Hydraulik II Skriptum - Department Wasser-Atmosphäre-Umwelt ...

HYDRODYNAMIK SEITE 103⎡⎢( − ψ)⇒⎢⎢⎣14 2n + 1 2( − Θ) gAj+1 2 1n n1 β Q jg Q j n + 1n +− +⎥Qn 1 2n 1 2( )j +h jΔtΔx+A2 + ⎥xj 2 C ARΔj ⎥1 44243 44444444 44444444⎦ 3 B2 jA2 j⎡n n n 1 2Q j 1g Q ⎤+j + 1Θ⋅gA⎢ ψ β +n+1 j+1 2+ +⎥Qn 1 2n 1 2( )j + 1 + h⎢ΔtΔx+A2 + ⎥xj+1 2 C ARΔ⎢⎣j+1 ⎥142431444444424444444⎦ 3 D2 j( − ψ)c2 jn+1 2⎤( 1 − Θ)( h − h )n+1j + 11 n Ψ gAn j+1 2n n n 1 2= Q j + Q j + 1 −j + 1 j + gA+j+1 2Is14Δ44444444444t ΔtΔx24444444444443E2 jj n j + 1j n j + 1j n j+ + 1 1 j n j+ +1 1jA2 Q + B2 h + C2 Q + D2 h = E2(4)Recurrence RelationA1 Q + 1+ B1 h + 1+ C1 Q + + + D1 h + + = E1(3)j n j j n j j n j 1 1 j n j 1 1jj n j + 1j n j + 1j n j+ + 1 1 j n j+ +1 1jn + 1j j n + 1= j + jA2 Q + B2 h + C2 Q + D2 h = E2(4)Substitution Q F h Geingesetzt in (3)( )( )j j n j + 1j j n j + 1j n j+ + 1 1 j n j+ +1 1jA1 Fh + G + B1 h + C1 Q + D1 h = E1j j j n +j 1j n j+ + 1 1 j n j+ +1 1j j jA1 F + B1 h + C1 Q + D1 h = E1 −A1Gn+1− C1j +−+−⇒ h =++++A F + B Q n 1D1jA F + B h n 1E1j A1jGjjj 1j 11j j1j1j j1jA1jFj + B1j1424314243142443H j I j J jn + 1jh = H Q + I h + J(5) und (6) in (4) eingesetztA2 Fh + 1 1+ G + B2 h + + C2 Q + + + D2 h + + = E2j n j+ + 1 1 j n j+ +1 1 j(6)j ( j n j j)j n j j n j 1 1 j n j 1 1j( A jFj B j)h n +2 + 2 j 1 + C2j Q n j+ + 1 1 + D2j h n j+ +1 1 = E2j−A2jGj( )( n + 1 n + 11) n + 1 n +A2j F j + B2 j H j Q j + 1 + I j h j + 1 + J j + C2 j Q j + 1 + D2 j h j + 1 = E2 j − A2 jGj( )+ + 1 1 + +1 1[ 2 + 2 + 2 ] + [( 2 + 2 ) + 2 ] = 2 − 2 − ( 2 + 2 )A F B H C Q A F B I D h E A G A F B Jj j j j j j n j j j j j j n j j j j j j j( )( )− − ( + )( A2jFj + B2j)Hj + C2j⇒ = − A2 F + B I − D++Q+++A F + B H + C h E A G A F B Jn j j2j j2j2n j2j j2j j211j jj 1j 12j j j j2j144442444431444442444443Fj+1nj+ + 1 1 j+ 1 n j+ + 1 1 j+1Q = F h + GG j+1+(5)Institut für Hydraulik und landeskulturelle Wasserwirtschaft Version 1.7