Page 45 - Fluid mechanics, heat transfer, and mass transfer

FLUID FLOW
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as 0.1–1 mm and can be as large as the smallest & Newton’s equation for laminar flow is
dimension of the turbulent stream.
& Flow inside an eddy is laminar because of its large t ¼ mðdU=dyÞ; ð2:1Þ
size.
where the shear stress is equal to the rate of change
& In turbulent flow, velocity is fluctuating in all
directions. of velocity times a proportionality factor, which is
viscosity. For turbulent flows, the equation is written
. What is Prandtl mixing length?
as
& Prandtl, in his work on turbulent shear stresses,
developed the concept of mixing length, which is t ¼ðm þ hÞðdU=dyÞ; ð2:2Þ
defined by assuming that eddies move in a fluid in a
where h is the eddy viscosity. The value of h is
manner similar to the movement of molecules in
variable for different flows, depending on the size
a gas. The eddies move a distance, called mixing
and velocity of the eddies, and for air and water it is
length (L), before they lose their identity.
much larger than m.
& Actually, the moving eddy or lump of fluid will
. What is meant by laminar and turbulent flows? Explain
gradually lose its identity. However, in the definition
with illustrations.
of Prandtl mixing length L, this small lump of fluid is
assumed to retain its identity while traveling the & In laminar flow, the fluid moves in layers called
entire length L and then lose its identity or get laminas, in streamlines. Laminar flow need not be
absorbed in the host region. in a straight line. For laminar flow, the flow follows
the curved surface of an airfoil smoothly, in layers.
. What is eddy viscosity?
The closer the fluid layers are to the airfoil surface,
& Eddy viscosity refers to the internal friction gener-
the slower they move. Moreover, the fluid layers slide
ated within the fluid as laminar flow becomes irreg-
over one another without fluid being exchanged
ular and turbulent as it passes over irregular solid
between the layers, that is, without lateral mixing
objects. Turbulent transfer of momentum by eddies
of the layers.
gives rise to the internal fluid friction, in a manner
& In turbulent flow, secondary random motions are
analogous to the action of molecular viscosity in
superimposed on the principal flow and there is an
laminar flow, but taking place on a much larger scale.
exchange offluid from one adjacent sector to another.
& Eddies are very efficient at transferring high-energy,
Also, there is an exchange of momentum such that
high-momentum fluid from the higher velocity parts
slow moving fluid particles speed up and fast moving
of the flow to the lower velocity parts of the boundary
particles give up their momentum to the slower
layer. This energy transfer between the boundary
moving particles and slow down themselves. Flow
layer and the main flow results in an increase in
moves instantaneously in many directions, but mean
apparent viscosity. The existence of eddies effective-
flow is in the downstream direction.
ly acts as a resistance to shear.
& Figure 2.1 illustrates change over from laminar
& It is important to remember that eddy viscosity is
to turbulent flow when an obstruction comes in its
neither a true viscosity nor a constant quantity, but
path.
that it is a useful concept helpful in solving problems
. Show the velocity profiles in a pipe for laminar and
related to sediment movement in turbulent flow.
turbulent flows by means of diagrams.
& In turbulent flows, there are local velocity gradients
& Laminar Flow:
that cause local stresses that work against the mean
velocity gradient. This results in a loss of energy from 2 2
the flow. v ¼ 2V½1 ðr =R ފ; ð2:3Þ

Changeover of laminar flow into turbulent eddies.
FIGURE 2.1