14 12/05

1. Identify the boundary layer variables $x$, $y$, $u$, and $v$ for the case of a circular cylinder of radius $r_0$ in terms of the cylindrical variables $r$, $\theta$, $v_r$, and $v_\theta$. Be careful of the difference between $r$ and the radius of the cylinder $r_0$.

2. Using the result of the previous question, write the true continuity equation in cylindrical coordinates from table C.3 in terms of the boundary layer coordinates and comment on the differences from the boundary layer continuity equation. Is the difference small if the boundary layer is thin? If it is thick?

3. According to what was said in class, if a circular cylinder is impulsively set into motion, the velocity profile in the boundary layer at small times is given by the Stokes layer profile $u=u_e\;{\rm erf}(y/\sqrt{4\nu t})$ where $u_e$ is the potential flow velocity just above the boundary layer, equal to $2U\sin\theta$, ($\theta$ measured clockwise from the front stagnation point.) And $y$ is, of course, the boundary layer coordinate. The corresponding velocity component in the potential flow, which hopefully in question 1 you found to be $v_\theta$, is given by $\partial\phi/r\partial\theta$ or $U(1+r_0^2/r^2)\sin\theta$ or $\frac12u_e(1+r_0^2/r^2)$. A composite profile can be formed by adding the two expressions and substracting the part $u_e$ that they have in common:
   
   $$u = u_e {\rm erf}\left(\frac{y}{\sqrt{4\nu t}}\right) + {\textstyle\frac12}u_e\left(1+\frac{r_0^2}{(r_0+y)^2}\right) - u_e$$
   
   
   Plot this composite velocity profile (1) near the front of the cylinder, $\theta=30^\circ$, (2) on top of the cylinder, $\theta=90^\circ$, and (3) near the rear stagnation point, $\theta=150^\circ$. Take $U=r_0=1$, $\nu=0.001$, and $t=1$. Plot first over the $y$-range from 0 to $3r_0$ and then from 0 to $0.15r_0$ to see the details. Use a spacing of no more than 0.003 in your $y$-values. You do not have to plot the first and third profiles slanted over 30 degrees, just plot them in the normal way (i.e. $y$ vertical and $u$ horizontal.) Note that the composite profile, like the true one, is not exactly constant and actually somewhat less than $u_e$ immediately above the boundary layer. The higher the Reynolds number, the smaller these errors, but the harder to see the profile.

4. According to Blasius, (the same one as from the flat plate, he was a student of Prandtl,) to better approximation, the velocity in the boundary layer is given by two terms,
   
   $$u_e {\rm erf}\left(\frac{y}{\sqrt{4\nu t}}\right) + t u_e ... ...}u_e}{{\rm d}x} \zeta_1'\left(\frac{y}{\sqrt{4\nu t}}\right)$$
   
   
   The second term is new. Add this second term to the composite solution above and replot. You should now see that flow reversal has occured at the rear of the cylinder, a first step in creating separation and the wake.

   BTW, the function $\zeta_1'$ as found by Blasius can be evaluated by saving the following code as a file zeta1p.m:

   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   %%% Blasius second function for impulsively started boundary layer flow. %%%
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
   % This really returns function zeta_1', which is the scaled second order
   % contribution to the velocity profile.  Argument eta may be an array.
   
   % start of the function definition
   function z1p = zeta1p(eta)
   
   % constant
   c1=1/sqrt(pi);
   
   % store the values of the exponential
   ex=exp(-eta.^2);
   
   % store the values of the complementary error function
   erc=erfc(eta);
   
   % compute function zeta_1' according to the result of Blasius
   z1p=-3*c1*eta.*ex.*erc+.5*(2*eta.^2-1).*erc.^2+2/pi*ex.^2+c1*eta.*ex...
   +2*erc-4/(3*pi)*ex+(3+4/(3*pi))*c1*(eta.*ex-.5/c1*(2*eta.^2+1).*erc);
   

   Also, ${\rm d}u_e/{\rm d}x$ is $u_e$ with the sine replaced by a cosine (and an $r_0$, but it is one). Only replot over the range from 0 to 0.15.

5. Finishing the class notes, derive the equations for the Blasius solution for steady laminar boundary layer flow along a flat plate. Do so by making the similarity assumption that $u=Uf'(\eta)$ where $\eta=y/\delta$, with $\delta=\delta(x)$ a typical boundary layer thickness. Use a separation of variables argument similar to the one we used in the Stokes second problem to show that $\delta=\sqrt{\nu x/U}$, except for a chosen constant that is not important, and that function $f$ satisfies the ODE:
   
   $$f''' + \frac12 f f'' = 0$$
   
   
   with boundary conditions:
   
   $$f(0)=f'(0)=0 \qquad f'(\infty)=1$$
   
   

6. Integrate the Blasius equation numerically to $\eta=10$. To do so, define the following computational variables:
   
   $$x_1 = f \qquad x_2 = f' \qquad x_3 = f''$$
   
   
   Verify that these variables satisfy the system of first order ordinary differential equations:
   
   $$x_1' = x_2 \qquad x_2' = x_3 \qquad x_3' = -\frac12 x_1 x_3$$
   
   
   To integrate this system numerically in Matlab or similar, you must define it numerically by creating a file blaseq.m containing:

   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   %% Definition of the first order system of ODE that represents the Blasius %%
   %%%% equation for the boundary layer along a semi-infinite flat plate. %%%%%%
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
   % Notations:
   %                   x(1) = f,   x(2) = f',   x(3) = f''
   % The initial condition for f'' at eta = 0 should be about 0.33206.  The
   % other two initial conditions are zero.
   
   % Return the derivatives of the x-values
   function xp = blaseq(eta,x)
   xp=[x(2); x(3); -.5*x(1)*x(3)];
   

   Then you can integrate the system using a statement like

   [eta_vals, x_vals] = ode45(@blaseq, [0 10], [0; 0; tau_wall]);
   

   where tau_wall is $f''(0)$, which physically corresponds to the scaled nondimensional wall shear. According to the book, its value is 0.33206. Verify that this is right by also trying the values 0.2 and 0.4 and see what is wrong then. (Hint: examine array x_vals.)

7. Plot the scaled nondimensional velocity profile in the normal way (i.e. $y/\delta$ vertical and $u/U$ horizontal.) Take the range of $y/\delta$ to be from 0 to 10. Hint: to get hold of column number $i$ of array x_vals, use x_vals(:,i).

8. The vorticity in a boundary layer can be approximated as
   
   $$\omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \approx - \frac{\partial u}{\partial y}$$
   
   
   since $v$ is small and $x$ is not. Show that the scaled vorticity $-\omega_z\sqrt{\nu x/U^3}$ corresponds to $f''(\eta)$. Plot the scaled vorticity profile. Does the boundary layer flow turn into a potential flow for large $y/\delta$ as the matching idea requires?