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. The equations for conservation of mass and momentum, when expressed in the general orthogonal x, y coordinate

system described above for a steady flow, will take the

following forms. __

The continuity equation,

Ie ( prTJ) + ^ (rrT) =

■3/^ . 7 > , 3TJ'

BP , Gu 5x

where Su = 1 r

dx^efflx^ * ay^efflx^

f

The momentum equation in y~direction;

^(piJrV) + ^ ( ?VrV) - ^ ( r ^ effg ) - J g r f r ^ f E )

¥ * sT

QV 1 r

dU-

^x^r /^eff dy^ + ^ r ^ / ^ e f f ^y^ 2/ e f f ( U s i n /3 + Vcosp )

“ Is

is: r x

cos, (2.3-5)

where U, V, P are time-mean velocities and static pressure. The full derivation of the momentum equations is given in Appendix A.3.

The momentum equations are obtained by assuming that the fluid is treated as obeying !Iewton!s viscosity lav/. For a turbulent flow, jUq^ accounts for both viscous

stress and Reynolds stress. By comparing equation (2.3-4) with equation (2.2-3)* one can write

OT. 3U. 7>U, 3U._____ ____

/ • e f f f e + ^ = / (:s q + ^ (2*3_6)

J d -1*

An appropriate model of turbulence is thus required to relate the turbulent stresses - pu.* u .r to some

i j known quantities throughout the flow field.

2.4 The Choice and Application of Two-Equation k - £ Model It was first proposed by Boussinesq in 1877 that the turbulent shear stress could be replaced by the product of the time-mean velocity gradient and the turbulent vis-

Thus, the effective viscosity in a turbulent flow is equal to the sum of the molecular viscosity-and the turbulent viscosity. Unlike the molecular viscosity which is the real property of the fluid, the turbulent

viscosity can become effective only when there is flow and its value varies from point to point in the flow

depending upon the turbulent structure at that particular location.

Many turbulence models have been proposed to relate ^ t to some quantities which can be determined. ... The out­

line of various models and their merits and shortcomings have been described in section 1.2.2. In the present studies of confined jet mixing and jet pump flows, owing to the interaction between the mixing shear region and the wall shear region, the length scale profile is unable to be

prescribedthroughout the flow field. The mixing length and one-equation models will not be able to predict these cosity^at , i.e.,

(2.4-1) Substituting Equation (2.4-1) into (2.3-6), one gets

flows satisfactorily. However, in view of the fact that the multi-equation models are far less established and more computing time is required, the choice of a two-

equation model is a compromise of accuracy and economics unless a more complicated multi-equations model is proved

to be necessary.

The Prandtl-Komogorov two-equation model states that the turbulent viscosity can be written as

^ = C « p i A (2.4-3)

P

2

2

where k = J(u* + v1 + w f ), 1 is the length scale and Cyu is a constant, k and.l are to be determined by their transport equations. However, it turns out that the

length scale itself is not the most appropriate dependent variable. Various workers have selected different com­ binations of m and n of a quantity km ln as their second dependent variable instead of using 1 itself. (See Table 1.2-1). A quantity, called turbulence energy dissipation rate £ , first proposed by Harlow and

Nakayama (1 9 6 8) and subsequently favoured by many other workers is chosen as the second dependent variables in

the present work where

v 3/2

£ = (2.4-4)

The reasons for this choice are : (i) it is relative­ ly easy to derive the exact equation for £ ; (ii) £ appears

directly as an unknown in the transport equation for k; (iii) the effective turbulent Prandtl Number appeared in the £ -equation as a constant irrespective of the distance from the wall whereas for other combinations,

and Spalding (1973).

Furthermore, the k - £ model are well established and has been incorporated into standard compiiter code by

Gosman and Pun (1974) for solving turbulent recirculating flows. The model was widely tested and enjoyed satisfac­ tory predictions for a wide range of flows• Examples of such applications of k- £ model can be found in the works of Hanjalic (1970), Elghoboshi and Pun (1974)> Matthews and Whitelaw (1971) and Nielson (1973).

The k- and £- equations, when using a general orthogonal axisymmetric coordinate system described in section 2.3.1* may b'e expressed in the following form at high Reynolds numbers,

k-equation:

2

such as kl and k / l, this Is not so, as proved by Launder

Ski ff -2kv 3

<rk

“

crk

>y

where

(2*4-7)

These equations are modified from the cylindrical polar forms used by G-osman and Pun (1974)* They differ in the expression of the turbulent energy production term G.

The derivation of G, equation (2*4-7) is given in Appendix A.4.

By-combining equations (2*4-3) and (2*4-4)* ju^ related to k and £ as

It is now possible to obtain the five unknown varia­ bles, namely, U, Y, P, k, £ by solving five simultaneous equations (2.3-3), (2*3-4), (2.3-5), (2.4-5) and (2.4-6) with the help of the auxilliary equations (2*4-2) and

(f e must be prescribed to complete the specification of the model. At high Reynolds, these constants are given the values listed in Table 2*4-1 as recommended by

launder and Spalding (1973) and Gosman and Pun (1974)• This set of values has been widely used in various flow problems and is generally accepted for flows of plane jets, mixing layers and the plane and axisymmetric wall flows.

j u t = C ^ p k 2/ £ (2.4-8)

(2.4-8)

The values of the constants ar ,

°/< c d °1 °2 ° k <*£

0.09

1.00

1.44

1.92

1.00

1.21

Table 2.4-1 The values of the constants used in the

In the-present study of jet pump flows, these values are chosen for the whole flow field without any modifica­ tion.

2.5 Modification of the Model for !Near W a l l1 Flow