Page 71 - Mechanical Engineers' Handbook (Volume 4)

60 Fluid Mechanics

V 2 1 p 1 V 2 2 p 2
z z 2
1
2g g 2g g
The negative sign balances the increase in velocity and pressure with radius.
The differential equations of motion for a viscous fluid are known as the Navier–Stokes
equations. For incompressible flow the x-component equation is
2
2
2
u u u u 1 p v u u
u
u v w X
t x y z x x 2 y 2 z 2
with similar expressions for the y and z directions. X is the body force per unit mass.
Reynolds developed a modified form of these equations for turbulent flow by expressing
each velocity as an average value plus a fluctuating component (u u u and so on).
These modified equations indicate shear stresses from turbulence ( u v , for example)
T
known as the Reynolds stresses, which have been useful in the study of turbulent flow.
6 FLUID ENERGY

The Reynolds transport theorem for fluid passing through a control volume states that the
heat added to the fluid less any work done by the fluid increases the energy content of the
fluid in the control volume or changes the energy content of the fluid as it passes through
the control surface. This is
—

Q Wk done (e ) dV e (V dS)
t control control
volume surface
and represents the first law of thermodynamics for control volume. The energy content
includes kinetic, internal, potential, and displacement energies. Thus, mechanical and thermal
energies are included, and there are no restrictions on the direction of interchange from one
form to the other implied in the first law. The second law of thermodynamics governs this.

6.1 Energy Equations
With reference to Fig. 17, the steady flow energy equation is

V 2 1 V 2 2
1 p v gz u q w 2 p v gz u 2
2 2
2
1
1
1 1
2 2
in terms of energy per unit mass, and where is the kinetic energy correction factor:

Figure 17 Control volume for steady-flow energy equation.