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A two-parameter model of turbulence, and its application to free jets

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Abstract

A model of turbulence is investigated in which the Reynolds stress appearing in the momentum equation is calculated from the expression\(\overline {u'\upsilon '} = - \sqrt k L(\partial U/\partial y)\); the kinetic energy, k, and the length scale,L, of turbulence are determined from differential transport equations for these quantities. These equations are solved for various free-jet situations, and the empirical constants involved are adjusted so as to give best agreement between predictions and experimental results. The plane mixing layer, the plane jet and the radial jet can be predicted with a single set of constants; for the round jet a different set has to be used. This suggests a dependence of the otherwise universal constants on the ratio ofL to the radiusr. Comparisons are presented of predicted and measured rates of spread, profiles forU, k and\(\overline {u'\upsilon '} \), and energy balances. For most cases the agreement is within the experimental accuracy.

Zusammenfassung

Im hier vorgeschlagenen Turbulenzmodell wird die Reynoldsspannung, die in der Bewegungsgleichung auftritt, mit Hilfe der Beziehung\(\overline {u'\upsilon '} = - \sqrt k L(\partial U/\partial y)\) berechnet. Die kinetische Energiek und der LÄngenma\stabL der Turbulenz werden durch Transport-differentialgleichungen für diese Grö\en bestimmt. Das Modell wird auf verschiedene Freistrahlen angewendet. Die empirischen Konstanten werden so geWählt, da\ die bestmögliche über-einstimmung zwischen Berechnungen und Versuchsergebnissen erzielt wird. Die ebene Mischungsschicht, der ebene Freistrahl und der Radialstrahl können mit dem gleichen Satz von Konstanten behandelt werden. Für den runden Strahl müssen dagegen andere Konstanten verwendet werden. Dies spricht dafür, da\ die sonst universellen Konstanten vom VerhÄltnis vonL zum Radiusr abhÄngen. Ein Vergleich von Rechen-ergebnissen und experimentellen Befunden über die Strahlausbreitung, die Profile vonU, k and\(\overline {u'\upsilon '} \) sowie die turbulente Energiebilanz wird mitgeteilt. Die übereinstimmung ist im allgemeinen von der Güte der Versuchsgenauigkeit.

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Abbreviations

C′s :

constants

F(n) :

energy spectrum

j :

exponent (equal to zero for plane flows and equal to unity for axisymmetric flows)

k :

kinetic energy of turbulence

L :

length scale

l :

mixing length

n :

wave number

r :

radius (=y)

Re :

Reynolds number

U, V :

velocities

′u, v′, w′ :

fluctuating velocites

x, y :

coordinates

δ :

jet width

ɛ :

dissipation ofk

Ν :

kinematic viscosity

ϱ :

density

σ :

effective Prandtl/Schmidt numbers (=constants)

C:

characteristic

E:

outer boundary of the jet

I:

inner boundary of the jet

m:

maximum max, min extreme values at a cross section

t:

turbulent

1/2:

position where mean velocity is one-half the maximum velocity

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Rodi, W., Spalding, D.B. A two-parameter model of turbulence, and its application to free jets. Warme- und Stoffubertragung 3, 85–95 (1970). https://doi.org/10.1007/BF01108029

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