108

MEKHANIKA ZHIDKOSTI I GAZA, JANUARY-FEBRUARY

1966

THE TURBULENT JET IN A CROSS FLOW T. A. Girshovich Mekhanika Zhidkosti i Gaza, Vol. 1, No. 1, pp. 151-153, 1966 Solutions of the problem of a jet in a cross flow usually reduce to determining the curved centerline of the jet [1-6]. We present below an approximate formulation of the problem of a two-dimensional jet issuing from an infinitely thin slot and propagatIng at some angle to an unbounded stream. The axial velocity, outer and inner boundaries, and other parameters of the jet are determined. From the solution, as a particular case, we obtain the known results for an ordinary submerged jet. The primary difficulty in posing the problem on the jet in a cross flowlis the specification of the boundary conditions. At the suggestion of Abramovich, it is considered that on the outer (facing into the stream) edge of the jet the velocity and pressure are equal to the velocity and pressure obtained for flow past a solid wall having the same form as the curved centerline of the jet. In other words, the velocity and pressure are obtained from a consideration of the flow about a half-body formed by two jets issuing at the angles ~a0 to the incident stream. The axis of symmetry of this half-body is parallel to the direction of the incident stream. On the inner (downstream) edge of the jet the velocity in the first approximation is considered to be equal to zero and the static pressure constant. We note that this boundary condition approximates the flow in the stagnant region downstream of the jet, where, as shown by experiments, circulatory zones develop which somewhat increase the expansion of the jet.

radius of curvature of the wall is considerably greater than the thickness of the boundary'layer, the equations of the boundary layer in the curvilinear coordinates selected above will have the form Ou

~ ~ t

Ou

+~-N

Op

u2

t

Op

p

o~ + 3 - - - N

'

Ov

Ou

p 0~ = n ( ~ ) '

t "Oz

o-V+-~

(1)

=~

Here R(x) is the radius of curvature; R > 0 if the wall is convex outward. To solve the problem we must solve the system of equations (1) with the following boundary conditions: 0u

u=us,

-~-~0,

v = O,

u~ Ou -~=0,

u=0,

P--pswithy=St, with y = 0 i

um

(2)

p= 0 withy=6~

(pressure is measured from the pressure in the region downstream of the jet 150). Introducing the new v a r i a b l e i = 1 for the upper half of the jet~ 2 for the lower half of the jet/.

~h = ~-i

(3)

and following the method of [7], we represent the distributed tangential stresses in form Ti = B o -~ BI~Ii -~- B2~li~ + Ba~ll3 ,

(4)

The coefficients of this polynomial are determined from the following boundary conditions: .z',

0~ Ti = O, The size of the stagnant region diminishes with reduction of the inclination of the jet to the stream. It is evident that at some small angle of inclination of the jet to the stream the circulatory zones disappear. On the othethand, at some large angle of inclination of the jet to the cross flow the radius of curvature of the jet centerline in its initial segment becomes comparable with the jet width. In both cases, the problem formulation adopted here will be invalid. The limiting angles for which it is acceptable must be determined experimentally. In addition, we make the following assumptions: 1) the curved jet centerline is a streamline; 2) the tangential stresses are equal to zero on the jet centerline; 3) the mixing length is constant across the curved jet centerline. The problem is solved in curvilinear orthogonal coordinates: we take the curved jet centerline as the abscissa axis, the normal to this centerline as the ordinate axis. This choice of coordinate system enables us to use the boundary layer equation for the jet in a cross flow. Thus, the problem on the jet boundary layer in a stream with variable wake velocity is solved in curvflinear coordinates. The solution is obtained using the integral method familar in boundary layer theory; this method has been adapted to jet theory by G~evskK [7]. Assume a two-dimensional turbulent jet issues at an angle to an infinite stream flowing out of an infinitely thin slot. In this case, the jet may be considered as boundary layer. A diagram of such a jet with all the notation is shown in the figure. It has been shown [8] that if the

~-

~ PSi

[u.,.,.'+

"q=O,

t ( 0p ~

]

-P- \ ' - ~ @ = 0 J ~ Ap8i with ~h = 0

~ 0

(5)

withBi~i,

Thus, we obtain the following expression for rt: 9 i = pStA1h (t - -

(6)

~0 m .

The velocity profile is determined using the Prandtl formula for the tangential stresses (it is clear that for very great curvature of the jet the Prandtl formula requires revision) 9, =

(7)

T P12 Ou

After equating expression (6) to expression (7), we find

Ou

on~

( ~

) 'l' (fli'/. __ Tl'/.). "

Integrating the resulting equation from 0 to ~t gives the following formula for the velocity profile in the jet:

u=um--

~

l2

]

i3

5

Here u m is the velocity on the jet centerline,

FLUID

DYNAMICS,

VOLUME

I, NUMBER

I

109

From the first condition (2) and from (8) we obtaIn the equation

u,~--%,~=l-St~:

l~ )

Equation (15) may be solved using the method of successive approximations. For the k-th approximation we have the equation

(9)

"

x~28

G,~ Joint solution of (8) and (9) enables us to obtain the relative profile of the differential velocity u--us

'

5 3 i -- ~ %'/~ + ~ ~l~'/= 9

i

(10)

We see from (10) that the longitudinal and transverse pressure gradiens do not affect the form of the relative velocity profile. This fact has been confirmed experimentally for the axlsymmetric jet [9]. Expression (9) for the inner and outer parts of the jet is written separately as follows:

\ l~ ]

Um--U $ = ~-

,

.

9

(11)

From (11) after simple transformations we obtain the simple algebraic relation between the ordinates of the inner and outer edges of the jet

61 = - - 88

i

_ _

uS)

~

Ih'

(12)

We see from (12) that for zero velocity of the cross flow we have the relation 61 = -6 2, i.e., we obtain the conventional submerged jet. To obtain the three unknown quantities um, 51, and 62 we have equation (12) and one of the equations (11). The third equation is obtained by integrating across the inner part of the jet the fixst equation of system (1) with the use of the continuity equation. As a result, we find

~ ~

R

~l~/'ln

-~

-

-

0

8/~ (8~, l-lP B-a

d~ 9 (16)

In approximate calculations, we can take 6 2 equal to 6 for the conventional submerged jet with some increase of the coefficient c. Then the remaining characteristics of the jet are found easily from relations (12) and (14). For the complete solution of the problem, we must find the form of the jet centerline. To do this, we can make use of a condition which has not yet been used, which is that the pressure on the outer boundary of the jet must be equal to the pressure obtained from an infinite flow about the jet considered as a solid curved wall. This condition has the form: P~

Um='~

2R~ ~s

oD R~ 0o/~82, l~_i ~ ~ 1 -

33 62

--61

[ 7

use ] 15 u~

I@2]

The flow about a solid curvilinear wall may be obtained, for example, by the method of superposition of flows. The imposition of a uniform flow parallel to the x 1 axis on the flow from a system of sources (sinks) distributed continuously along this same axis in some region with the intensity q(g) gives the stream function of the total flow of the form t

~-Yooyl -t- ~ -

b t q (~) arc tg xl~.yt ~ d~, a

(18)

Equating the stream function ~ to zero gives the equation of the wall (jet centerline). Using (18) we calculate the velocity and pressure on the wall and assume that they are equal to the velocity and pressure on the jet boundary. Then the resulting expressions are substituted in (17), from which we then determine the distribution of the sources

q(G

(13) REFERENCES

0

The value of the constant K2 = mK is unknown. It is apparent that there will not be great error in considering Ks to be half of the kinematic momentum of the jet (m = 1/2). The value of Ks may be determined more precisely from the solution for the initial segment of a two-dimensional jet issuing from a slot of finite width. As a result of the integration in expression (13), with the use of the second equation (I) we obtain

~

/
-

N-~ \ ~

5 62)-I 84

R

"

(14)

When R ---* ~o equation (14) gives the expression for the velocity on the centerline of the conventional submerged jet, Substituting (14) into the first equation (11) and certain transformations lead to the following differential equation for the inner boundary 52 of the jet: e~s/8~2/~~(Wn~ - - s/s~8~R- 1 ) _ 0. t t 33~2 R' R2 d8/~8~R -I --Win ~/~82~B-~) -

-

Whence with R --~ co 6s, = (_225/8)gz = - c or 62 = -cx, i.e., we obtain the known equation of the boundary of the turbulent submerged jet.

I. G. N. Abramovlch, Theory of Turbulent Jets [in Russian], GOStekhizdat, 1960. 2. V. V. Baturin and L A. Shepelev, "Air curtains," Otoplenie i ventilyatsiya, no. 5, 1936. 3. S. V. Vakhlamov, "Calculation of jet trajectory in a cross

flow," Inzh.-fiz. zh., vol. 7, no. i0, 1964. 4. M. S. Volynskii, Shape of Liquid Jets in Gas Streams [in Russian], Oborongiz, 1988. 5. V~ S. Ivanov, "On the form of the centerline of an aixsymmetric fan jet in a cross flow," Izv. VUZ. Aviatsionnaya tekhnika, no, 4, 1963. 6. V. A. Kosterin and g, V. Rzhevskii, "On the calculation of the trajectory and range of fan and dual plane jets in bounded plane flow," Izv. VUZ. Aviatsionnaya tekhnika, no. 1, 1964. 7. A. S. Ginevskii, "Turbulent wakes and jets in a cross flow with longitudinal pressure gradient," Izv. AN SSSR, OTN, Mekhanika i mashinostroenie, no. 2, 1959. 8. H. Schlichting, Boundary Layer Theory [Russian translation], Izd. inostr, lit., 1956. 9. G. F. Keller and W. D. Baines, "The round turbulent jet in a cross-wind," L Fluid Mechn,, 15, Arpil, 1963.

20 March 1965

Moscow