Stefan LLEWELLYN SMITH
EBUII 574
x23475
http://mae.ucsd.edu/~sgls
Homework III
Due
Friday January 31, 2020, in class (or before).
Problems (MYO is the textbook; 8th edition)
MYO 8.19
MYO 8.24
MYO 8.26 (I would skip the optional part)
MYO 8.31
MYO 8.40
MYO 8.57
MYO 8.63
Write a paragraph about Colebrook, who has a formula named
after him. Find some more recent work by Barenblatt and
co-authors on flow in pipes. What do these papers claim?
Comments
This homework covers Sections 8.2–8.4 of the book.
Section 8.2 revisits fully developed laminar flow. You already saw
Poiseuille flow in MAE 101A. This is chance to reproduce the
important scaling between pressure drop and diameter. We will also
consider the balance of forces on a section of fluid in the pipe and
consider the effect of gravity. Developing an understanding how to
obtain the flow profile both from a force balance and the
Navier-Stokes equation is useful for later. Dimensional analysis can
also give the appropriate pressure drop, but not the profile.
Finally we will look at the energy balance in this laminar flow.
Turbulent flow is critical to engineering applications. The Reynolds
number is greater than 4000 in many situations. As explained in
section 8.3, the relation of shear stress to average velocity
gradient is no longer one of simple proportionality. Instead,
Reynold stress terms contribute. These represent the average effect
of products like u'v', which are related to the transport of
momentum by eddies. Empirical theories for these terms exist, but
they require matching to experiments and can have limited ranges of
validity. The structure of turbulent flow in a pipe is more complex
than for laminar flow. There is a (thin) viscous sublayer near the
wall in which viscosity is important and the velocity is
approximately linear. Then there is an overlap region in which the
velocity changes logarithmically. This is sometimes called "the law
of the wall." Finally, there is an outer region. In a pipe,
power-law profiles are used as empirical fits.
We will see later in the course what happens for flow over a body.
In that case, the flow is not fully developed and the boundary layer
grows downstream. Experiments show that the transition to turbulence
happens for larger Re.
Section 8.4 returns to the dimensional analysis of Section 8.2, but
for general flow. This leads to a relation between pressure drop,
length of pipe over diameter, Reynolds number and (non-dimensional)
roughness that depends on the friction factor. The head loss in the
energy equation can be related to the friction factor. The Moody
chart plots friction factor as a function of Reynolds number and
roughness. There are empirical fits to the curves such as the
Colebrook formula. The losses caused by flow through a long straight
pipe are called major losses. Other losses come e.g. from fittings
such as valves, bends, expansions, etc. They are called minor losses
and can be parameterized. Finally, non-square ducts can be
accommodated by defining a hydraulic radius. These results are
combined to relate quantities in applications, in particular length,
diameter head loss and flow rate (see Section 8.5). In some
situations a nonlinear equation has to be solved. This is less of an
issue these days with calculators and computers.