Comprehensive Physiology Wiley Online Library

Aerodynamic Theory

Full Article on Wiley Online Library



Abstract

The sections in this article are:

1 Steady Flow in a Straight Tube
1.1 Fully Developed Laminar Flow
1.2 Flow in the Entrance Region
2 Calculation of Pressure Drop
3 Dimensional Analysis—the Similarity Principle
4 Turbulent Flow in a Straight Tube
5 Effects of Changes in Geometry on Pressure and Flow
5.1 Changes in Cross‐Sectional Area—Bernoulli's Theorem
5.2 Flow in a Curved Tube
6 Flow in Branched Tubes
6.1 Laminar Inspiratory Flow
6.2 Laminar Expiratory Flow
6.3 Turbulent Flow in Branched Tubes
6.4 Pressure Drop in Branched‐Tube Systems
7 Unsteady Flow
Figure 1. Figure 1.

Development of velocity profile with distance along a tube; thickness (δ) of boundary layer increases. Numbers are values Of x/dRe at which corresponding profiles occur, where x is distance from entrance, d is tube diameter, V is fluid velocity, is average fluid velocity, and Re is Reynolds number (see Eq. ).

Adapted from Prandtl and Tietjens
Figure 2. Figure 2.

Qualitative picture of flow downstream of a single symmetric bifurcation with Poiseuille flow in the parent tube. Lower branch indicates direction of secondary motions, new boundary layer, and separation region. Upper branch indicates velocity profiles in plane of the junction (solid line) and in normal plane (dashed line).

Figure 3. Figure 3.

Moody plot of friction factor (CF) against tracheal Reynolds number (Re) for inspiratory flow in a cast of major airways of human bronchial tree. Solid lines have slopes of −1, –½, and 0.

Adapted from Slutsky et al.
Figure 4. Figure 4.

Change in velocity profile shape as flow enters a region of pipe of smaller radius (top) and larger radius (bottom). A1, A2: cross‐sectional areas at stations 1 and 2.

From Pedley et al.
Figure 5. Figure 5.

Flow separation at an expansion. Note turbulence generated at edge of jet.

Figure 6. Figure 6.

A: secondary motions develop when fluid flows in a curved tube, with flow in center of tube directed toward outside of bend and returning near walls. B: axial velocity profile in plane of the bend is also distorted from Poiseuille flow (upstream) to a form having a peak near the outside wall (downstream). C: profile in transverse plane is distorted to an M shape. D: note initial skew in velocity profile when entry‐flow profile is flat.

Figure 7. Figure 7.

Asymmetric bifurcation showing some quantities that must be specified to define flow uniquely. , average velocity; d, diameter; θ, angle size; S, possible sites of flow separation.

Figure 8. Figure 8.

Streamlines in steady flow in a T junction when flow rates in the 2 daughter tubes are comparable. Solid line, streamline near wall, remaining close to it; dashed line, streamline near center line of parent tube; dasheddotted line, streamline between the two; S, sites of flow separation.

Figure 9. Figure 9.

Secondary motions generated in parent tube of a single bifurcation during expiratory flow.

From Schroter and Sudlow


Figure 1.

Development of velocity profile with distance along a tube; thickness (δ) of boundary layer increases. Numbers are values Of x/dRe at which corresponding profiles occur, where x is distance from entrance, d is tube diameter, V is fluid velocity, is average fluid velocity, and Re is Reynolds number (see Eq. ).

Adapted from Prandtl and Tietjens


Figure 2.

Qualitative picture of flow downstream of a single symmetric bifurcation with Poiseuille flow in the parent tube. Lower branch indicates direction of secondary motions, new boundary layer, and separation region. Upper branch indicates velocity profiles in plane of the junction (solid line) and in normal plane (dashed line).



Figure 3.

Moody plot of friction factor (CF) against tracheal Reynolds number (Re) for inspiratory flow in a cast of major airways of human bronchial tree. Solid lines have slopes of −1, –½, and 0.

Adapted from Slutsky et al.


Figure 4.

Change in velocity profile shape as flow enters a region of pipe of smaller radius (top) and larger radius (bottom). A1, A2: cross‐sectional areas at stations 1 and 2.

From Pedley et al.


Figure 5.

Flow separation at an expansion. Note turbulence generated at edge of jet.



Figure 6.

A: secondary motions develop when fluid flows in a curved tube, with flow in center of tube directed toward outside of bend and returning near walls. B: axial velocity profile in plane of the bend is also distorted from Poiseuille flow (upstream) to a form having a peak near the outside wall (downstream). C: profile in transverse plane is distorted to an M shape. D: note initial skew in velocity profile when entry‐flow profile is flat.



Figure 7.

Asymmetric bifurcation showing some quantities that must be specified to define flow uniquely. , average velocity; d, diameter; θ, angle size; S, possible sites of flow separation.



Figure 8.

Streamlines in steady flow in a T junction when flow rates in the 2 daughter tubes are comparable. Solid line, streamline near wall, remaining close to it; dashed line, streamline near center line of parent tube; dasheddotted line, streamline between the two; S, sites of flow separation.



Figure 9.

Secondary motions generated in parent tube of a single bifurcation during expiratory flow.

From Schroter and Sudlow
References
 1. Adler, M. Strömung in gekrümmten Rohren. Z. Angew. Math. Mech. 51: 257–275, 1934.
 2. Agrawal, Y., L. Talbot, and K. Gong. Laser anemometer study of flow development in curved circular pipes. J. Fluid Mech. 85: 497–518, 1978.
 3. Berger, C., P. Calvet, and C. Jacquemin. Structure D'écoulements de gaz dans des systèmes tubulaires bifurques. Toulouse, France: Cent. Etud. Rech. Toulouse, 1972. (Report.).
 4. Brech, R., and B. J. Bellhouse. Flow in branching vessels. Cardiovasc. Res. 7: 593–600, 1973.
 5. Caro, C. G., T. J. Pedley, R. C. Schroter, and W. A. Seed. Mechanics of the Circulation. Oxford, UK: Oxford Univ. Press, 1978.
 6. Chang, H. K., and O. A. El eMasry. A model study of flow dynamics in human central airways. Pt. I. Axial velocity profiles. Respir. Physiol. 49: 75–95, 1982.
 7. Clarke, S. W., J. G. Jones, and D. R. Oliver. Factors affecting airflow through branched tubes. Bull. Physio‐Pathol. Respir. 8: 409–428, 1972.
 8. Collins, W. M., and S. C. R. Dennis. The steady motion of a viscous fluid in a curved tube. Q. J. Mech. Appl. Math. 28: 133–156, 1975.
 9. Currie, I. G. Fundamental Mechanics of Fluids. New York: McGraw‐Hill, 1974.
 10. Dean, W. R. The streamline motion of fluid in a curved pipe. Philos. Mag. 7 (5): 673–695, 1928.
 11. Dekker, E. Transition between laminar and turbulent flow in human trachea. J. Appl. Physiol. 16: 1060–1064, 1961.
 12. Douglass, R. W., and B. R. Munson. Viscous energy dissipation in a model of the human bronchial tree. J. Biomech. 7: 551–557, 1974.
 13. Hardin, J. C., J. C. Yu, J. L. Patterson, and W. Trible. The pressure/flow relation in bronchial airways on expiration. In: Biofluid Mechanics, edited by D. J. Schneck. New York: Plenum, 1980, vol. 2, p. 39–55.
 14. Hinze, J. O. Turbulence. New York: McGraw‐Hill, 1959.
 15. Horsfield, K., and G. Cumming. Angles of branching and diameters at branches in the human bronchial tree. Bull. Math. Biophys. 29: 245–259, 1967.
 16. Horsfield, K., and G. Cumming. Morphology of the bronchial tree in man. J. Appl. Physiol. 24: 373–383, 1968.
 17. Isabey, D., and H. K. Chang. Steady and unsteady pressure‐flow relationships in central airways. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 51: 1338–1348, 1981.
 18. Isabey, D., and H. K. Chang. A model study of flow dynamics in human central airways. Pt. II. Secondary flow velocities. Respir. Physiol. 49: 97–113, 1982.
 19. Ito, H. Friction factors for turbulent flow in curved pipes. Trans. ASME Ser. D 81: 123–134, 1959.
 20. Jaffrin, M. Y., and T. V. Hennessey. Pressure distribution in a model of the central airways for sinusoidal flow. Bull. Physio‐Pathol. Respir. 8: 375–390, 1972.
 21. Jaffrin, M. Y., and P. Kesic. Airway resistance: a fluid mechanical approach. J. Appl. Physiol. 36: 354–361, 1974.
 22. Lew, H. S., and Y. C. Fung. On the low‐Reynolds number entry flow into a circular cylindrical tube. J. Biomech. 2: 105–119, 1969.
 23. Lutz, R. J., J. N. Cannon, K. B. Bischoff, and R. L. Dedrick. Wall shear stress distribution in a model canine artery during steady flow. Circ. Res. 41: 391–399, 1977.
 24. McConalogue, D. J., and R. S. Srivastava. Motion of fluid in a curved tube. Proc. R. Soc. London Ser. A 307: 37–53, 1968.
 25. Olson, D. E. Fluid Mechanics Relevant to Respiration: Flow Within Curved or Elliptical Tubes and Bifurcating Systems. London: Imperial College, 1971. Dissertation.
 26. Olson, D. E., L. D. Iliff, and M. F. Sudlow. Some aspects of the physics of flow in the central airways. Bull. Physio‐Pathol. Respir. 8: 391–408, 1972.
 27. Owen, P. R. Turbulent flow and particle deposition in the trachea. In: Circulatory and Respiratory Mass Transport, edited by G. E. W. Wolstenholme and J. Knight. London: Churchill, 1969, p. 236–252. (Ciba Found. Symp. 69.).
 28. Pacome, J. J. Structures D'écoulement et pertes de charge calculée dans le modèle D'arbre bronchique de Weibel. Toulouse, France: Paul Sabatier Univ., 1975. Dissertation.
 29. Pedley, T. J. Viscous boundary layers in reversing flow. J. Fluid Mech. 74: 59–79, 1976.
 30. Pedley, T. J. Pulmonary fluid dynamics. Annu. Rev. Fluid Mech. 9: 229–274, 1977.
 31. Pedley, T. J. The Fluid Mechanics of Large Blood Vessels. Cambridge, UK: Cambridge Univ. Press, 1980.
 32. Pedley, T. J., R. C. Schroter, and M. F. Sudlow. Energy losses and pressure drop in models of human airways. Respir. Physiol. 9: 371–386, 1970.
 33. Pedley, T. J., R. C. Schroter, and M. F. Sudlow. The prediction of pressure drop and variation of resistance within the human bronchial airways. Respir. Physiol. 9: 387–405, 1970.
 34. Pedley, T. J., R. C. Schroter, and M. F. Sudlow. Flow and pressure drop in systems of repeatedly branching tubes. J. Fluid Mech. 46: 365–383, 1971.
 35. Pedley, T. J., R. C. Schroter, and M. F. Sudlow. Gas flow and mixing in the airways. In: Lung Biology in Health and Disease. Bioengineering Aspects of the Lung, edited by J. B. West. New York: Dekker, 1977, vol. 3, chapt. 3, p. 163–265.
 36. Poiseuille, J. L. M. Recherches experimentales sur le mouvement des liquides dans les tubes de très petits diamètres. C. R. Acad. Sci. 11: 961–967, 1041–1048, 1840.
 37. Prandtl, L. The Essentials of Fluid Dynamics. Glasgow: Blackie, 1952.
 38. Prandtl, L., and O. G. Tietjens. Applied Hydro‐ and Aeromechanics. New York: Dover, 1957.
 39. Reynolds, O. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Philos. Trans. R. Soc. London 174: 935–982, 1883.
 40. Rouse, H. (editor). Engineering Hydraulics. New York: Wiley, 1950.
 41. Schlichting, H. Boundary Layer Theory (6th ed.). New York: McGraw‐Hill, 1968.
 42. Schroter, R. C., and M. F. Sudlow. Flow patterns in models of the human bronchial airways. Respir. Physiol. 7: 341–355, 1969.
 43. Slutsky, A. S., G. G. Berdine, and J. M. Drazen. Steady flow in a model of human central airways. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 49: 417–423, 1980.
 44. Snyder, B., and M. J. Jaeger. Lobar flow patterns in a hollow cast of canine central airways. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 54: 749–756, 1983.
 45. Stehbens, W. E. Turbulence of blood flow. Q. J. Exp. Physiol. 44: 110–115, 1959.
 46. Talukder, N. An investigation on the flow characteristics in arterial branchings. (Abstract). Mech. Eng. 97: 81, 1975.
 47. Talukder, N., and R. M. Nerem. Flow characteristics in vascular graft models. In: Digest Int. Conf. Mech. Med. Biol., 1st, Aachen, 1978, vol. 7, p. 281–284.
 48. Taylor, G. I. The criterion for turbulence in curved pipes. Proc. R. Soc. London Ser. A 124: 243–249, 1929.
 49. Weibel, E. R. Morphometry of the Human Lung. Heidelberg: Springer‐Verlag, 1963.
 50. Zeller, H., N. Talukder, and J. Lorenz. Model studies of pulsating flow in arterial branches and wave propagation in blood vessels. AGARD Conf. Proc. 65: 15–15. 8, 1970.

Contact Editor

Submit a note to the editor about this article by filling in the form below.

* Required Field

How to Cite

T. J. Pedley, Jeffrey M. Drazen. Aerodynamic Theory. Compr Physiol 2011, Supplement 12: Handbook of Physiology, The Respiratory System, Mechanics of Breathing: 41-54. First published in print 1986. doi: 10.1002/cphy.cp030304