Page 64 - Mechanical Engineers' Handbook (Volume 4)

4 Fluid Kinematics 53

fluid particles rotate about their own centers or not. Flow is uniform if the velocity does not
change in the direction of flow. If it does, the flow is nonuniform. Laminar flow exists when
there are no lateral motions superimposed on the mean flow. When there are, the flow is
turbulent. Flow may be intermittently laminar and turbulent; this is called flow in transition.
Flow is considered incompressible if the density is constant, or in the case of gas flows, if
the density variation is below a specified amount throughout the flow, 2–3%, for example.
Low-speed gas flows may be considered essentially incompressible. Gas flows may be con-
sidered as subsonic, transonic, sonic, supersonic, or hypersonic depending on the gas speed
compared with the speed of sound in the gas. Open-channel water flows may be designated
as subcritical, critical, or supercritical depending on whether the flow is less than, equal to,
or greater than the speed of an elementary surface wave.

4.1 Velocity and Acceleration

In Cartesian coordinates, velocity components are u, v, and w in the x, y, and z directions,
respectively. These may vary with position and time, such that, for example, u dx/dt
u(x, y, z, t). Then

u u u u
du dx dy dz dt
x y z t
and

du udx udy udz u
a
x
dt xdt ydt zdt t
Du u u u u
u v w
Dt x y z t
The first three terms on the right hand side are the convective acceleration, which is zero
for uniform flow, and the last term is the local acceleration, which is zero for steady flow.
In natural coordinates (streamline direction s, normal direction n, and meridional direc-
tion m normal to the plane of s and n), the velocity V is always in the streamline direction.
Thus, V V(s, t) and
V V
dV ds dt
s t
dV V V
a V
s
dt s t
where the first term on the right-hand side is the convective acceleration and the last is the
local acceleration. Thus, if the fluid velocity changes as the fluid moves throughout space,
there is a convective acceleration, and if the velocity at a point changes with time, there is
a local acceleration.

4.2 Streamlines
A streamline is a line to which, at each instant, velocity vectors are tangent. A pathline is
the path of a particle as it moves in the fluid, and for steady flow it coincides with a
streamline.