In this chapter, the partial differential equations governing the basic lav/s of conservation of mass and momentum for a incompressible viscous fluid are first
described. The equations, when apply to a turbulent flow, require the additional terms to account for the fluctuating components of the variables. A two-equation k- £ turbulence model which provides informations for the extra terms is incorporated into the time-mean differen tial equations to form a complete mathematical model for the two-dimensional axisymmetric turbulent flows. Appro priate boundary, conditions which simulate the practical
jet pump situation in order to obtain realistic prediction are discussed.
2.1 The Equations of Motion for an Incompressible Yiscous Pluid
The derivation of the equations of motion based on the basic laws of conservation are readily available in many standard text books on fluid mechanics such as
Schlichting (1960) and Hinze (1975)• The equations, according to Hinze (1975)? when expressed in a tensor notation, using Cartesian coordinates takes the following forms:
Continuity: ~ + ^ 7 j ~ 0 (2.1-1)
Momentum equation in x^-direction: Du.
PdF = ^ x T ^ j i * Fi (2.1-2)
3 = 1, 2, 3
where ( f . . is the stress in the x.-direction operates in a
ji l
plane which is perpendicular to the direction x.. F. is an external force per unit volume acting on the fluid in x^-direction.
For an incompressible fluid, - 2 - cT.. = _ + _2_
■2>Xj 31 '^ x i b x.. equation (2.1-2) can be written as
' „ ( ^ i + H h ) d x.a
Du. ^ >u. -ju. -
l ^ / < H 7 + Scf)
i ~ D L D i J
p — —i. ~
— _j.
\ Dt b x . b x . + F ± (2.1-3)
d = 1» 2, 3
where p is the static pressure and j x is the dynamic
viscosity of the fluid. Equations (2.1-1) and (2.1-3) are usually referred to as the Havier-Stokes equations which form the basis of the whole theory of viscous fluid mecha nics.
2.2 The Need for Turbulence Modelling -- v •
The equations of motion described in section 2.1 are generally applicable to laminar flows but not turbu lent flows. In brief, a turbulent flow is defined as an irregular fluid motion in which the various quantities show a random variation with time and space coordinates. Turbulent flows can occur when fluids flow through
conduits (turbulent pipe flow), pass over solid bodies (wake), or when neighbouring stream of the fluids with different velocities pass over one another (jet mixing). At present, one is unable to obtain solution for the time- dependent turbulent flow field using existing computers. Fortunately, it is possible to describe turbulent flow with distinct average values of various quantities such
as velocity, pressure and temperature, etc. If a turbu lent flow field is quasi-steady, averaging with respect to time can be Lised. But for a homogeneous turbulent flow field, averaging with respect to space is preferred. In most of the engineering problems, time-averaged values
are more useful for engineers and designers.
The instantaneous values of velocity and pressure can
be written as -
u i = U i * u i l (2.2-1)
and p «'P + p« (2.2-2)
where U. and P are the time-mean values and u . 1, p f are
1 i
the fluctuating values.
' The equations of motion for the average values in turbulent flow were first derived by Osborne Reynolds. He substituted the instantaneous values of u^ and p into the equation (2.1-3) to give the following form.
DU. ^ -s ^U. ?>U._____ ____
p — 1 j. JS— n( — ~ ~— 1) - puJut ) V D t ‘Ox. *ax. / V2x. 7>x. } \ l D 1 J L J ^ « + P. (2.2-3) l 3 = 1/ 2, 3
Compare this equation with the original momentum equation (2.1-3), it can he seen that the extra-terms - pu. ’u . ’
1 J are required to add to the viscous stresses in order that the instantaneous variables can he substituted by their time-mean values. Because Reynolds was the first person to derive the equation for turbulent flow in this form, the turbulent terms-p'u. !u a r e often called Rey-
j- j nolds stresses.
To solve equation (2,2-3), the terms - p u ^ ’u m u s t be known. Since there is no direct way of knowing the magnitude of these terms, a mathematical model to relate effect with'known quantities is therefore required. Thus, a model of turbulence, in the words of Launder and Spalding
(1 9 7 2) will !propose a set of equations which, when solved with the mean-flow equations, allows calculation of the relevant correlations and so simulates the behaviour of real fluid in important respects1.
2,3 The Differential Equations of Conservation Applied to Two-Dimensional Axisymmetrical Flows
2,3*1 The Coordinate System
Before making any attempt to express any equation for a particular flow configuration, an appropriate coordinates system must be chosen. In this thesis, owing to the fact that fluid flows take place at various flow components, the most general two-dimensional orthogonal axisymmetrical
coordinate system is used. Fig. 2.3-1 illustrates such a coordinate system in which the coordinates x and y characterise the members of two orthogonal families of surfaces of revolution, r and r are the radii of cur-
x y
vature for x and y surfaces intercepting at point P and r is the distance from P to the axis of symmetry.
Axis of symmetry
Fig.2.3-1 T h e .Orthogonal Axisymmetric Coordinate System.
The merit of using such a general arbitrary ortho gonal coordinate system is that the coordinates can be so chosen that all the flow boundaries are parallel to the grid surfaces. In the present investigation, a typical jet pump flow field consists of (i) an annular entrance region,(ii) a cylindrical mixing tube and (iii) a diffuser. By using the coordinate system outlined above a grid pattern can be devised to accommodate all the three
flow regions as shorn in F i g.20'3-2 mixing tube Axis of symmetry secondary / inlet primary ini et
Fig.2*3-2 The coordinate system applied to jet pump configuration
In general,' r . r and r are function of x and y< x y
In the uniform mixing tube region, r = o°
x
r - oo
y
r = y
In the diffuser region,
rx = 0 0 r = x + x
y
o
r = (x + x o )sin|6 (2.3-1) (2.3-2)where x Q and p are given in Figure 2.3-3 and their values