Frictional and inertial forces


Fig.I-6
- these are cases of the right-hand column in Fig.I-2: boundary layer, mixing layer, submerged jet, etc.. Their typical feature is the fact that fluid is slowed down by friction in the shear layer. As an example, Fig.I-6 presents the case of the submerged jet where fluid, initially accelerated in the nozzle, is slowed down in the downstream direction. It tends, nevertheless, to continue in the original motion due to the action of the inertial force. It is not possible to evaluate this inertial force acting on the fluid particle in the form of the infinitesimally small parallelpiped in Fig.I-6 simply as = , where is the mass of the particle, because the flow in the present case is steady - so that velocities, such as the component , do not change in time. What happens, however, is advective transport of momentum: while = / is zero, there is nonzero ( / ).( / = ( / ). . Since the flow is two-dimensional, changes in directions of both co-ordinates should be taken into account. The elementary inertial force should be written:
= [ ( / ) + ( / ) ]
- and this is in equilibrium with friction force = (/) =
= (/) (/ )
Inserting these expressions to the both sides of force equilibrium equation, we obtain the Prandtl equation:
( / ) + ( / ) = ( / )
... which Prandtl obtained (1904) in a different way, by simplification of general Navier-Stokes equations. Even though the Prandtl equation is more simple, it is still a partial differential equation, the solution of which cannot be undertaken in the present course. It will be used, instead, for deriving the basic information about a laminar jet: dependence of its maximum velocity and its width on the downstream distance . We can assume power-law dependences:
and
The jet width , of course, is no simply measurable distance. Velocity decreases asymptotically towards zero at the jet edges asymptotically, theoretically reaching the zero value at infinity. To get some measurable quantity, it is e.g. useful to define the momentum width - the width of an equivalent jet with rectangular velocity profiles, having velocity , constant everywhere in the profile. It was derived in chap. [H] that unless the jet meets an obstacle, its momentum flow rate is constant. For the assumed rectangular profile, the momentum flow rate may be evaluated as

(under the assumption of planar flow, with very large - theoretically infinite - transverse dimension . Because of the constancy,

and because there is

the momentum rate conservation leads to the condition for the exponents:
2 + = 0

The other condition follows Prandtl's equation. Its advective terms must vary with the downstream distance in a manner that is identical to the variation of the diffusion term. Considering how some of the expressions vary - there is, for example ,
and it is also evident that


we get for variations of the first advective term:

.. while for the diffusion term the variation with downstream distance is

Comparing the exponents in these approximations for the two terms leads to the other condition:
2 - 1 = - 2
The solutions satisfying the two contitions are:
= - 1/3, = 2/3.


A similar derivation may be done for the axisymmetric jet (issuing from a round hole nozzle), where the momentum condition is so that the corresponding condition for the exponents is and the resultant values of the exponents are:



Going to another page: click
This is page Nr. I04 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
Any comments and suggestions concerning this text may be mailed to the author to his address tesar@fsid.cvut.cz

WWW server administrators: Jiri Kvarda, Zdenek Maruna ...... Contact: webmaster@vc.cvut.cz
Last change : 4.04.1997