Ioannis Vardoulakis N.T.U. Athens

Transcription

1 ENSHMG March 10-14, 2008 an EU SOCRATES short course on Engineering Continuum Mechanics: traffic flow and shallow water waves Ioannis Vardoulakis N.T.U. Athens ( 1

2 natural phenomena mathematical modeling-the continuum assumption the elementary traffic-flow theory St. Venant s shallow-water theory shallow-water model of granular flows 2

3 the continuum assumption < ρ > zl, = ρ( z) Leonhard Euler

4 L. Prandtl and O.G. Tietjens, Fundamentals of Hydro- and Aerodynamics, Dover,

5 The Lagrangian Approach to continuum mechanics emphasizes the particulate (material) description of a motion The pathlines of car-particles flowing in a highway: Red paths belong to receding particles; white paths to approaching particles. In Fluid Mechanics these lines are called the pathlines. We notice however this snapshot corresponds to a photograph with long exposure. Thus long exposures yield the Lagrangian view, whereas short exposures the Eulerian view 5

6 The Eulerian Approach to continuum mechanics emphasizes the spatial description of a motion Ground-wind velocities meteorologic map of a given place at a given instant. In Fluid Mechanics the integral curves of this velocity field are called the streamlines v = v( x, y, t) 6

7 The Elementary Traffic-Flow Theory (eulerian) (M.J. Lighthill and G.B. Whitham 1955) 7

8 empirical constitutive law 8

9 the celerity 9

10 the traffic flow problem 10

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13 The traffic light problem 13

14 the expansion fan 14

15 the shock wave 15

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18 1 st Modification: Viscosity correction (the driver looks ahead; stabilizing) 2 nd Modification: Reaction time (destabilizing) 18

19 Viscosity correction If the flow ahead is getting denser or looser, the driver is able to adjust the speed of the vehicle accordingly: v = V( ρ ν ρ ) ρ x ν= v c 0 2 ρ 2ρ ρ ρ + vmax 1 =ν ( BURGER Eq. ) 2 t ρmax x x 19

20 Canonical form of the Burger equation 2ρ C= vmax 1 ρmax 2 C C C + C =ν t x x 2 20

21 Reaction time v = V ( ρ(x,t τ) ) V( ρ) τv ( ρ) ρ t ρ t + C ρ x x ( ρ) = τ ρv ( ρ) ρ t ρ = ρ * + ρ~ (x,t) ρ * ρ * * ρ c τρ V(ρ ) + = t x x t 2 21

22 Linear stability analysis ρ ~ = Re ( exp(ikx + st) ), i = 1 s = 2 * * * τk ρ cv '(ρ ) ikc ( * * ) 1+ τkρ V'(ρ ) 2 with V <0 instability is predicted at heavy traffic conditions (c<0) 22

23 St. Venant s Shallow-Water Theory (eulerian) De Barr Saint-Venant, A.J.C. (1850). Mémoire sur des formules nouvelles pour la solution des problèmes ralitfs aux eaux courantes. C.R. Acad. Sc., Paris, 31, 283. De Barr Saint-Venant, A.J.C. (1871). Théorie du movement non-permanent des eaux avec applications aux crues des rivieres et a l' introduction des marées dans leurs lit. C.R. Acad. Sc., Paris, 73,

24 The "shallow water theory" is traced originally to St. Venant in the mid 19 th century and its successful application to open channel flow and to river dynamics is extensively presented in standard textbooks. Within the shallow water theory the governing mass and momentum balance equations of continuum mechanics are averaged over space and time in an appropriate manner so that simplified and mathematically more tractable differential equations are derived. Space averaging is done over the height of the flowing mass, whereas time averaging is done primarily in order to account for the effect of fluctuations at the interface between the flowing mass and its stationary base. 24

25 Shallow-water limit 25

26 Storage Equation ( hv) = x h t 26

27 Momentum Balance h ρ gh = w x ρ w hv 27

28 the material time derivative φ = Dφ/ Dt 28

29 Kinematic water-waves h h v + v + h = t x x v v h + v + g = t x x 0 0 v = V( h) 29

30 Kinematic water-waves h t h + ch ( ) = 0 x c h = 3ceq heq 2 3 c eq = gh eq 30

31 Hyperbolic problems 31

32 The breaking-dam problem 32

33 Tsunamis (=harbor waves) 33

34 Long wave approaching a slopping beach (tsunami) 34

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39 the tsunami c= gh 0 39

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42 Question Given that for a traffic-flow problem the velocity-density relationship is linear determine the traffic flow density so that the car flow is optimal 42