Advertisement

Dimensional Analysis, Similitude and Model Experiments

  • Kolumban HutterEmail author
  • Yongqi Wang
  • Irina P. Chubarenko
Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

Abstract

The goal of this chapter is to explain how natural processes can be reproduced at much smaller scale—in laboratory experiments. Physical ideas of dimensional analysis are laid down, and Buckingham’s Pi-theorem is formulated. Some typical models in geophysical fluid mechanics are discussed—the so-called Reynolds, Froude, Mach, Rossby models, in which the corresponding Pi-products (called, correspondingly, the Reynolds, Froude, etc. numbers) remain invariants. The procedures of dimensional analysis are applied to the instability of shear flows and demonstrate that the gradient Richardson number must be larger than 1/4, if the shear flow in a vertically stratified viscous fluid should maintain stability. Other explicit applications of dimensional analysis are in the theory of self-similar structures in turbulent boundary layers at large Reynolds numbers and in the spreading of an oil spill on a still water surface. We demonstrate how results from laboratory experiments on heat transfer by forced convection and on subaquatic density currents can be used to deduce inferences at the prototype scale, using invariance of the Pi-products describing such problems. A popular method to obtain such downscaling rules is inspectional analysis. It is based on the assumption that a set of functional equations (partial or ordinary differential equations and initial and boundary conditions) describes the process under study adequately. The laboratory copy of the process in Nature then needs to keep the Pi-products, exhibited by the functional equations, invariant for complete similitude of the mapping. This procedure is demonstrated for the Navier-Stokes-Fourier-Fick equations, for the tidal equations in estuaries and for a rotating laboratory study of circulation dynamics in Lake Constance. This last example demonstrates how difficult it is to obtain reliable responses of the water motion in a homogeneous or stratified water body subject to wind and/or inflows from tributaries, for which the Froude and Rossby numbers are the minimum two invariants of the downscaling operation, which must be observed to guarantee approximate similitude.

List of Symbols

Roman

Symbols

\(A\)

Cross sectional area; Area of a body submerged in a fluid perpendicular to the approaching velocity

\(a = \textstyle {\frac{1}{2}}(\nu _\mathrm{{top}} - \nu _{\text {bottom}})\)

See Eq. (30.63)

\(a_{ij}, i \le m, j \le N\)

Coefficients of dimensional matrix

\({\mathcal A} = [H]/[L]\)

Aspect ratio

\({\mathcal A}_{m,p}\)

Aspect ratio in model and prototype

\(b\)

Coefficient in a parameterization of \(\nu _{\text {vert}}^\text {turb}\) (see Eq. (30.63)), channel width

\({\mathbb B}r = (\rho \nu V^{2})/(\kappa \varDelta T)\)

Brinkmann Number

\(C_D, C_D^{\prime }, C_D^{\prime \!\prime }\)

Dimensionless drag coefficient

\(C_{h}\)

Chezi drag coefficient

\({\mathbb C}h\)

Chezi number

\(c\)

Phase velocity, specific heat

\(c = c_r + \mathrm{{i}}c_i\)

Complex velocity

\(c_{\alpha }\)

Mass concentration of constituent \(\alpha \)

\(c_p\)

Specific heat at constant pressure

\(c_w\)

Wind drag coefficient

\(c_s\)

Solid concentration

\(D\)

Particle diameter,—of a pipe

\(D, D_0\)

Diffusivity, reference diffusivity

\(D^{\alpha \beta }\)

Species cross diffusivities

\({\mathfrak d}\)

Nominal diameter of a sediment grain

\({\mathfrak d}^{*}\)

Dimensionless particle diameter

\({\varvec{E}}\)

Deviator of \({\varvec{D}} \;(= {\varvec{D}} - \textstyle {\frac{1}{3}}(\mathrm{{tr}}{\varvec{D}}) {\varvec{I}})\)

\({\mathbb E}d\)

Dissipation number

\({\mathbb E}u\)

Euler number, pressure coefficient

\({\mathbb E}k\)

Ekman number

\(f\)

Symbol for an unspecified function

\([f]\)

Order of magnitude value of \(f\)

\({\mathcal F}_x, {\mathcal F}_y, {\mathcal F}_z\)

Cartesian components of the viscous body force \({\varvec{{\mathcal F}}}\)

\({\mathbb F}r = V^{2}/(Dg)\)

Froude number

\(({\mathbb F}r)_d = (\varDelta \rho /\rho ){\mathbb F}r\)

Densimetric Froude number

\(\mathcal {g}, {\varvec{g}}\)

Gravity constant, gravity vector

\({\mathfrak g}\)

Gibbs free energy, free enthalpy

\(H\)

Thickness of a fluid layer

\({\mathcal H}(x)\)

Heaviside function \(= 1 { for } x \ge 0, = 0 { for } x < 0\)

\(h, h_{1,2}\)

Depths of fluid layers, heights of the free surface and interface of the layers

\({i}\)

Imaginary unit \( {i} \equiv \sqrt{-1}\)

\({\varvec{j}}_{\alpha }\)

Diffusive mass flux

\(K\)

Modulus of a force \({\varvec{K}}\)

\(k\)

Surface roughness of a pipe, wavenumber, turbulent kinetic energy

\(k_1, k_2,..., k_n\)

Exponents in the representation of a physical quantity in terms of dimensional products.

\(L\)

Symbol for length units

\(\ell \)

Length of a pipe discharging water from a vessel

\(M\)

Symbol for mass units

\({\mathbb M} = V/c\)

Mach number

\(m, m^{\prime }\)

Masses of two bodies subject to Newtons law of gravitation

\({\mathbb N} = \alpha D/\kappa \)

Nusselt number

\(n\)

Manning roughness coefficient

\(P\)

Pressure number, Work done by the external pressure on the perturbation velocity

\({\mathbb P}\)

Pressure ratio

\({\mathbb P}e = {\mathbb R}e{\mathbb P}r\)

Peclet number

\({\mathbb P}e^{tracer} = {\mathbb S}{\mathbb R}e\)

Tracer Peclet number

\({\mathbb P}r \equiv {\mathbb P}r^{th} = \nu /\kappa \)

Prandtl number

\(({\mathbb P}r^{tracer} = [V][L])/([{\mathcal D}])\)

Prandtl (Schmidt number of tracer diffusion

\({\mathbb P}roud = [\nu ]^{2}[L]^{2}/(g[H]^{5})\)

Proudman number

\(p\)

Pressure

\(p^{\prime }\)

Pressure perturbation

\(p^\text {air}\)

Air pressure

\(Q\)

Volume flux through a cross section, volume discharge of a density flow, rate of energy transfer from the mean flow by the Reynolds stress \(\tau _{13}\)

\(Q_j, 0 \leqslant j \leqslant N\)

Unspecified physical variable in a laboratory experiment

\(q\)

Source of volume flux

\({\varvec{q}}\)

Heat flux vector

\(R\)

Hydraulic radius

\({\mathbb R}a\)

Radiation number

\({\mathbb R}e = vD/\nu \)

Reynolds number

\({\mathbb R}e^{*}\)

Critical particle Reynolds number, outer boundary layer Reynolds number

\({\mathbb R}e-{\ell }\)

Viscous sub-layer Reynolds

\({\mathbb R}i^\text {grad}\)

Gradient Richardson number

\({\mathbb R}i^{c}\)

Critical Richardson number

\({\mathbb R}i^\text {jump}\)

Jump Richardson number

\(r\)

Specific radiation source

\(s\)

Auxiliary velocity variable, specific entropy

\(s(t)\)

Mean diameter of an oil spill on a lake

\(\langle \,s^{2}\rangle \)

Standard deviation of \(s(t)\)

\(S_{E}\)

Slope of the energy gradient line

\({\mathbb S} = [\nu ]/([{\mathcal D}^\text {spec}])\)

Schmidt number

\({\mathbb S}t = [L]/([V][\tau ])\)

Strouhal number

\(T\)

Symbol for time units, kinetic energy of the fluctuating motion, temperature (Kelvin or Celsius)

\({\varvec{t}}^\text {wind}\)

Wind shear traction

\({\varvec{t}}^{R}\)

Viscous Cauchy stress

\({\mathbb T}h\)

Temperature number

\(U\)

Down-slope velocity

\(U(z)\)

Basic state of \(x\)-velocity component

\(u = U(z) + u^{\prime }\)

Velocity component in \(x\)-direction

\(u^{\prime }\)

Perturbation verlocity of \(u\)

\(u^{*} = \sqrt{\tau /\rho }\)

Shear stress velocity

\(V\)

Symbol for velocity, Potential energy of a water particle vertically displaced from equilibrium level

\(v\)

Velocity component in \(y\)-direction

\({\varvec{v}}\)

Velocity vector

\({\varvec{v}}^{\prime }\)

Perturbation of \({\varvec{v}}\)

\({\varvec{W}}\)

Wind velocity at the lake surface

\(W\)

Total complex velocity

\({\mathbb W} = \rho V^{2}D/\sigma \)

Weber number

\(w\)

Velocity component in \(z\)-direction

\(w^{\prime }\)

Perturbation of \(w\)

\(x_i\)

Cartesian coordinates, independent variable of a dimensionally homogeneous function

\(y\)

\(y\)-coordinate of a Cartesian coordinate system

\(z\)

\(z\)-coordinate of a Cartesian coordinate system

Greek

Symbols

\(\alpha \)

Heat transfer coefficient, parameter in Baranblatt et al. self-similar turbulent boundary layer in shear flows, see Eq. (30.122)

\(\beta \)

Exponent of the Reynolds number in a parameterization of the heat transfer coefficient

\(\gamma \)

Exponent of the Brinkmann number in a parameterization of the heat transfer coefficient

\(\varGamma \)

Gravitational constant

\(\varDelta \)

\(=(\rho _s/\rho 1)\) or \(\varDelta = \rho _2/\rho _1\)

\(\varDelta \,p\)

Pressure difference

\(\varDelta \,T\)

Temperature difference

\(\delta \)

Boundary layer thickness

\(\delta (x)\)

Dirac delta function, defined as \(\int _{- \infty }^{\infty } f(x)\delta (x-x_0)\text {d}x = f({x_0})\)

\(\varepsilon \)

Specific dissipation rate of turbulent kinetic energy

\(\zeta (x,t)\)

Vertical free surface displacement

\(\zeta _{I}(x,t)\)

Vertical interface displacement

\(\zeta \)

Bulk viscosity

\(\zeta \)

Perturbation of \(\zeta \)

\(\eta \)

Shear viscosity

\(\theta _c\)

Critical Shields parameter

\(\kappa \)

Thermal conductivity

\(\kappa = 0.4\)

von Kármán constant

\(\lambda = \lambda ({\mathbb R}e, k/D)\)

Turbulent drag function in pipe flow

\(\lambda _L = \bar{L}/L\)

Ratio of the scale length of a laboratory model and the corresponding scale length in the prototype

\(\lambda _V = \bar{V}/V\)

Ratio of the scale velocity of the model to that of the prototype

\(\lambda _{f} = \bar{f}/f\)

Scale of the variable \(f\)

\(\nu \)

Kinematic viscosity, volume fraction

\(\nu _s\)

Solid volume fraction

\(\nu _{\text {vert}}^\text {turb}\)

Turbulent kinematic viscosity

\(\nu _\mathrm{{top}}, \nu _{\text {bottom}}\)

Top/bottom kinematic viscosities

\(\nu _{\text {vert}}\)

Kinematic viscosity according to Eq. (30.64)

\(\varPi , \varPi _i, \varPi ^{*}, \varPi _y, \varPi _{x_i}\)

Various \(\varPi \)-products

\(\pi _{\alpha }\)

Production rate of constituent \(\alpha \)

\(\rho \)

Mass density (general, of the mixture)

\(\rho _0\)

Reference density, equilibrium density

\(\rho _f, \rho _s\)

Fluid, solid mass densities

\(\bar{\rho }\)

Linear combination of \(\rho \) and \(\rho _s\)

\(\sigma \)

Stress, surface tension, dimensionless density anomaly

\(\tau \)

Shear stress

\(\tau = \sqrt{2gh}/(2\ell )t\)

Dimensionless time

\(\tau _c\)

Critical shear stress

\(\varphi = v/\sqrt{2gh}\)

Dimensionless velocity

\(\phi = u(z)/u{*}\)

Dimensionless universal boundary layer velocity profile

\(\phi \)

Inclination angle of a lake shore

\(\Phi \)

Specific viscous dissipation rate

\({\varvec{\omega }}, \omega \)

Angular velocity

\({\varvec{\varOmega }}, \varOmega \)

Earths angular velocity

Miscellaneous

Symbols

\(\frac{{\text {d}}(\bullet )}{{\text {d}}t} = (\bullet )^{\bullet }\)

Material time derivative, keeping the reference configuration fixed

\(\frac{\partial (\bullet )}{\partial \,t}\)

Time derivative, keeping the spatial position fixed

\(\times \)

Multiplication sign in general text; vector product in \({\mathbb R}^{3}\)

\([\![f]\!] = f^{+} f^{-}\)

Jump of \(f\) across a singular surface

\([f]_R = [\bar{f}]/[f]\)

Ratio of \(\bar{f}\) in the model to \(f\) in the prototype

References

  1. 1.
    Auerbach, M and Ritzi, M.: Die Oberflächen- und Tiefenströme des Bodensees. IV der Lauf des Rheinwassers durch den Bodensee in den Sommermonaten. Arch. Hydrobiol., 32, 409–433 (1938)Google Scholar
  2. 2.
    Baker, D.J.: A technique for the precise measurements of small fluid velocities. J. Fluid Mech., 573–575 (1966)Google Scholar
  3. 3.
    Barenblatt, G. I.: Dimensional analysis. Sc Sordon & Breach Science Publishers, New York etc., 135 p. (1987)Google Scholar
  4. 4.
    Barenblatt, G. I., Chorin, A. J. and Prostokishin, V. M.: Self-similar intermediate structures in turbulent boundary layers at large Reynolds numbers. J. Fluid Mech., 410, 263–283 (2000)Google Scholar
  5. 5.
    Birkhoff, G.: Hydrodynamics, A study in Logic, Fact and Similitude, Dover, Publ. New York (1955)Google Scholar
  6. 6.
    Buckingham, E.: On physically similar systems; Illustrations of the use of dimensional equations. Phys. Rev. 4(4), 35–376 (1914)Google Scholar
  7. 7.
    Buechi, P. and Rumer, Jr.: Wind-induced circulation pattern in a rotating model of Lake Erie. Proc. 12th Conf. Great Lakes Res., 406–414 (1969)Google Scholar
  8. 8.
    Carlson, D. E.: On some new results in dimensional analysis. Arch. Rational Mech. Anal. 68, 191–220 (1978)Google Scholar
  9. 9.
    Carlson, D. E.: A mathematical theory of physical units, dimensions and measures. Arch. Rational. Mech. Anal. 70, 289–305 (1979)Google Scholar
  10. 10.
    Colebrook, C.F.: Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws. J. Institution of Civ. Engrs., London, 11, 133–136 (1938–39)Google Scholar
  11. 11.
    Drazin, P.G.: The stability of a shear layer in an unbounded heterogeneous inviscid fluid. J. Fluid Mech. 4, 214–224 (1958)Google Scholar
  12. 12.
    Drazin, P. G. and Howard, N. L.: Hydrodynamic stability of parallel flow of inviscid fluid. In ‘Advances of Applied Mechanics’, Academic Press, New York, 9, 1–89 (1966)Google Scholar
  13. 13.
    Einstein, H.A.: The bedload function for sediment transportation in open channel flow. Technical Bulletin Nr 1026. U.S. Dept. Agriclture, Washington DC. (1950)Google Scholar
  14. 14.
    Esch, R. E.: Stability of parallel flow of a fluid over a slightly heavier fluid. J. Fluid Mech. 12, 192–208 (1962)Google Scholar
  15. 15.
    Goldstein, S.: On the stability of superposed streams of fluid of different densities. Proc. Royal Soc. London, A132, 524–548 (1931)Google Scholar
  16. 16.
    Görtler, H.: Dimensionsanalyse, Springer Berlin etc., 247 p, (1975)Google Scholar
  17. 17.
    Graf. W.H.: Hydraulics of Sediment Transport, Mc Graw Hill, New York, pp. 513 (1966), Water Resources Publ. (1984)Google Scholar
  18. 18.
    Harleman, D.R.F., Holley, Jr. E.R., Hoopes, J.A. and Rumer, Jr. R.: The Great Lakes and Michigan. Great Lakes Research Division, Spec. Report Nr. 3, University Michigan, 26–28 (1961)Google Scholar
  19. 19.
    Harleman, D.R.F., Bunker, R.M. and Hall, J.B.: Circulation and thermocline development in a rotating lake model. Proc 7th Conf. Great Lakes Res. Publ. 11, 340–356 (1964)Google Scholar
  20. 20.
    Harleman, D.R.F.: Physical hydraulic models. In: Estuarine Modeling. An Assessment. EPA, Water Pollution Control Res. Ser., 16070. DZV 02/71 (1971)Google Scholar
  21. 21.
    Harleman, D.R.F., Holley, E., Hoopes, J. and Rumer, R.: Physical Limnology and Meteorology. The feasibility of a dynamic model study of Lake Michigan. Mich. Univ. Great Lakes res. Div. Publ., 9, 51–67 (1962)Google Scholar
  22. 22.
    Howard. N. L.: Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512 (1961)Google Scholar
  23. 23.
    Hutter, K.: Fluid- und Thermodynamik - Eine Einführung. Springer Verlag, Berlin etc, second edition, 445 p. (2002)Google Scholar
  24. 24.
    Hutter, K. and Luca, I.: A global view of sediment transport in alluvial systems. Mitt. des Inst. für Wasserbau und Wasserwirtschaft, Tech. University Munich, Nr 127, 81–165 (2013)Google Scholar
  25. 25.
    Hutter, K and Jöhnk, K.: Continuum methods and physical modeling - Continuum mechanics, dimensional analysis, turbulence, Springer Verlag, Berlin, etc., 635 p (2004)Google Scholar
  26. 26.
    Kármán, Th. von: Mechanische Ähnlichkeit von Turbulenz, Proc. 3rd Intl. Congress for Applied Mechanics (Ed. C. W. Oseen. & W. Weibull), .1, 85–93 (1930) AB Sveriges Litografisska Trycckenier, StockholmGoogle Scholar
  27. 27.
    Keulegan, G.H.: Wind tides in small closed channels. Bur. Stabnd. J. Res. 46, 358–381 (1951)Google Scholar
  28. 28.
    Kiefer, F.: Naturkunde des Bodensees, 209 p. Jan Thorbecke Verlag KG, Sigmaringen (1972)Google Scholar
  29. 29.
    Langhaar, H. L.: Dimensional analysis and model theory. John Wiley and Sons, New York, 166 p (1964)Google Scholar
  30. 30.
    Leblond, P. H. and Mysak, L. A.: Waves in the Ocean Elseviers Oceanographic Series. 602 pp. Elsevier Scientific Publ. Company, Amsterdam, Oxford, New York (1978)Google Scholar
  31. 31.
    Li, C.Y.: Hydraulic model study of surface and subsurface wind-driven currents in Lake Ontario. Ph. D. thesis, State University, New York at Buffalo (1973)Google Scholar
  32. 32.
    Li, C.Y., Kiser, K.M. and Rumer, R.R.: Physical model study of circulation patterns in Lake Ontario. Limnol. & Oceanogr. 20, 323–337 (1975)Google Scholar
  33. 33.
    Liggett, J.A.: Unsteady circulation in shallow, homogeneous lakes. J. Hydraul. Div. Am. Soc. Civ. Engr. 95(HY4), 1273–1288 (1969)Google Scholar
  34. 34.
    Maiss, M.: Schwefelhexafluoroid (SF) als Tracer für Mischungsprozesse im westlichen Bodensee. Ph. D. Dissertation Ruprecht-Karls-Universität, Heidelberg (1992)Google Scholar
  35. 35.
    Meyer-Peter, E. and Müller, R.: Formulas for bedload transport. Report on Second Meeting of IAHR, Stockholm, Sweden, 39–64 (1948)Google Scholar
  36. 36.
    Miles, J. W.: On the stability of heterogeneous shear flows. J. Fluid Mech., 10, 496–508 (1961)Google Scholar
  37. 37.
    Moody, L.F.: Friction factors for pipe flow. Trans. Amer. Soc. Mech. Engrs. 66, 671–684 (1944)Google Scholar
  38. 38.
    Nikuradse, J.: Untersuchungen über die Geschwindigkeitsverteilung in turbulenten Strömungen. Dissertation Göttingen (1926). VDI-Forschungsheft, 281, Berlin, p 44 (1926)Google Scholar
  39. 39.
    Nikuradse, J.: Gesetzmässigkeiten der turbulenten Strömung in glatten Rohren. Forschungsarb Ing-wesen. 356, p. 36 (1932)Google Scholar
  40. 40.
    Nikuradse, J.: Strömungsgesetze in rauhen Rohren. Forschungsarb. Ing-wesen, 361, p. 22 (1933)Google Scholar
  41. 41.
    Oman, G.: Das Verhalten des geschichteten Zürichsees unter äusseren Windlasten - Resultate eines numerischen Modells Mitteilung Nr 60 der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie an der ETH Zürich, pp. 1–185 (1982)Google Scholar
  42. 42.
    Okubo, A.: Some speculations on oceanographic diffusion diagrams. Rapp. P-v Run. Const. Explor. Mer., 167, 77–85 (1974)Google Scholar
  43. 43.
    Plate, E. J.: Water surface velocities induced by wind shear Proc. Am. Soc. Civ. Eng., 96, 295–312 (1970)Google Scholar
  44. 44.
    Prandtl. L.: Zur turbulenten Ströhmung in Rohren und längs Platten. Ergeb. Aerodyn. Versuch., Series 4, Goettingen, 18–29 (1932)Google Scholar
  45. 45.
    Prandtl, L.: Neuere Ergebnisse der Turbulenzforschung. Zeitschrift VDI, 77, 105–113 (1933)Google Scholar
  46. 46.
    Rayleigh, J. W. S. (Lord): On thee stability, or instability of certain fluid motions. Proc. Lond. Math. Soc. 9, 57–70 (1980)Google Scholar
  47. 47.
    Rumer, R.R. and Robson, L.: Circulation studies in a rotating model of Lake Erie. Proc. 11th Conf. Great Lakes Res., 487–495 (1968)Google Scholar
  48. 48.
    Rumer, R.R. and Hoopes, J.A.: Modeling Great Lakes circulations. In: Water Environment and Human Needs, Mass. Inst. Techn. (1970)Google Scholar
  49. 49.
    Schmalz, J.: Die Oberflächen- und Tiefenströme des Bodensees. III: Der Weg des Rheinwassers im Bodensee. Schr. Ver. Gesch. Bodensee, 60, 154–210 (1932)Google Scholar
  50. 50.
    Schwab, D.: Simulation and forecasting Lake Erie storm surges. Monthly Weather Reviews, 106, 1476–1487 (1978)Google Scholar
  51. 51.
    Scotti, R. S. and Corcos, G.M.: Measurements on the growth of small disturbances in a stratified shear layer. Radio Science, 4, 1309–1313 (1969)Google Scholar
  52. 52.
    Shields, A.: Anwendung der Ähnlichkeitsmechanik und der Turbulenzforschung. Mitt. Der Preussischen Versuchsanstalt für Wasserbau und Schiffsbau, Heft 26, Berlin (1936)Google Scholar
  53. 53.
    Shiau, J.C. and Rumer, R.R.: Adjustment of friction in hydraulic models of lakes. Proc. Am. Soc. Civ. Eng. 99, 2251–2262 (1973)Google Scholar
  54. 54.
    Simons, T.J.: Circulation Models in Lakes and Inland Seas. Canadian Bulletin of fisheries and Aquatic Sciensces Bulletin, 203 Ottawa (1980)Google Scholar
  55. 55.
    Spurk, J. H.: Dimemsionsanalyse in der Strömungslehre. Springer Verlag, Berlin etc., 270 p (1992)Google Scholar
  56. 56.
    Squire, H. B.: On the stability of three-wwimensional disturbances of viscous flow between parallel walls. Proc. R. Soc. Lond. 142A, 621–628 (1933)Google Scholar
  57. 57.
    Stewart, K.: Tracing inflows in a physical model of Lake Constance. J. Great Lakes Res., 14(4), 466–478 (1988). Internatl. Assoc. Great Lakes Res, 1988Google Scholar
  58. 58.
    Stewart, K. and Rumer, R.R.: Mass oscillation study of Cayuga Lake, N.Y. Proc 13th Conf. great Lakes Res. 540–551 (1970)Google Scholar
  59. 59.
    Stewart, K. and Hollan E.: Physical model study of Lake Constance. Schweiz. Z. Hydrol. 46(1), 5–40 (1984)Google Scholar
  60. 60.
    Synge, J.L.: The stability of heterogeneous liquids. Trans. R. Soc. Can, 27(III), 1–18 (1933)Google Scholar
  61. 61.
    Taylor, G. I.: Effect of variation of density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. 132A, 499–523 (1931)Google Scholar
  62. 62.
    Van Rijn, L.C.: Sediment pick-up functions J. Hydraul. Engineering, ASCE 110(10), 1494–1502 (1984a)Google Scholar
  63. 63.
    Van Rijn, L.C.: Sediment transport, Part II. J. Hydraul. Engineering, ASCE 110(11), 1613–1641 (1984b)Google Scholar
  64. 64.
    Van Rijn, L.C.: Unified view of sediment transport by currents, and waves. I Initiation of motion, bed roughness and bed load transport. J. Hydraulk. Eng. ASACE 133(6), 649–667 (2007)Google Scholar
  65. 65.
    Vetsch, D. Numerical simulation of sediment transport with mesh-free methods Doctoral dissertation, laboratory of Hydraulics, hydrology and Glaciology, Swiss Federal Institute of Technology, Zurich, p. 200 (2012)Google Scholar
  66. 66.
    Wasmund, E.: Die Strömungen im Bodensee, verglichen mit bisher in Binnenseen bekannten Strömungen. Int. Review ges. Hydrobiol. Hydrogr. 19, 21–155 (1928)Google Scholar
  67. 67.
    Yalin, M.S.: Mechanics of Sediment Transport, Second Edition Pergamon Press, Oxford (1977)Google Scholar
  68. 68.
    Yalin, M.S. and da Silva A.M.F.: Fluvial processes, IAHR, intntl. Assoc. Hydraul. Engr. and Research (2001)Google Scholar
  69. 69.
    Yih, C. S.: Stability of of two-dimensional parallel flows for three-dimensional disturbances. Quart. Applied. Math. 12, 434–435 (1955)Google Scholar
  70. 70.
    Yih, C. S.: On stratified flows in a gravitational field. Tellus  9, 220–228 (1957)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Kolumban Hutter
    • 1
    Email author
  • Yongqi Wang
    • 2
  • Irina P. Chubarenko
    • 3
  1. 1.c/o Laboratory of Hydraulics, Hydrology and Glaciology at ETHZürichSwitzerland
  2. 2.Chair of Fluid Dynamics, Department of Mechanical EngineeringTU DarmstadtDarmstadtGermany
  3. 3.P.P. Shirshov Institute of OceanologyRussian Academy of SciencesKaliningradRussia

Personalised recommendations