Page 23 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

8 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

Stokes has denoted this total derivative by a, the operator a being
equal to 2 +v-grad = 2 + {grad ()|-v, due to the obvious equality
(v - grad) v = (grad Vv) + Vv, Vv + grad or [gradQ] - v being the so-called
convective part of #.

When all the motion parameters, expressed in Eulerian coordinates,
do not depend explicitly on time, the respective motion is called steady
or permanent. Obviously, the steady condition is 4 = 0 or, equivalently,
D _
fi = Vv grad.
Conversely, if time appears explicitly, the motion is unsteady or non-
permanent.
Before closing this section we should make precise the notions of tra-
jectories (pathlines), streamlines and streamsurfaces, vortex lines and
vortex tubes, circulation and the concept of stream function as well.

1.2.1 Trajectories

In general the trajectory (pathline) is the locus described by a material
point (particle) during its motion. The trajectories will be the integral
curves (solutions) of the system

_
day _ dwz =dt (in Cartesian coordinates)
des
=
=
V1 V2 V3
or of the system

dz! daz? = dz3 , “ ;
SHE ay HET Hd (in curvilinear coordinates),
v v v

where v = vz (zi, t )ix = v*(z*,t ) a x, v* being the so-called contravari-
ant components of the velocity v in the covariant base vectors a, of the
considered curvilinear system.

Obviously, at every point of a trajectory the velocity vector is neces-
sarily tangent to the trajectory curve. At the same time we will sup-
pose again the regularity of the velocity field v(r,t), to ensure the exis-
tence of the solution of the above system (in fact the vectorial equation
oe =v/(r,t) ). A detailed study of this system, even in the case when
some singular points occur (for instance, the “stagnation points” where

v(r,t) = 0), has been done by Lichtenstein [84].

1.2.2 Streamlines and Streamsurfaces
For a fixed time ¢ , the streamlines and the streamsurfaces are the
curves and, respectively, the surfaces in the motion field on which the

velocity vector is tangent at every point of them. A streamsurface could
be considered as a locus of streamlines.