Hostname: page-component-7c8c6479df-fqc5m Total loading time: 0 Render date: 2024-03-29T13:39:15.042Z Has data issue: false hasContentIssue false

The three-dimensional distributions of tangential velocity and total-temperature in vortex tubes

Published online by Cambridge University Press:  29 March 2006

C. U. Linderstrm-Lang*
Affiliation:
Research Establishment Ris, Roskilde, Denmark

Abstract

The axial and radial gradients of the tangential velocity distribution are calculated from prescribed secondary flow functions on the basis of a zero-order approximation to the momentum equations developed by Lewellen. It is shown that secondary flow functions may be devised which meet pertinent physical requirements and which at the same time lead to realistic tangential velocity gradients.

The total-temperature distribution in both the axial and radial directions is calculated from such secondary flow functions and corresponding tangential velocity results on the basis of an approximate turbulent energy equation. The method employed for the solution of this equation stresses the equivalence of the vortex tube to counter-current systems with transverse diffusion such as distillation columns and heat exchangers.

An availability function is derived that permits the evaluation of vortex tube performance on the basis of velocity data.

Turbulent diffusivities resulting from the quantitative use of the tangential velocity approximation are shown to agree with those derived from the total-temperature calculations.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, O. L. 1963 Theoretical effect of Mach number and temperature gradient on primary and secondary flows in a jet-driven vortex. UAC-Research Lab. Report no. RTD-TDR-63-1098.Google Scholar
Benjamin, T. Brooke 1962 Theory of the vortex breakdown phenomenon J. Fluid Mech. 14, 593.CrossRefGoogle Scholar
Bruun, H. H. 1967 Theoretical and experimental investigation of vortex tubes. Dept. of Fluid Mech. Report no. 67-1, Danish Techn. Univ.Google Scholar
Bruun, H. H. 1969 Experimental investigation of the energy separation in vortex tubes J. Mech. Engng Sci. 11, 567.Google Scholar
Cohen, K. 1951 The Theory of Isotope Separation. McGraw-Hill.Google Scholar
Deissler, R. G. & Perlmutter, M. 1960 Analysis of the flow and energy separation in a turbulent vortex Int. J. Heat Mass Transfer, 1, 173.Google Scholar
Fulton, C. D. 1950 Ranque's tube Refrig. Engng. 58, 473.Google Scholar
Gulyaev, A. I. 1966 Vortex tubes and the vortex effect (Ranque effect) Soviet Phys. tech. Phys. 10, 1441. (Russ. orig. Zh. Tekh. Fiz. 35 (1965), 1869.)Google Scholar
Hartnett, J. P. & Eckert, E. R. G. 1957 Experimental study of the velocity and temperature distribution in a high velocity vortex-type flow Trans. ASME. 79, 751.Google Scholar
Hilsch, R. 1946 Die Expansion von Gasen im Zentrifugalfeld als Klteprozess Z. fr Naturf. 1, 208.Google Scholar
Kassner, R. & Knoernschild, E. 1948 Friction laws and energy transfer in circular flow. Report PB-110936, parts I and II.Google Scholar
Keyes, J. J. 1961 Experimental study of flow and separation in vortex tubes with application to gaseous fission heating J. Am. Rocket Soc. 31, 1204.Google Scholar
Lay, J. E. 1959 An experimental and analytical study of vortex-flow temperature separation by superposition of spiral and axial flows, part 1 Trans. ASME. 81, 202.Google Scholar
Lewellen, W. S. 1962 A solution for three-dimensional vortex flows with strong circulation J. Fluid Mech. 14, 420.Google Scholar
Lewellen, W. S. 1964 Three-dimensional viscous vortices in incompressible flow. Ph. D. thesis, Univ. of California (University Microfilms, Inc., Ann Arbor, order no. 64-8331).Google Scholar
Lewellen, W. S. 1965 Linearized vortex flows A.I.A.A. J. 3, 91.Google Scholar
Linderstrm-Lang, C. U. 1970a Vortex tubes with weak radial flow. Part I. Calculation of the tangential velocity and its axial gradient. Ris Report no. 216.Google Scholar
Linderstrm-Lang, C. U. 1970b Vortex tubes with weak radial flow. Part II. Calculation of the three-dimensional temperature distribution. Ris Report no. 217.Google Scholar
Linderstrm-Lang, C. U. 1970c Vortex tubes with weak radial flow. Part III. Calculation of the performance and estimation of the turbulent diffusivity. Ris Report, no. 218.Google Scholar
Ragsdale, R. G. 1961 Applicability of mixing length theory to a turbulent vortex system. NASA TN D-1051.Google Scholar
Reynolds, A. J. 1961 Energy flows in a vortex tube J. Appl. Math. Phys. 12, 343.Google Scholar
Rosenzweig, M. L., Lewellen, W. S. & Ross, D. H. 1964 Confined vortex flows with boundary-layer interaction. Report no. ATN-64(9227)-2. AD 431844.Google Scholar
Rosenzweig, M. L., Ross, D. H. & Lewellen, W. S. 1962 On secondary flows in jetdriven vortex tubes J. Aero. Sci. 29, 1142.Google Scholar
Scheller, W. A. & Brown, G. M. 1957 The Ranque-Hilsch vortex tube Ind. Engng Chem. 49, 1013.Google Scholar
Scheper, G. W. 1951 The vortex tube; internal flow data and a heat transfer theory Refrig. Engng. 59, 985.Google Scholar
Shapiro, A. H. 1954 Compressible Fluid Flow, II. New York: The Ronald Press Co.Google Scholar
Sibulkin, M. 1962 Unsteady, viscous, circular flow. Part 3. Application to the Ranque-Hilsch vortex tube J. Fluid Mech. 12, 269.Google Scholar
Stone, W. S. & Love, T. A. 1950 An experimental study of the Hilsch tube and its possible application to isotope separation. Oak Ridge Nat. Lab. Report ORNL, 282.Google Scholar
Suzuki, M. 1960 Theoretical and experimental studies on the vortex tube Sci. Pap. Inst. Phys. Chem. Research, Tokyo, 54, 43.Google Scholar
Takahama, H. 1965 Studies on vortex tubes Bull. JSME. 8, 433.Google Scholar
Takahama, H. & Kawashima, K.-I. 1960 An experimental study of vortex tubes Mem. Faculty Engng Nagoya. Univ. 12, 227.Google Scholar