mapping of distance along the plate and time is substituted into the expression for the diffusion of momentum to a stationary plate that was introduced into a uniformly translating fluid at $t=0$ .

This first approximation neglects the transverse velocity and assumes that the time since passage of the front of the plate corresponds to the average velocity of the fluid in the boundary layer. Although this solution is quite close to the exact, Blasius solution, it does not satisfy the continuity equation. We now derive the second approximation by application of the continuity equation.

$\begin{array}{c}{v}_y=-{\int }_0^y\frac{\partial v_x}{\partial x}dy\approx -{\int }_0^y\frac{\partial v_x^{\left(1\right)}}{\partial x}dy\\ =\frac{U_{\infty }}{\sqrt{U_{\infty }x/\nu }}\left[1-exp\left(\frac{-y^2}{8\nu x/U_{\infty }}\right)\right]\end{array}$

The limiting value of the transverse velocity with this approximation is,

$\begin{array}{c}lim\\ y\to \infty \end{array}{v}_y=\frac{U_{\infty }}{\sqrt{U_{\infty }x/\nu }}=\frac{U_{\infty }}{\sqrt{{\mathrm{Re}}_x}}$

This limiting value of the transverse velocity differs from the Blasius solution only by a coefficient of 0.865.

The transverse velocity results in convection of momentum away from the wall. If the convection velocity was constant, then its effect can easily be taken into account with the solution to the convection-diffusion equation. However, the transverse convection increases from zero at the wall to the limiting value in the free stream. Thus it makes more sense to use the average transverse velocity between the wall and at a point in the boundary layer for substitution into the solution of the convective-diffusion equation.

$\begin{array}{ccc}\hfill {\overline{v}}_y& =& \frac{{\int }_0^y{v}_{y}dy}{y}\hfill \\ & =& \frac{U_{\infty }}{\sqrt{U_{\infty }x/\nu }}\left[1-\frac{\sqrt{\pi }}{2}\frac{\sqrt{8\nu x/U_{\infty }}}{y}\mathrm{erf}\left(\frac{\mathrm{y}}{\sqrt{8\nu x/U_{\infty }}}\right)\right]\hfill \end{array}$

The second approximation will include the transverse convection by substituting the average transverse velocity into the convective-diffusion solution.

$\begin{array}{ccc}\hfill v_x^{\left(2\right)}& =& U_{\infty }\mathrm{erf}\left(\frac{y-{\overline{v}}_{y}x/U_{\infty }}{\sqrt{8\nu x/U_{\infty }}}\right)\hfill \\ & =& U_{\infty }\mathrm{erf}\left\{\frac{\mathrm{y}}{\sqrt{8\nu x/U_{\infty }}}-\frac{1}{\sqrt{8}}\left[1-\frac{\sqrt{\pi }}{2}\frac{\sqrt{8\nu x/U_{\infty }}}{y}\mathrm{erf}\left(\frac{\mathrm{y}}{\sqrt{8\nu x/U_{\infty }}}\right)\right]\right\}\hfill \end{array}$

The first and second approximations are compared with the Blasius exact solution and quadratic and cubic approximation in the following figure.

The inclusion of the transverse convection made an insignificant improvement in the velocity profile. Thus it will be neglected in the following. The drag coefficient is calculated from the first approximation.