Abstract
A novel Orr–Sommerfeld-like equation for gravity-driven turbulent open-channel flows over a granular erodible bed is here derived, and the linear stability analysis is developed. The whole spectrum of eigenvalues and eigenvectors of the complete generalized eigenvalue problem is computed and analyzed. The fourth-order eigenvalue problem presents singular non-polynomial coefficients with non-homogenous Robin-type boundary conditions that involve first and second derivatives. Furthermore, the Exner condition is imposed at an internal point. We propose a numerical discretization of spectral type based on a single-domain Galerkin scheme. In order to manage the presence of singular coefficients, some properties of Jacobi polynomials have been carefully blended with numerical integration of Gauss–Legendre type. The results show a positive agreement with the classical experimental data and allow one to relate the different types of instability to such parameters as the Froude number, wavenumber, and the roughness scale. The eigenfunctions allow two types of boundary layers to be distinguished, scaling, respectively, with the roughness height and the saltation layer for the bedload sediment transport.
Similar content being viewed by others
References
Andreotti B., Fourriere A., Ould-Kaddour F., Murray B., Claudin P.: Giant aeolian dune size determined by the average depth of the atmospheric boundary layer. Nature 457(7233), 1120–1123 (2009). doi:10.1038/nature07787
Best J.: The fluid dynamics of river dunes: a review and some future research directions. J. Geophys. Res. 110, F04S02 (2005)
Blumberg P.N., Curl R.L.: Experimental and theoretical studies of dissolution roughness. J. Fluid Mech. 65, 735–751 (1974)
Butler K.M., Farrel B.F.: Three-dimensional optimal perturbations in viscous shear flows. Phys. Fluids A 4, 1637–1650 (1992)
Camporeale, C., Ridolfi, L.: Nonnormality and transient behavior of the de Saint-Venant-Exner equations. Water Resour. Res. 45(8), (2009)
Canuto C., Hussaini M., Quarteroni A., Zang T.: Spectral Methods. Fundamentals in Single Domains. Springer, Berlin (2006)
Canuto C., Hussaini M., Quarteroni A., Zang T.: Spectral Methods. Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, Berlin (2007)
Carling P., Richardson K., Ikeda H.: A flume experiment of the development of subacqueous fine-gravel dunes from a lower-stage plane bed. J. Geophys. Res. Earth Surf. 110, F04S05 (2005)
Colombini M.: Revisiting the linear theory of sand dune formation. J. Fluid Mech. 502, 1–16 (2004)
Colombini M., Stocchino A.: Finite-amplitude river dunes. J. Fluid Mech. 611, 283–306 (2008)
Criminale W., Jackson T., Joslin R.: Theory and Computation in Hydrodynamic Stability. Cambridge University Press, Cambridge (2003)
Davis P., Rabinowitz P.: Methods of Numerical Integration. Academic Press, London (1984)
Devauchelle O., Malverti L., Lajeunesse E., Lagree P.Y., Josserand C., Thu-Lam K.D.N.: Stability of bedforms in laminar flows with free surface: from bars to ripples. J. Fluid Mech. 642, 329–348 (2010). doi:10.1017/S0022112009991790
DiPrima R., Habetler G.: A completeness theorem for non-selfadjoint eigenvalue problems in hydrodynamic stabiilty. Arch. Rat. Mech. Anal. 32, 218–227 (1969)
Dongarra J., Straughan B., Walker D.: Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems. Appl. Numer. Math. 22(4), 399–434 (1996)
Drake T., Shreve R., Dietrich W., Whiting P., Leopold L.: Bedload transport of fine gravel observed by motion-picture photograhy. J. Fluid Mech. 192, 193–217 (1988)
Drazin P., Reid W.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1981)
Dufek J., Bergantz G.W.: Suspended load and bed-load transport of particle-laden gravity currents: the role of particle-bed interaction. Theor. Comput. Fluid Dyn. 21(2), 119–145 (2007). doi:10.1007/s00162-007-0041-6
Engelund F., Fredsoe J.: Sediment ripples and dunes. Ann. Rev. Fluid Mech. 14, 13–37 (1982)
Exner, F.M.: Uber Die Wechselwirkung Zwischen Wasser und Geschiebe in Flussen, pp. 165–180. Sitzer Akad. Wiss, Wien (1925)
Feltham D.L., Worster M.G.: Flow-induced morphological instabilty in mushy layer. J. Fluid Mech. 391, 337–357 (1999)
Fourriere A., Claudin P., Andreotti B.: Bedforms in a turbulent stream: formation of ripples by primary linear instability and of dunes by nonlinear pattern coarsening. J. Fluid Mech. 649, 287–328 (2010). doi:10.1017/S0022112009993466
Fredsoe J.: On the development of dunes in erodible channels. J. Fluid Mech. 64, 1–16 (1974)
Giannakis D., Fischer P.F., Rosner R.: A spectral Galerkin method for the coupled Orr-Sommerfeld and induction equations for free-surface MHD. J. Comp. Phys. 228(4), 1188–1233 (2009). doi:10.1016/j.jcp.2008.10.016
Golub G., Welsch J.: Calculation of Gauss quadratures rules. Math. Comp 23, 221–230 (1969)
Grosch C., Salwen H.: The stability of steady and time-dependent plane Poiseuille flow. J. Fluid Mech. 34, 177–194 (1968)
Gustavsonn L.H.: Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260 (1991)
Guy, H., Simons, D., Richardson, E.: Summary of alluvial channel data from flume experiments. Prof. paper 462-I. US Geol. Survey (1966)
Huppert H.E., Dade W.B.: Natural disasters: explosive volcanic eruptions and gigantic landslides. Theor. Comp. Fluid Dyn. 10(1–4), 201–212 (1998)
Kirchner, N.: Computational aspects of the spectral Galerkin fem for the Orr-Sommerfeld equation. Int. J. Numer. Meth. Fluids 119–137 (2000)
Kleinhans M.: Sorting in grain flows at the lee side of dunes. Earth Sci. Rev. 65(1–2), 75–102 (2004). doi:10.1016/S0012-8252(03)00081-3
Kopriva D.: Implementing Spectral Methods for Partial Differential Equations. Springer, Berlin (2009)
Langlois V., Valance A.: Formation of two-dimensional sand ripples under laminar shear flow. Phys. Rev. Lett. 94(24), 1–4 (2005)
Lanzoni, S., Siviglia, A., Frascati, A., Seminara, G.: Long waves in erodible channels and morphodynamic influence. Water Resour. Res. 42(6) (2006). doi:10.1029/2006WR004916
Melenk J., Kirchner N., Schwab C.: Spectral Galerkin discretization for hydrodynamic stability problems. Computing 65, 97–118 (2000)
Meyer-Peter, E., Müller, R.: Formulas for bed-load transport. In: Proceedings 2nd Meeting IAHR, pp. 39–64 (1948)
Nezu I., Rodi W.: Open-channel flow measurements with a laser Doppler Anemometer. J. Hydraul. Eng. ASCE 112, 335–355 (1986)
Olsson P.J., Henningson D.S.: Optimal disturbance growth in water table flow. Stud. Appl. Maths 94, 183–210 (1995)
Orszag S.: Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, 689 (1971)
Packman A., Brooks N.: Hyporheic exchange of solutes and colloids with moving bed forms. Water Resour. Res. 37(10), 2591–2605 (2001)
Parker, G.: 1D sediment transport morphodynamics (2003). http://vtchl.uiuc.edu/people/parkerg/morphodynamics_e-book.htm
Parker G., Seminara G., Solari L.: Bed load at low shields stress on arbitrarily sloping beds: alternative entrainment formulation. Water Resour. Res. 39, 1183 (2003)
Pope S.: Turbulent Flows. Cambridge University Press, Cambridge (2000)
Reddy S., Schmid P., Henningson D.: Pseudospectra of the Orr-Sommerfeld operator. SIAM J. Appl. Math. 53(1), 15–47 (1993)
Reddy S.C., Henningson D.S.: Energy growth in viscous channel. J. Fluid Mech. 252, 209–238 (1993)
Richards K.: The formation of ripples and dunes on an erodible bed. J. Fluid Mech. 99, 597–618 (1980)
Schmid P.: Nonmodal stability theory. Ann. Rev. Fluid Mech. 39, 129–162 (2007)
Schmid, P.J., Henningson, D.S.: Stability and transition in shear flows. Appl. Math. Sci., vol. 142. Springer, New York (2001)
Sekine M., Kikkawa H.: Mechanics of salting grains. J. Hydraul. Engng. ASCE 118, 536–558 (1992)
Seminara G.: Stability and morphodynamics. Meccanica 33, 59–99 (1998)
Seminara G.: Fluvial sedimentary patterns. Ann. Rev. Fluid Mech. 42, 43–66 (2010). doi:10.1146/annurev-fluid-121108-145612
Shen J.: Efficient spectral-galerkin methods I. Direct solvers for the second and fourth order equations using Legendre polynomials. SIAM J. Sci. Comput. 15(6), 1489 (1994)
Swarztrauber P.: On computing the points and weights for Gauss-Legendre quadrature. SIAM J. Sci. Comput. 24(3), 945–954 (2002)
Szegö, G.: Orthogonal Polynomials. American Mathematical Society, Providence (1939)
Trefethen L.N., Embree M.: Spectra and Pseudospectra. Princeton University Press, Princeton (2005)
Vanoni V.: Sedimentation Engineering. ASCE Manual and Reports on Engineering Practice. ASCE, New York (1975)
Whitham G.B.: Linear and Nonlinear Waves. Wiley-Intercience, New York (1974)
Wong M., Parker G.: Reanalysis and correction of bed-load relation of Meyer-Peter and Müller using their own database. J. Hydraul. Engng. ASCE 132, 1159–1168 (2006)
Yakimiw E.: Accurate computation of weights in classical Gauss–Christoffel quadrature rules. J Comput. Phys. 129, 406–430 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Malik
Rights and permissions
About this article
Cite this article
Camporeale, C., Canuto, C. & Ridolfi, L. A spectral approach for the stability analysis of turbulent open-channel flows over granular beds. Theor. Comput. Fluid Dyn. 26, 51–80 (2012). https://doi.org/10.1007/s00162-011-0223-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00162-011-0223-0