Nonparallel spatial stability of shallow water flow down an inclined plane of arbitrary slope P. Bohorquez & R. Fernandez-Feria E. T. S. Ingenieros Industriales, Universidad de Málaga, Spain For further information Introduction F IGURE 1depicts the flow after the sudden release of a dam over an inclined plane of arbitrary slope (Bohorquez & Fernandez-Feria 2006). We have found for longer times (see Fig. 2), when the kinematic wave approximation is broadly valid, the unexpected development of roll waves – the flow suddenly changes from a quasi-uniform and quasi-steady state to a strong non-uniform and unsteady state. However, there is no previous evidence in experimental or theoretical research (i.e., Chanson 2006) of these hydrodynamic instabilities. What is the stability threshold in this nonparallel flow? Is the Jeffreys (1925) instability condition ‘‘Froude larger than 2’’ still valid?... And what happens to the asymptotic solution of Hunt (1982)? F IGURE 1. Sketch of the dam-break problem over an inclined plane and water depth profiles for several times,  0 =10m,  =20° and k s =1mm. Note the longer time the more uniform profile. Figure extracted from Bohorquez & Fernandez-Feria (2006) F IGURE 2. Velocity profi- les at different instants of time for the same parame- ters as Fig.1. We observe the spontaneous formation of roll waves. Figure extracted from Bohorquez & Fernandez-Feria (2006) Please contact or More information on this and related projects can be obtained at This research has been supported by a fellowship (PB) from the Ministerio de Educación y Ciencia of Spain, and by the COPT of the Junta de Andalucía (Spain; Ref. 807/ ) Acknowledgments Statement of the problem F IGURE 3. Sketch of coordinates and variables:  is the angle between the bed and the horizontal, t is the time, X is the coordinate along the bed,  is the depth of water measured along the coordinate Y perpendicular to the bed, U is the depth-averaged velocity component along X and f is the Darcy-Weisbach friction factor. All the magnitudes in these equations have been non-dimensionalized with respect to a length scale  0, corresponding to some initial depth, and a velocity scale U 0 =(g  0 ) 1/2, where g is the acceleration due to the gravity. We analyse the linear, spatial stability of the basic flow including the first order effects of the streamwise gradient of both the velocity V(X,t) and the water depth H(X,t). After performing a local spatial stability analysis (Fernandez-Feria 2000) we obtain a set of four homogeneous linear equations We consider here the one-dimensional flow over a constant slope bed. In the shallow water approximation, the dimensionless equations for the mass conservation and momentum in the direction of the flow can be written as (see Fig. 3) yielding a dispersion relation of the form det(F)  D(a,  ; , f, V, V X, H, H X ) = 0 that determines the complex wavenumber a(X)  (X)+i  (X) for a given real frequency . The real part  (X) is the local exponential growth rate, and the imaginary part  (X) is the local wavenumber. Parallel flow: roll waves The flow is stable (  0) for any value of the frequency  if Fr>Fr c. Nonparallel flow: kinematic wave approximation F IGURE 6 shows the neutral curves in the plane {Fr,  } for several values of  >0, while Fig. 7 depicts contour lines of constant growth rate for a particular value of  >0. There are marked differences with the neutral curves for the parallel case (Fig. 4). Firstly, the flow is always stable independently of the Froude number for very small frequencies [i.e., for  0. This critical Froude number tends to zero as  decreases, though the frequency  c1 also vanishes as   0. As in the parallel case, the flow is unstable for almost any frequency when Fr>2 (except for very small frequencies, as commented on above). For high frequencies the stability region shrinks to disappear for very high . The very high frequencies, like the very small ones  <  , are too extreme to be physically meaningful. Results When the kinematic wave approximation is satisfied the number of parameters in the dispersion relation is subtantially reduced with the change of variable that follows: F IGURE 4. Neutral curves (σ=0) and contour lines for a constant growth rate σ in the plane {Fr,  }.Red lines: σ =.001,01,.2,.4,.6,.8, 1,1.2,2,3. Blue lines: σ =-.001,-.01,-.05,-.1,-.15,-.2,-.25. F IGURE 5. Spatial evolution of a perturbation (5·10 -5 %) in a background parallel flow with Fr=2.5, V=1 and θ=1° at several times. F IGURE 6. Neutral curves (σ=0) for different values of  >0 (as indicated) in the plane {Fr,  }. The critical points (Fr c1,  c1 ) and (Fr c2,  c2 ) are shown for one of the curves. F IGURE 7. Contour lines for a constant growth rate σ in the plane {Fr,  } for  = Black line: σ=0. Red lines: σ =.001,.01,.2,.4,.6,.8,1,1.2, 2,3. Blue lines: σ =-.001,-.01,-.05,-.1,-.15,-.2,-.25. We have found the spontaneous (i.e., with the only forcing of the round-off numerical noise) formation of roll waves in our numerical simulations of the dam-break problem in an inclined channel for Froude numbers larger than 2.5 (Fig. 8), and for Froude larger than 2.1 when disturbances with much higher amplitude than the numerical noise are introduced upstream. However, we have not found the formation of roll waves for Froude numbers less than 2. Bohorquez, P. & Fernandez-Feria, R Transport of suspended sediment under the dam-break flow on an inclined bed of arbitrary slope. J. Hydr. Res. (submitted) Chanson H Analytical solutions of laminar and turbulent dam-break waves. Riverflow 2006, Lisbon, Portugal. Jeffreys, H The flow of water in an inclined channel of rectangular section. Phil. Mag., 49, Hunt B Asymptotic solution for dam-break problem. J. Hydr. Div. ASCE, 108, Fernandez-Feria R Axisymmetric instabilities of Bödewadt flow. Phys. Fluids, 12, References F IGURE 8. Velocity profiles U(X) for different instants of time in the dam-break problem for Fr=2.5,  0 =1m and  =1°. Comparation of the numerical simulations (line) with Hunt’s solution (Hunt 1982, circles) THESE ARE ROLL WAVES!!!