CHAPTER 8
ADDITIONAL SUBJECTS IN
FUNDMENTALS OF FLOW
Dr . Ercan Kahya
Vorticity for the z axis:
When vorticity is zero,
irrotational flow
Figure shows how
the wood chips float
Case (a): rotational flow
Case (b): irrotational flow
Although liquid makes a rotary movement, its microelements
always face the same direction without performing rotation.
In natural vortices such as hurricanes, tornados, eddying water currents has a
basic structure with a forced vortex at the center and free vortex at the periphery.
Given closed curve s, the integrated vs’ along this same curve is
called circulation Γ . Taking counterclockwise rotation positive,
Circulation dΓ of elemantary rectangle ABCD (area dA)
& the circulation is equal to the product of vorticity by area. Integration gives
Stokes theorem: surface integral of vorticity equal to the circulation
If there is no vorticity inside a closed curve, then circulation around it is zero.
Flow of viscous fluid
Mass flow rate at inlet & outlet sections in x- & y-directions
Fluid mass stored in the fluid element per unit time (x-dir.) :
Fluid mass stored in the fluid element per unit time (y-dir.) :
Mass change in unit time (right hand side) :
Continuity Equation
or
Unsteady flow & compressible fluid
& for real and ideal fluid
For steady flow and incompressible fluid:
For steady flow and incompressible fluid for axially
symetric flow using cylindrical coordinates:
For axially symetric flow using cylindrical coordinates:
The strict solutions obtained for N-S equations to date are
only for some special flows. Two such examples are:
Flow of a viscous fluid between two parallel plates
A flow in a long circular pipe is a parallel flow of axial symmetry.
In this case, it is convenient to use the Navier-Stokes equation using
cylindrical coordinates
Under the same conditions as in the previous section, N-S
equation simplifies to