# A model for periodic structures in turbulent boundary layers

## Abstract

Many experimental studies emphasize the importance of periodic recognizable flow patterns for the transport process in turbulent flow. In this paper a model is formulated for the large scale part of the turbulent motion. The experimental observation that the structures in the outer region run in phase with the bursting cycle in the wall layer forms the basis of the model. The wall layer, where viscous stresses are important and the outer region where the inviscid approximation holds, are treated separately. The small scale part of the turbulent motion, which is assumed to be important in localized regions only (bursts regions), couples the wall region and the outer region.

The mean wall shear stress calculated with this model agrees reasonably well with the empirical formulae for the friction coefficient even for the more complex case of the transpired boundary layer. The main conclusion of the model calculations is that the transport of momentum can be very well explained in terms of turbulent structures. The model clearly illustrates how momentum is transported in three stages: (i) Thin elongated layers near the wall slow down as the result of viscous forces. (ii) The retarded fluid-is ejected in localized regions or bursts. (iii) The large scale motion in the outer region takes over the transport.

In this paper special attention will be given to the function of the longitudinal vortices in the wall layer. In turns out that they hardly influence the turbulent exchange, but that they are very important for the creation of locally unstable regions. It is believed that the strength of the longitudinal vortices is influenced by the large scale structures in the outer region. By this mechanism the large scales in the outer region can influence the burst frequency.

In a discussion some ideas are presented about what this can mean for special flow phenomena as: drag reduction by polymer solutions or along compliant walls and rapid shear stress change along curved walls.

## Keywords

Wall Shear Stress Large Eddy Simulation Turbulent Boundary Layer Outer Region Wall Region## Preview

Unable to display preview. Download preview PDF.

## References

- Acton, E., (1976). The modelling of large eddies in a two-dimensional shear layer, J. Fluid Mech., 76, 561.zbMATHCrossRefADSGoogle Scholar
- Bark, F.H., (1975). On the wave structure of the wall region of a turbulent boundary layer, J. Fluid Mech., 70, 229.CrossRefADSGoogle Scholar
- Beljaars, A.C.M., (1979). A model for turbulent exchange in boundary layers, Ph.D. thesis, Eindhoven University of Technology, Eindhoven, Netherlands.Google Scholar
- Beljaars, A.C.M., Krishna Prasad, K. and Vries, D.A. de, (1980). A structural model for turbulent exchange in boundary layers, submitted to J. Fluid Mech.Google Scholar
- Blackwelder, R.F. and Eckelmann, H., (1979). Streamwise vortices associated with the bursting phenomenon, J. Fluid Mech., 24, 577.CrossRefADSGoogle Scholar
- Blackwelder, R.F., and Kaplan, R.E., (1972). The intermittent structure of the wall region of a turbulent boundary layer, Univ. S. Calif. Rep. USCAE, 1-22.Google Scholar
- Blackwelder, R.F., and Kaplan, R.E., (1976). On the wall structure of turbulent boundary layers, J. Fluid Mech., 76, 89.CrossRefADSGoogle Scholar
- Blackwelder, R.F., and Kovasznay, L.S.G., (1972). Time scales and correlation in a turbulent boundary layer, Phys. Fluids, 15, 1545.CrossRefADSGoogle Scholar
- Bradshaw, P., (1973). Effects of streamline curvature on turbulent flow. AGARDograph No. 169.Google Scholar
- Bradshaw, P., (1973). The strategy of calculation methods for complex turbulent flows, Imperial College Aero report 73-05.Google Scholar
- Brodkey, R.S., Wallace, J.M., and Eckelmann, H., (1974). Some Properties of Truncated Turbulence Signals in Bounded Shear Flows. J. Fluid Mech., 63, 209.CrossRefADSGoogle Scholar
- Brown, G.L., and Roshko, A., (1974). On density effects in turbulent mixing layers, J. Fluid Mech., 64, 775.CrossRefADSGoogle Scholar
- Brown, G.L., and Thomas, A.S.W., (1977). Large structures in a turbulent boundary layer, Phys. Fluids, 20, S243.CrossRefADSGoogle Scholar
- Bushnell, O.M., J.N. Hefner and R.L. Ash, (1977). Compliant wall drag reduction for turbulent boundary layers. Phys. Fluids, 20, S31.CrossRefADSGoogle Scholar
- Cebeci, T., and Smith, A.M.O., (1974). Analysis of turbulent boundary layers, Academic Press, New York.zbMATHGoogle Scholar
- Corino, E.R., and Brodkey, R.S., (1969). A visual investigation in the wall region of turbulent flow, J. Fluid, Mech., 37, 1.CrossRefADSGoogle Scholar
- Deardorff, J.W., (1970). A numerical study of three-dimensional turbulent channel flow at large Reynolds number, J. Fluid Mech., 41, 453.zbMATHCrossRefADSGoogle Scholar
- Eckelmann, H., (1974). The structure of the viscous sublayer and the adjacent wall region in a turbulent channel flow, J. Fluid Mech., 65, 439CrossRefADSGoogle Scholar
- Einstein, H.A., and Li, H., (1956). The viscous sublayer along a smooth boundary, J. Engng. Mech. Div. Am. Soc. Civ. Engrs., 82 (EM2), 945.Google Scholar
- Falco, R.E., (1977). Coherent motion in the outer region of turbulent boundary layers, Phys. Fluids, 20, S124.CrossRefADSGoogle Scholar
- Fendell, F.E., (1972). Singular perturbation and turbulent shear flow near walls, J. Astronautical Sc., 20, 129.ADSGoogle Scholar
- Fortuna, G. and Hanratty, T.J., (1972). The influence of drag-reducing polymers on turbulence in the viscous sublayer, J. Fluid Mech., 53, 575.CrossRefADSGoogle Scholar
- Hanratty, T.J., (1956). Turbulent exchange of mass and momentum with a boundary, A.I.Ch.E.Je, 2, 359.Google Scholar
- Hinze, J.O., (1975). Turbulence, McGrawhill, New York, 2nd ed.Google Scholar
- Johnston, J.P., (1972). The Suppression of shear-layer turbulence in rotating systems. ARARD Conf. Proc. 93.Google Scholar
- Kim, H.T., Kline, S.J., and Reynolds, W.C., (1971). The production of turbulence near a smooth wall in a turbulent boundary layer, J. Fluid Mech., 50, 133.CrossRefADSGoogle Scholar
- Kim, J. & Moin, P., (1979). Large eddy simulation of turbulent channel flow — Illiac IV Calculation, AGARD Symposium on Turbulent Boundary Layer-Experiment, Theory, and Modelling.Google Scholar
- Kline, S.J., (1968). Discussion, Proc. of AFOSR-IFP-Stanford Conf. on computation of Turbulent Boundary Layers. Ed: S.J. Kline et al., Vol. 1 p. 527.Google Scholar
- Kline, S.J., Reynolds, W.C., Schraub, F.A. and Rundstadler, P.W., (1967). The structure of turbulent boundary layers. J. Fluid Mech., 30, 741.CrossRefADSGoogle Scholar
- Kovasznay, L.S.G., Kibens, V., and Blackwelder, R.F., (1970). Large scale motion in the intermittent region of a turbulent boundary layer, J. Fluid Mech., 41, 283.CrossRefADSGoogle Scholar
- Landahl, M.T., (1965). A wave-guide model for turbulent shear flow, NASA, CR-317.Google Scholar
- Landahl, M.T., (1967). A wave-guide model for turbulent shear flow, J. Fluid Mech., 29, 441.zbMATHCrossRefADSGoogle Scholar
- Landahl, M.T., (1975). Wave breakdown and turbulence, SIAM, J. Appl. Mech., 28, 735.CrossRefzbMATHGoogle Scholar
- Laufer, J., and Badri Narayanan, M.A., (1971). The mean period of the production mechanism in a boundary layer, Phys. Fluids, 14, 182.CrossRefADSGoogle Scholar
- Mager, A., (1964). Three-dimensional laminar boundary layers in “Theory of Laminar flows”, Ed. F.K. Moore, Princeton University Press.Google Scholar
- Mellor, G.L., (1972). The large Reynolds number asymptotic theory of turbulent boundary layers. Int. J. Engng. Sci., 10, 851.CrossRefMathSciNetGoogle Scholar
- Mizushina, T., and Usui, H., (1977). Reduction of eddy diffusion for momentum and heat in viscoleastic fluid flow in a circular tube. Phys. Fluids, 20, S100.CrossRefADSGoogle Scholar
- Nychas, S.G., Hershey, H.C., and Brodkey, R.S., (1973). A visual study of turbulent flow, J. Fluid Mech., 61, 513.CrossRefADSGoogle Scholar
- Offen, G.R., and Kline, S.J., (1974). Combined dye-streak and hydrogen bubble visual observations of a turbulent boundary layer, J. Fluid Mech., 62, 223.CrossRefADSGoogle Scholar
- Offen, G.R., and Kline, S.J., (1975). A proposed model of the bursting process in turbulent boundary layers, J. Fluid Mech., 70, 209.CrossRefADSGoogle Scholar
- Orszag, S.A., (1978). Prediction of compliant wall drag reduction-Part II, Cambridge Hydrodynamics Report 11.Google Scholar
- Praturi, A.K., and Brodkey, R.S., (1978). A stereoscopic visual study of coherent structures in turbulent shear flow, J. Fluid Mech., 89, 251.CrossRefADSGoogle Scholar
- Rajagopalan, S., and Antonia, R.A., (1979). Some properties of the large structures in a fully developed turbulent duct flow, Phys. Fluids, 22, 614.CrossRefADSGoogle Scholar
- Ramapriyan, B.R. and B.G. Shivaprasad, (1977). Mean flow measurements in turbulent boundary layers along mildly curved surfaces. AIAA Journal, 15, 189.ADSCrossRefGoogle Scholar
- Ramapriyan, B.R., and B.G. Shivaprasad, (1978). The structure of turbulent boundary layers along mildly curved surfaces, J. Fluid Mech., 76, 561.Google Scholar
- Schumann, U. (1975). Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli, J. Comp. Phys., 18, 376.zbMATHCrossRefADSMathSciNetGoogle Scholar
- Senda, M., Susuki, K. and Sats, T., (1979). Turbulence structure related to the heat transfer in a turbulent boundary layer with injection, 2nd symposium on Turbulent Shear Flows, London, 1979.Google Scholar
- Shen, S.F., (1964). Stability of laminar flows, in: Theory of laminar flows, edited by F.K. Moore, Princeton Univ. Press, New Jersey.Google Scholar
- Shubert, G., and Corcos, G.M., (1967). The dynamics of turbulence near a wall according to a linear model, J. Fluid Mech., 29, 113.CrossRefADSGoogle Scholar
- Sternberg, J., (1962). A theory for the viscous sublayer of a turbulent flow, J. Fluid Mech., 13, 241.zbMATHCrossRefADSGoogle Scholar
- Sternberg, J., (1968). Discussion, Proc. of AFOSR-IFP-Stanford Conf. on computations of turbulent boundary layers. Ed: S.J. Kline et al., Vol. 1, p. 411.Google Scholar
- Stuart, J.T., (1965). The production of intense shear layers by vortex stretching and convection, AGARD Rep. 514.Google Scholar
- Tennekes, H., and Lumley, J.L., (1974). A first course in turbulence, MIT Press, Cambridge.Google Scholar
- Virk, P.S., (1975). Drag reduction fundamentals, AICHE J., 21, 625.CrossRefGoogle Scholar
- Willmarth, W.W., (1975). Structure of turbulence in boundary layers, Adv. App. Mech., vol. 15, p. 159., Academic Press.CrossRefGoogle Scholar
- Willmarth, W.W., and Woodridge, G.E., (1962). Measurements of the fluctuating pressure at the wall beneath a thick turbulent boundary layer, J. Fluid Mech., 14, 187.zbMATHCrossRefADSGoogle Scholar
- Witting, H., (1958). Ober den Einfluss der Strömlinien-Krümmung auf die stabilität Laminarer Strömingen, Arch. Rat. Mech. Anal., 2, No. 3, 243.zbMATHCrossRefMathSciNetGoogle Scholar