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Numerical solution of laminar jet mixing with and without free stream

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Abstract

In part A, systematic numerical solutions of two-dimensional and axisymmetrical laminar jet of an incompressible fluid with and without free stream have been obtained. For the case without free stream, the exact numerical solutions have been obtained for different initial velocity profiles and compared with experimental results and similarity solutions. For the case with free stream, the numerical solutions have been compared with the linearized analytical solutions. In part B, an approximate numerical solution for the axial velocity distribution of the three-dimensional laminar jet is proposed. The accuracy of the method has been determined from the exact solutions of the two limiting cases of three-dimensional jets, i.e. the two-dimensional and axisymmetric case of part A. Methods to improve the results due to the approximation are discussed.

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Abbreviations

a :

radius of circular exit

b, xxx0062; :

half-width of jet with and without cross flow respectively

c 2, c r :

momentum coefficient for two-dimensional and axisymmetric jet respectively

c u :

correction factor for central excess velocity

D :

half-width of the shorter side of rectangular jet exit

h :

spacing in x-direction

J 0, J 1 :

Bessel function of zeroth and first order of the first kind respectively

k :

spacing in z-direction

L :

aspect ratio of rectangular jet exit

M 2, M r :

total momentum of two-dimensional and axisymmetric jet respectively

p :

spacing in y-direction

Re, xxx0052;e :

Reynolds number defined U * j D/ν and Ū*a/ν respectively

u*, v*, w* :

excess velocity components along x*-, y*- and z*-direction respectively

u, v, w :

nondimensional excess velocity components defined by u*/U * j , Rev*/U *j and Rew*/U * j respectively

u m, ū m :

nondimensional maximum excess velocity over a station with and without cross flow respectively

u i :

nondimensional excess velocity distribution at jet exit

U *0 :

uniform velocity of moving stream

U 0 :

nondimensional uniform velocity of moving stream (U *0 /U * j )

U * j :

excess velocity at the center of jet exit

Ū*:

mean excess velocity at jet exit

x*, y*, z* :

axis of Cartesian coordinate

x, y, z :

nondimensional distance defined by x*/ReD, y*/D and z*/D respectively

ξ :

nondimensional parameter defined by (10)

δ :

constant

ν :

kinematic viscosity of fluid

ρ :

density of fluid

0:

axisymmetrical jet

∞:

two-dimensional jet

L :

three dimensional jet with aspect ratio L

-:

without cross flow

*:

refers to the physical quantity

References

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  4. Flugge-Lotz, I., and F. G. Blottner, Numerical Solution of Compressible Boundary Layer, Div. of Eng. Mech., Stanford University, TR 131, January, 1962.

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  8. Pai, S. I., and T. Y. Hsieh, Numerical Solution of Laminar Jet Mixnig With and Without Free Stream, Tech. Note BN-627, University of Maryland, November 1969.

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Authors and Affiliations

  1. Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, 20742, College Park, Maryland, USA

    S. I. Pai & T. Hsieh

Authors
  1. S. I. Pai
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  2. T. Hsieh
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Additional information

This research was supported in part by the United States Army Material Command Harry Diamond Laboratories under Contract No. DAAG 39-69-C-0036. The computer time was supported by National Aeronautics and Space Administration Grant NSG-398 to the Computer Science Center of the University of Maryland.

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Pai, S.I., Hsieh, T. Numerical solution of laminar jet mixing with and without free stream. Appl. Sci. Res. 27, 39–62 (1973). https://doi.org/10.1007/BF00382475

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  • DOI: https://doi.org/10.1007/BF00382475

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