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G. SEMINARA Dipartimento di Ingegneria Ambientale, Università di Genova, Italy Coworkers: M. Colombini, B. Federici, M. Guala, S. Lanzoni, N. Siviglia, L. Solari, M. Tubino, D. Zardi, G. Zolezzi, Morphodynamic Influence and related issues Major issue In what directions does morphodynamic influence (i.e. perturbations of bottom topography and/or river alignement) propagate in a river reach ? Plan of the talk 1. The settled case of 1-D perturbations. 2. The case of 2-D perturbations : free bars. 3. The case of 2-D perturbations : forced bars Can a bend (or other geometrical constraint) affect bed topography upstream ? 4. Plan form perturbations in meandering rivers: Can meander evolution be upstream influenced? 1. THE SETTLED CASE OF 1D PERTURBATIONS Sketch of the channel and notations Assume : wide rectangular cross section Q, U Y Y0 Qs h x Y b Independent Variables : x , t Unknown functions : Y , h , U FORMULATION OF THE PROBLEM Governing equations Y UY 0 t x (Continuity of the liquid phase ) U U h U 2 U g 2 0 t x x C Y (Conservation of Momentum) h Qs 1 p 0 t x (Continuity of the solid phase ) The morphodynamic response of the channel to small initial perturbations of bed elevation Seek solution h,h, Y , Qs ,U h0 ,h0 , Y0 , Qs0 ,U0 h1,h1, Y1, Qs1,U1 Basic uniform state Small perturbation Make the formulation dimensionless U1 Y1 h1 x t u , y , h , X 2 , T 2 U0 Y0 Y0 C Y0 C Y0 U 0 Linearize governing equations and reduce 2 3 u u 1 u u u 2 2 u 1 2 2 3 2 3 0 3 t t x t F0 x x F0 x 3 2 Parameters 2 U F02 0 gY0 Froude number b Qso 1 U 0Y0 Rat io between solid and fluid discharge Seek solutions for perturbations in the form of normal modes u f expik x ct 2 k wavenumber L cr kr wavespeed ci ki growth rate Dispersion relationship 1 2 3 i 2 ik 1 i ik 3 k 1 2 ik 2 0 F0 F0 3 2 THREE MODES Typically ~ O (10-3 - 10-4 ), hence expand j 1,2,3 0 j 1 j [O0] k2 01 k i ik 1 2 F0 k2 02 k i ik 1 2 F0 03 0 [O] 11 ................... 12 ................... 1 3k ik 1 2 3 F0 k i 2 2 F0 1 9k 2 k 4 1 2 F0 2 13 Hydrodynamic modes: F0 < 2 they are both stable Hydrodynamic modes: F0 > 2 one mode is unstable (roll waves) In the short (inertial) wave limit (k large): F0 > 1 both modes migrate downstream F0 < 1 one mode migrates upstream In the long wave limit (k small) both modes migrate downstream Morphodynamic mode: Invariably stable for any Froude number F0 = 0.5 : Downstream migration F0 = 1 : No Migration Long waves (k <<1) are Weakly damped and nearly non migrating F0 = 2.5 : upstream Migration Upstream morphodynamic influence in supercritical flows: Fully non linear numerical solution (Siviglia, 2005) F0 = 2.4 = 0.028 Short perturbation: Propagation is very fast! Downstream morphodynamic influence in subcritical flows: Fully non linear numerical solution (Siviglia, 2005) F0 = 0.51 = 0.001 Short perturbation: Propagation is still fast but less so as is smaller! - Non linear effects generate fronts Growthrate and wavespeed tend to vanish as k → 0 Fully non linear numerical solution (Siviglia, 2005) F0 = 0.51 Damping is very weak! = 0.001 Propagation Is very slow! 2. 2-D PERTURBATIONS: free bars Free bars arise spontaneously whenever bed topography is unstable to 2-D perturbations of spatial scale of the order of channel width Under what conditions do they form? Incipient conditions for bar formation determined by classical linear stability analysis: Bars form provided the width to depth ratio of the channel b exceeds critical value bc (e.g. Blondeaux and Seminara, JFM, 1985) In the case of alternate bars we find: b c = b c t * , ds ) - t* : average Shields stress of the mean flow - ds relative roughness of the mean flow Multiple row bars form for higher values of b A MORE DELICATE PROBLEM RELATED TO THE ISSUE OF MORPHODYNAMIC INFLUENCE: How does bar growth occur? For values of b larger than bc any perturbation e.g. located at some cross section of the channel leads to sand wave which migrates with amplitude growing in space and time But: • does the perturbation spread both upstream and downstream? • does it eventually reach an equilibrium (possibly periodic) state? Absolute versus Convective instability (Briggs, 1964 and Bers,1975 in plasma physics, Huerre & Monkewitz, 1990 in hydrodynamics) Absolute instability an impulse response propagates for large times at all points in the flow Convective instability an impulse response decays to zero for large times at all points in the flow: disturbances are convected away as they amplify Is bar instability convective or absolute ? (Federici and Seminara, JFM, 2003) Numerical simulations (Federici and Colombini, 2003) If initial perturbation of bed topography is not persistent localized initial perturbation of bed topography 3-DSpatially view t=0 s t=500 n Initial perturbation of bed topography randomly distributed in space t=500 t=1000 t=1500 Because of the convective nature of bar instability: Need persistent small perturbation of bed topography in the initial crosssection of the channel b = 8 (bc=5.6) t* 0.057 ds= 0.053 t=500 t=650 only if the domain is sufficiently long!!! Forced development of bars leads to equilibrium amplitude t=750 Varying the amplitude of the initial perturbation does not influence the equilibrium amplitude does influence the spatial position at which equilibrium amplitude is reached (— ) perturbation amplitude = 0.001 (----) perturbation amplitude = 0.002 The frequency of the perturbation at initial cross-section does not influence the spatial position at which the equilibrium amplitude is reached if monochromatic, it influences the equilibrium amplitude (—) perturbation frequency = 7.9*10-4 (----) perturbation frequency = 5.7*10-4 linearly most unstable forcing a discrete spectrum containing (10) 20 harmonics of equal amplitudes with frequencies obtained from the linear dispersion relationship in the unstable range. Temporal evolution of bar wavenumber l Temporal evolution of bar wavespeed Numerical simulation of the laboratory experiment H-2 of Fujita & Muramoto (1985) bars hardly develop uniformly along the whole reach a more developed bar always forms upstream and downstream bars have decreasing heights Wave group All bars migrating downstream amplify The tail of the wave group remains in the upstream reach Was an equilibrium amplitude reached? Numerical simulation of the laboratory experiment H-2 of Fujita & Muramoto (1985) b = 10 (bc=7) t* 0.064 ds = 0.047 t=145’ t=240’ t=290’ t=480’ Hence: A persistent perturbation is needed to reproduce the mechanism of formation and development of bars in straight channels correctly. Bars evolve spatially and reach an equilibrium amplitude that is independent of the amplitude and frequency of the initial perturbation Bars lengthen and slow down as they grow in amplitude 2-D informations are propagated downstream The distance from the initial cross section where equilibrium amplitude is reached does depend on the intensity of the initial perturbation. Length of the channel in laboratory experiments must be large enough for equilibrium conditions to be reached : Uncertainty on significance of values of bar amplitude, wavelength and wavespeed reported by different authors 3. Forced Bars Forced bars arise as a response of bed topography i) to variations of channel geometry, e.g. channel curvature ii) to perturbed boundary conditions Fundamental question: does the presence of a bend affect bed topography In the downstream and/or upstream reach ? (Struiksma et al., 1985, J. Hydr. Res. Zolezzi and Seminara, 2001, J. Fluid Mech. Zolezzi et al., 2005, J. Fluid Mech.) FORMULATION (Zolezzi and Seminara, 2001) NOTATIONS Curvature C (s) s Curvature ratio Width ratio ASSUMPTIONS • slowly varying approach • wide channel n0 1 b >> 1 The exact solution of the linear problem of fluvial morphodynamics characteristic exponents Upstream-downstream influence integration constants Required to fit Boundary condtns. Local effect of curvature MORPHODYNAMIC INFLUENCE: The 4 characteristic exponents Dominant UPSTREAM INFLUENCE 3 exponentially GROWING solutions + - bbR 3 exponentially DECAYING solutions Dominant DOWNSTREAM INFLUENCE Validation of the theory of upstream overdeepening U-FLUME EXPERIMENTS • Reproduce sub- and superresonant conditions • Measure temporal bed evolution • Obtain steady bed topography by time-averaging Laboratory of D.I.A.M. - University of Genova PATTERN OF STEADY COMPONENT OF BED TOPOGRAPHY IN U- CHANNELS UNDER SUPERRESONANT CONDITIONS (Zolezzi, Guala & Seminara, 2005, JFM) PATTERN OF STEADY COMPONENT OF BED TOPOGRAPHY IN U- CHANNELS UNDER SUPERRESONANT CONDITIONS (Zolezzi et al., 2005, JFM) PATTERN OF STEADY COMPONENT OF BED TOPOGRAPHY IN U- CHANNELS UNDER SUBRESONANT CONDITIONS (Zolezzi et al., 2005, JFM) BIFURCATION AS A PLANIMETRIC DISCONTINUITY W. Bertoldi, A. Pasetto, L. Zanoni, M. Tubino Department of Civil and Environmental Engineering, Universtiy of Trento, Italy INITIAL STAGE: MIGRATING BARS FINAL STAGE: STEADY BARS Under super-resonant conditions (b > bR) upstream influence 4. Downstream and upstream influence In the plan form evolution Of meandering channels Fundamental questions: Is bend instability convective or absolute ? In what directions do wavegroups migrate? (Lanzoni, Federici and Seminara , 2005) b=15, t * = 0.3, dune covered bed Plan form response of initially straight channel to small random perturbations: Free boundary conditions Subresonant : instability is convective and meander groups migrate downstream planform configurations after several neck cut offs Superresonant : instability is convective and meander groups migrate upstream Conclusion and main message The direction of propagation of 1-D morphodynamic information Changes as the critical value of the Froude number (F0=1) is crossed Role of the Froude number somewhat taken by the aspect ratio of the channel b when the propagation of 2-D morphodynamic information is considered Superresonant channels display features quite different from those of subresonant channels Field verification of this framework urgently needed : a challenge for geomorphologists ? The end Why does the “standard model” (Ikeda & al., 1981) not predict upstream influence ? Local effect of curvature Downstream influence •Only one characteristic exponent l1=2 b Cf0 •Only downstream influence STEADY BED TOPOGRAPHY IN U- CHANNELS : SUPERRESONANT BED PROFILES AVERAGED AT THE INNER AND OUTER BANKS (Zolezzi et al., 2005, JFM) Run U2 STEADY BED TOPOGRAPHY IN U- CHANNELS : SUPERRESONANT BED PROFILES AVERAGED AT THE INNER AND OUTER BANKS (Zolezzi et al., 2005, JFM) Run U3 STEADY BED TOPOGRAPHY IN U - CHANNELS : SUBRESONANT BED PROFILES AVERAGED AT THE INNER AND OUTER BANKS (Zolezzi et al., 2005, JFM) Run D2 Bend instability:Linear theory (Blondeaux &Seminara, 1985) -Bend instability selects near resonant wavenumber bR 10 b 25 bR 10 Subresonant meanders migrate downstream while superresonant meanders migrate upstream b 5 b 25 b 5 Planimetric response of initially straight channel to small random perturbations: Periodic boundary conditions Subresonant Superresonant Planimetric response of initially straight channel to small random perturbations: Free boundary conditions Subresonant : instability is convective and meander groups migrate downstream Superresonant : instability is convective and meander groups migrate upstream Highly Superresonant : instability is absolute and meander groups migrate upstream The third (morphodynamic) mode C > 0 Fo > 1 C< 0 Fo < 1 C = 0 Fo = 1 1 1 2 2 F0 Re13 k c 2 2 k F0 1 9 k 2 1 2 F0 k2 Im13 F02 C → 0 as k→ 0 C → - /(F02 -1) as k → ∞ i) Growth rate is negative for any value of Fo and k 3 1 9 k 2 1 2 F0 2 ii) Damping tends to vanish for very long waves. MORPHODYNAMIC INFLUENCE: LINEAR THEORY Fourier expansion in n IV order ordinary problem for um(s) 4 Characteristic exponents lmj