By N. Rajaratnam
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Extra resources for Turbulent Jets
1 (b) Fig. 1-6. Mixing length and kinematic eddy viscosity for plane turbulent free jets (Zijnen, 1958a). 40. 50 and C , a value of zero, giving thereby: uJU0 = 3 . 5 0 / m 0 [ 1-98] Concerning the length scale b , we have b = C , x . 097. 114. 10 for convenience. 065. 35 035 r 0 (C) (d) Fig. 1-7a--d. 005 010 015 x x 020 025 030 25 X X (f 1 x Fig. 1-7. Turbulence characteristics of plane turbulent free jets (Heskestad, 1965). /bo Albertson et al. 44. If E is the kinetic energy of the flow at any section, we could show that: [ 1-1001 26 where E , is the kinetic energy at the nozzle.
It is interesting to find that the velocity profiles are indeed similar. This observation has been confirmed by the experiments of Corrsin (1946)’ Hinze and Zijnen (1949)’ Albertson et al. (1950) and others. In the following sections we will develop methods to predict the velocity profiles as well as the variation of the velocity and length scales u, and b. 2 EQUATIONS O F MOTION In this section we will develop the equations of motion for the circular turbulent jet. We start with the Reynolds equations in the cylindrical system ( r , 4, z ) and apply the boundary-layer approximations since the jet occupies only a small width in the transverse direction.
At any section, from dimensional considerations, we could write I b or 1 = Ob = PC,x = Cx where 0 and C are unknown constants. T = pc2X2(aU/ar)2 Let C 2 = a3 where a is another constant, let [ 2-51] where q5 = r/ax. Let us now bring in the Stokes stream function, $, which is defined as: [2-521 [ 2-53] a rv = - - - ( u m a 2 x 2 ~ ) ax [2-541 We have: urn/ U o = C1/ ( x / r o ) or: urn = C,Uoro/x = n / x [ 2-55] where n is a dimensional factor, independent of x . e. e. F ' - ( F / @ ) = 0, F ' = F/q5, F ( 0 ) = 0 (2) 0 ; Fr(q5)/q5 = 0 ; F ' ( m ) = 0 (3) m;q5 = m ; u = Tollmien obtained a series solution of [2-601, (see Abramovich (1963) for a description of the solution) and Table 2-1 gives the results regarding the distribution of u / u , and v/u,.