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.
Similar content being viewed by others
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
References
Batchelor, G. K., and A. A.Townsend: Decay of Turbulence in the Initial Period. Proc. Roy. Soc. A193 (1948) and Decay of Turbulence in the Final Period, Proc. Roy. Soc. A190 (1948).
Beckwith, I. E., and D. M.Bushnell: Detailed Description and Results of a Method for Computing Mean and Fluctuating Quantities in Turbulent Boundary Layers, NASA TN D-4815 (1968).
Bradbury, L. J. S.: The Structure of a Self-Preserving Turbulent Plane Jet. J. Fluid Mech. 23, part 1 (1965) pp. 31/64.
Bradshaw, P., D. H. Ferriss, andN. P. Atwell: Calculation of Boundary-Layer Development using the Turbulent Energy Equation. J. Fluid Mech. 28, part 3 (1967) pp. 593/616.
Chou, P. Y.: On Velocity Correlations and the Solution of the Equations of Turbulent Fluctuation. Quart. Appl. Math. 3, 38 (1945).
Chou, P. Y.: Pressure Flow of a Turbulent Fluid between two Infinite ParallelPlanes. Quart. Appl. Math. 3 (1945) p. 198.
Du P.Donaldson, C.: A Computer Study of an Analytical Model of Boundary Layer Transition. AIAA-Paper No. 68–38 (1968).
Emmons, H. W.: Shear Flow Turbulence. Proc. 2nd U. S. National Congress App. Mech., ASME (1954).
Glushko, G. S.: Turbulent Boundary Layer on a Flat Plate in an Incompressible Fluid. Izv. Akad. Nauk SSSR, Mekh. No. 4 (1965) p. 13.
Gosman, A. D., W. M. Pun, A. K. Runchal, D. B. Spalding, andM. W. Wolfshtein: Heat and Mass Transfer in Recirculating Flows. London: Academic Press, 1969.
Harlow, F. H., andP. I. Nakayama: Turbulent Transport Equations. The Phys. of Fluids 10, 11 (1967) p. 2323.
Harlow, F. H., and P. I.Nakayama: Transport of Turbulence Energy Decray Rate. University of California Rept. LA-3854 (1968).
Harlow, F. H.: Transport of Anisotropic or Low-Intensity Turbulence. University of California Rept. LA-3947 (1968).
Harlow, F. H., and C. W.Hirt: Generalised Transport Theory of Anisotropic Turbulence. University of California Rept. LA-4086 (1969).
Heskestad, G.: Hot-Wire Measurements in a Plane Turbulent Jet. J. of Appl. Mech., Dec. 1965 (1965) p. 1.
Heskestad, G.: Hot-Wire Measurements in a Radial Turbulent Jet. J. of Appl. Mech. (1966) p. 417.
Kolmogorov, A. N.: C. R. Acad. Sci. USSR 30, 301, and 32, 16 (1941).
Kolmogorov, A. N.: Equations of Turbulent Motion of an Incompressible Fluid. Itv. Ak. Nauk SSR, Seria fizicheska VI 1942, No. 1/2 (1942) pp. 56/58.
Laufer, J.: The Structure of Turbulence in Fully-Developed Pipe Flow. NACA Rep. 1174 (1954).
Liepmann, H. P., and J.Laufer: Investigations of Free Turbulent Mixing. NACA TN 1257 (1957).
Maydew, R. C., and J. F.Reed: Turbulent Mixing of Axisymmetric Compressible Jets (in the Half-Jet Region) with Quiescent Air. SANDIA Corp., Aerothermodynamics, Sc-4764 (RR) March (1963).
Nee, V. W., andL. S. G. Kovasznay: Simple Phenomenological Theory of Turbulent Shear Flows. The Phys. of Fluids 12, 3 (1969) p. 473.
Ng, K. H., and D. B.Spalding: Some Applications of a Model of Turbulence to Boundary Layers near Walls. To be published.
Prandtl, L.: über die ausgebildete Turbulenz, ZAMM 5 (1925) p. 136.
Prandtl, L.: Bemerkungen zur Theorie der freien Turbulenz. ZAMM 22 (1942) pp. 241/243.
Prandtl, L., and K.Wiegkabdt: über ein neues Formelsystem für die ausgebildete Turbulenz. Nachr. Akad. Wiss., Göttingen, Math.-phys. Kl., 1945, p. 6.
Reichardt, H.: GesetzmÄ\igkeiten der freien Turbulenz, VDI-Forschungsheft 414 (1942).
Robins, A.: The Structure and Development of a Plane Turbulent Free Jet. Ph. D. Thesis, Imperial College, to be completed.
Rotta, J.: Statistische Theorie nichthomogener Turbulenz. Zeitsch. f. Physik, 129 pp. 547/572, and 131. pp. 51/77 (1951).
Rotta, J.: über eine Methode zur Berechnung turbulenter Scherströmungen. Aerodynamische Versuchsanstalt Göttingen Rept. 69A 14 (1969).
Spalding, D. B., andS. V. Patankar: Heat and Mass Transfer in Boundary Layers. London: Morgan Grampian, 1967.
Spalding, D. B.: Heat Transfer from Turbulent Separated Flows. J. Fluid Mech. 27, part 1 (1967) pp. 97/109.
Spalding D. B.: The Calculation of the Length Scale of Turbulence in some Boundary Layers remote from Walls. Imperial College, Mech. Eng. Dept., TWF/TN/31 (1967).
Sunyach M., and F.Mathieu: Zone de mélange d'un jet plan. To be published in Intern. J. Heat Mass Transfer.
Taillard A., M. Sunyach, andJ. Mathieu: étude d'un jet plan. C. R. Acad. Sc. Paris, t. 264 (1967) pp. 527/530.
Townsend, A. A.: Self-Preserving Flow inside a Turbulent Boundary-Layer. J. Fluid Mech. 22 (1965) p. 773.
Trubchikov, B. Y.: A Thermal Method of Measuring Turbulence in Wind Tunnels. TSAGI Rept. No. 372, Moscow (1938).
Tuve, G. L.: Air Velocity in Ventilating Jets. Heating, Piping, and Air Conditioning, Jan. (1953).
Wolfshtein, M. W.: Convection Processes in Turbulent Impinging Jets. Ph. D. Thesis, Imperial College (1967).
Wygnanski, I., and H. E.Fiedler: Some Measurements in the Self-Preserving Jet. Boeing Sct. Res. Lab. Doc. D1-82-0712 (1968).
Wygnanski, I., and H. E.Fiedler: The Two Dimensional Mixing Region. To be published.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01108029