Page 192 - Bird R.B. Transport phenomena

176 Chapter 5 Velocity Distributions in Turbulent Flow

(b) Show that, if fl is below some threshold value ft , the angle 0 is zero. Above the thresh-
thr
old value, show that there are two admissible values for 0. Explain by means of a carefully
drawn sketch of 0 vs. П. Above fl label the two curves stable and unstable.
thr
(c) In (a) and (b) we considered only the steady-state operation of the system. Next show that
the equation of motion for the sphere of mass m is
Jin
mL^ = mQ, L sin 0 cos 0 - mg sin 0 (5С.З-2)
2
dt
Show that for steady-state operation this leads to Eq. 5C.3-1. We now want to use this
equation to make a small-amplitude stability analysis. Let 0 = 0 + в и where 0 is a steady-
O
O
state solution (independent of time) and 0, is a very small perturbation (dependent on
time).
(d) Consider first the lower branch in (b), which is 0 = 0. Then sin 0 = sin 0! « 6 and cos 0 =
O }
cos 0] ~ 1, so that Eq. 5B.2-2 becomes
(5С.З-3)
ш1
We now try a small-amplitude oscillation of the form в^ = АЩе~ \ and find that
(5C.3-4)
Now consider two cases: (i) If fl 2 < g/L, both co and a>_ are real, and hence 6 } oscillates; this
+
2
indicates that for П < g/L the system is stable, (ii) If П > g/L, the root ш^ is positive imagi-
2
2
nary and e~ 10)t will increase indefinitely with time; this indicates that for П > g/L the system
is unstable with respect to infinitesimal perturbations.
(e) Next consider the upper branch in (b). Do an analysis similar to that in (d). Set up the
equation for 0^ and drop terms in the square of 0] (that is, linearize the equation). Once again
ш
try a solution of the form 9 } = АЩе~ *}. Show that for the upper branch the system is stable
with respect to infinitesimal perturbations.
(f) Relate the above analysis, which is for a system with one degree of freedom, to the prob-
lem of laminar-turbulent transition for the flow of a Newtonian fluid in the flow between two
counter-rotating cylinders. Read the discussion by Landau and Lifshitz 5 on this point.
5D.1 Derivation of the equation of change for the Reynolds stresses. At the end of §5.2 it was
pointed out that there is an equation of change for the Reynolds stresses. This can be derived
by (a) multiplying the fth component of the vector form of Eq. 5.2-5 by v- and time smoothing,
(b) multiplying the ;th component of the vector form of Eq. 5.2-5 by v\ and time smoothing, and
(c) adding the results of (a) and (b). Show that one finally gets

p — v V = -p(vV • Vv) - p(vV • Vvf - p{V • v'vV)
Dt

+
- (v'Vp'] - (v'V//) + 4- J v ' W + {v'VV) j (5D.1-1)
Equations 5.2-10 and 11 will be needed in this development.
5D.2 Kinetic energy of turbulence. By taking the trace of Eq. 5D.1-1 obtain the following:

§ (V • \ V4') - (V • ^V) + /x(v' • W ) (5D.2-1)
t P
Interpret the equation. 6

5 L. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, Oxford, 2nd edition (1987), §§26-27.
6 H. Tennekes and J. L. Lumley, A First Course in Turbulence, MIT Press, Cambridge, Mass. (1972), §3.2.