Outline Notes: Nonlinear hyperbolic systems Shallow water - PDF document

Outline Notes: • Nonlinear hyperbolic systems • Shallow water equations • Shock waves and Hugoniot loci • Integral curves in phase plane • Compression and rarefaction R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 Shallow water equations Notes: h ( x, t ) = depth u ( x, t ) = velocity (depth averaged, varies only with x ) Conservation of mass and momentum hu gives system of two equations. mass flux = hu , momentum flux = ( hu ) u + p where p = hydrostatic pressure h t + ( hu ) x = 0 � hu 2 + 1 � 2 gh 2 ( hu ) t + = 0 x Jacobian matrix: � 0 1 � � f ′ ( q ) = , λ = u ± gh. gh − u 2 2 u R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.1] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.1] Shallow water equations Notes: h t + ( hu ) x = 0 = ⇒ h t + µ x = 0 � hu 2 + 1 � 2 gh 2 ( hu ) t + = 0 = ⇒ µ t + φ ( h, µ ) x = 0 x where µ = hu and φ = hu 2 + 1 2 gh 2 = µ 2 /h + 1 2 gh 2 . Jacobian matrix: � ∂µ/∂h ∂µ/∂µ � � 0 1 � f ′ ( q ) = = , ∂φ/∂h ∂φ/∂µ gh − u 2 2 u Eigenvalues: λ 1 = u − λ 2 = u + � � gh, gh. Eigenvectors: � � � � 1 1 r 1 = r 2 = u − √ gh u + √ gh , . R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.1] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.1]

Shallow water equations Notes: Hydrostatic pressure: Pressure at depth z > 0 below the surface is gz from weight of water above. Depth-averaged pressure is � h p = gz dz 0 h = 1 � 2 gz 2 � � � 0 = 1 2 gh 2 . R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.1] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.1] Compressible gas dynamics Notes: In one space dimension (e.g. in a pipe). ρ ( x, t ) = density, u ( x, t ) = velocity, p ( x, t ) = pressure, ρ ( x, t ) u ( x, t ) = momentum. Conservation of: mass: ρ flux: ρu momentum: ρu flux: ( ρu ) u + p (energy) Conservation laws: ρ t + ( ρu ) x = 0 ( ρu ) t + ( ρu 2 + p ) x = 0 Equation of state: p = P ( ρ ) . (Later: p may also depend on internal energy / temperature) R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Chap. 14] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Chap. 14] Two-shock Riemann solution for shallow water Notes: Initially h l = h r = 1 , u l = − u r = 0 . 5 > 0 Solution at later time: R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Fig. 13.7] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Fig. 13.7]

Two-shock Riemann solution for shallow water Notes: � Characteristic curves X ′ ( t ) = u ( X ( t ) , t ) ± gh ( X ( t ) , t ) Slope of characteristic is constant in regions where q is (Shown for g = 1 so √ gh = 1 everywhere initially.) constant. Note that 1-characteristics impinge on 1-shock, 2-characteristics impinge on 2-shock. R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Fig. 13.8] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Fig. 13.8] An isolated shock Notes: If an isolated shock with left and right states q l and q r is propagating at speed s then the Rankine-Hugoniot condition must be satisfied: f ( q r ) − f ( q l ) = s ( q r − q l ) For a system q ∈ lR m this can only hold for certain pairs q l , q r : For a linear system, f ( q r ) − f ( q l ) = Aq r − Aq l = A ( q r − q l ) . So q r − q l must be an eigenvector of f ′ ( q ) = A . A ∈ lR m × m = ⇒ there will be m rays through q l in state space in the eigen-directions, and q r must lie on one of these. For a nonlinear system, there will be m curves through q l called the Hugoniot loci. R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Fig. 13.7] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Fig. 13.7] Hugoniot loci for shallow water Notes: � h � � � hu q = , f ( q ) = . hu 2 + 1 2 gh 2 hu Fix q ∗ = ( h ∗ , u ∗ ) . What states q can be connected to q ∗ by an isolated shock? The Rankine-Hugoniot condition s ( q − q ∗ ) = f ( q ) − f ( q ∗ ) gives: s ( h ∗ − h ) = h ∗ u ∗ − hu, ∗ − hu 2 + 1 s ( h ∗ u ∗ − hu ) = h ∗ u 2 2 g ( h 2 ∗ − h 2 ) . Two equations with 3 unknowns ( h, u, s ) , so we expect 1-parameter families of solutions. R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7]

Hugoniot loci for shallow water Notes: Rankine-Hugoniot conditions: s ( h ∗ − h ) = h ∗ u ∗ − hu, ∗ − hu 2 + 1 s ( h ∗ u ∗ − hu ) = h ∗ u 2 2 g ( h 2 ∗ − h 2 ) . For any h > 0 we can solve for � g � h ∗ h − h � u ( h ) = u ∗ ± ( h ∗ − h ) 2 h ∗ s ( h ) = ( h ∗ u ∗ − hu ) / ( h ∗ − h ) . This gives 2 curves in h – hu space (one for + , one for − ). R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] Hugoniot loci for shallow water Notes: For any h > 0 we have a possible shock state. Set h = h ∗ + α, so that h = h ∗ at α = 0 , to obtain � � � gh ∗ + 1 hu = h ∗ u ∗ + α u ∗ ± 2 gα (3 + α/h ∗ ) . Hence we have � h � � � � 1 � h ∗ = + α as α → 0 . � hu h ∗ u ∗ u ∗ ± gh ∗ + O ( α ) Close to q ∗ the curves are tangent to eigenvectors of f ′ ( q ∗ ) Expected since f ( q ) − f ( q ∗ ) ≈ f ′ ( q ∗ )( q − q ∗ ) . R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] Hugoniot loci for one particular q ∗ Notes: States that can be connected to q ∗ by a “shock” Note: Might not satisfy entropy condition. R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7]

Hugoniot loci for two different states Notes: “All-shock” Riemann solution: From q l along 1-wave locus to q m , From q r along 2-wave locus to q m , R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] All-shock Riemann solution Notes: From q l along 1-wave locus to q m , From q r along 2-wave locus to q m , R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] All-shock Riemann solution Notes: From q l along 1-wave locus to q m , From q r along 2-wave locus to q m , R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7]

2-shock Riemann solution for shallow water Notes: Given arbitrary states q l and q r , we can solve the Riemann problem with two shocks. Choose q m so that q m is on the 1-Hugoniot locus of q l and also q m is on the 2-Hugoniot locus of q r . This requires � 1 � g + 1 � u m = u r +( h m − h r ) 2 h m h r and � 1 � � g + 1 u m = u l − ( h m − h l ) . 2 h m h l Equate and solve single nonlinear equation for h m . R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] Hugoniot loci for one particular q ∗ Notes: Green curves are contours of λ 1 Note: Increases in one direction only along blue curve. R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Sec. 13.7] Hugoniot locus for shallow water Notes: States that can be connected to the given state by a 1-wave or 2-wave satisfying the R-H conditions: Solid portion: states that can be connected by shock satisfying entropy condition. Dashed portion: states that can be connected with R-H condition satisfied but not the physically correct solution. R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Fig. 13.9] R.J. LeVeque, University of Washington IPDE 2011, July 6, 2011 [FVMHP Fig. 13.9]