Chapter 5 & 6 - Separation

1. Problems

(i) Batchelor conjectured that the correct limit for the Nav ier-Stokes equations as $t\to \infty$ and then $R\to \infty$ of a recirculation region is a constant vorticity region of finite extent. Normally computations are based around letting $t\to \infty$ to obtain a steady solution and then allowing the Reynolds number to increase. Is it possible to compute solutions with the li mits interchanged, so $R\to \infty$ and then $t\to \infty$? One way this might be approached would be to take a viscous calculation at finite Reynolds number as the starting point for a time dependent inviscid (but rotational) calculation. How would a flow with a distribution of vorticity from a viscous calculation subsequently develop if viscosity were suddenly switched off? That is, from an initial distribution of vorticity $\omega_0(x,y)$ and corresponding stream function $\psi_0(x,y)$ solve

$$\omega_t+{\rm\bf u}\cdot\nabla\omega =0,\ \ \ \nabla^2\psi =-\omega ,$$ (1)

with appropriate boundary conditions giving $\psi=0$, $\omega_n=0$ ($n$ being normal derivative) on a body and uniform far field. It is difficult to see that how if this problem started with a recirculati ng region (such as from a viscous calculation of the steady vortex behind a cylinder) it would evolve by convection of vorticity alone to one where there were regions of constant vorticity. It would seem more plausible that the existing vorticity would be swept away into the wake and it would not be possible to achieve a Prandtl-Batchelor flow by this means.

(ii) In Brown & Stewartson(1969) it is claimed that logarithmic terms cancel so that the $u$ velocity is finite on the separation line (page 57 of their paper). It is very unusual for these normally meticulous authors not to reference a detailed justification of a remark such as this. Can you either track down a suitable reference or show this result yourself?

(iii) Modify Leigh's method using the initial profile

$$u(0,Y)=\left\{ \begin{array}{r@{\quad\quad} r} \sin kY,&kY<\pi /2,\\ 1,&kY\ge \pi /2.\end{array}\right.$$ (2)

and find a value $k$ such that the separation point is the same as predicted by Leigh for Howarth's starting profile,

$$\alpha_s =0.9585.$$ (3)

Then use Terrill's method to calculate the wall shear for a linearly decreasing velocity profile and compare the results from the two methods. Does Terrill's method predict the same separation point? What is your estimate of the singularity power $q$, where $\sigma\sim (\alpha_s-\alpha)^q$ near separation?

(iv) Leigh and Terrill each use a semi-implicit method to solve a set of non-linear equations. Apply Newton's method to solve Terrill's discrete non-linear equations. Is the computation more efficient?

(v) Implement Kawaguti's method to solve the Navier-Stokes equations. In Kawaguti & Jain the outer boundary condition was relaxed to have just the stream-function and vorticity zero. What effect does this have on the computed solution at $R_d=40$?

2. Software

(i) solvebrod.m calculate first $n$ Brodetsky coefficients $A_1,\ldots ,A_n$ for given separation angle
(ii) brodgen.f calculate flow using Brodetsky coefficients calculated by solvebrod.m
(iii) lcwcyl.f Woods' method for cylinder with constant pressure on free streamline
(iv) lcwcyl2.f Woods' extended model for cylinder with variable pressure on free streamline
(v) pohlhausen.f Pohlhausen's method to solve integral form of boundary layer equation
with either Heimenz observations, potential flow or linear variation for flow at the boundary layer edge
(vi) fskan.f determine existence of Falkner-Skan solutions by varying $f''(0)$ for fixed $\beta$
(vii) fskan1.f determine existence of Falkner-Skan solutions by varying $\beta$ for fixed $f''(0)$
(viii) burgers.f Burgers' method for either Heimenz or potential outer flow
(ix) leigh.f Leigh's method for calculating solution of boundary layer equation
(x) terrill.f Terrill's method for calculating solution of boundary layer equation
(xi) thom.f Thom's method to calculate solution of Navier-Stokes equations for flow about a cylinder

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