Consistent section-averaged shallow water equations with bottom friction Victor Michel-Dansac † , Pascal Noble † , Jean-Paul Vila † Tuesday, February 4th, 2020 † Institut de Mathématiques de Toulouse et INSA Toulouse 32 ème Séminaire CEA/GAMNI, Paris
Motivation: 2D/1D coupling for estuary simulation Gironde estuary: satellite picture Gironde estuary: 2D mesh 1/31
Existing approaches Regarding the shape of the river bed, as of now, lation; • fully 2D models are used but they are computationally costly. 1 see Bresch and Noble, 2007, in the context of laminar fmows 2 see Richard, Rambaud and Vila, 2017, in the context of turbulent fmows 3 see Decoene, Bonaventura, Miglio and Saleri, 2009 4 see Marin and Monnier, 2009 2/31 • the derivation of 1D models is well-understood 1,2 in the ideal case of a � -shaped channel; • for more complex shapes, the water surface of uniform station- ary fmows is recovered 3,4 using a empiric terms or data assimi-
Specifjcations of the 1D model The goal of this work is to develop a new model, based on the shallow water equations, that is: simulations, ocean model forcing, …); 3/31 • generic enough to not require empiric friction coeffjcients; • consistent with the 2D shallow water equations in the asymp- totic regime corresponding to an estuary or a river; • hyperbolic; • easily implementable (collaboration with the SHOM for fmood • able to handle the meanders of the river.
1. Governing equations 2. Asymptotic expansions 3. Transverse averaging 4. A zeroth-order model 5. Numerical treatment of real data 6. Numerical validation of the model on an academic test case 7. Conclusion and perspectives
The non-conservative 2D shallow water system x exponent Chézy friction coeffjcient • g is the gravity constant velocity C 2 4/31 y z water height: h ( x , y , t ) Z ( x , y ) : known river shape • u = ( u , v ) is the water h t + ∇ · ( h u ) = 0 • C h ( x , y ) is the (known) � � − ∇ Z − u � u � u t + u · ∇ u + g ∇ h = g h h p • p = 4 / 3 is the friction law
Introduction of reference scales: the topography y side view of the river y z x front view of the river z 0 x fmow from upstream to downstream; 5/31 Regarding the geometry, we assume that Z ( x , y ) = b ( x ) + φ ( x , y ) , where: • b ( x ) represents the main longitudinal topography, driving the • φ ( x , y ) represents small longitudinal and transverse variations. Thus, h + φ represents the altitude of the water surface. h ( x , y ) φ ( x , y ) b ( x ) ⊙ ⊙
Introduction of reference scales: the coordinates quantity coordinates transverse coordinates longitudinal 6/31 non-dimensional scale reference quantity dimensional Y X x ∈ ( 0 m, 60000 m ) X ∈ ( 0 , 30 ) X = 2000 m x = x y ∈ (− 100 m, 100 m ) Y = 100 m y = y Y ∈ (− 1 , 1 )
Non-dimensional form of the 2D shallow water system F 2 C 2 h p u v 2 v u F 2 R 2 1 C 2 h p u v 2 u 1 We introduce the following non-dimensional numbers to emphasize 7/31 the difgerent scales of the fmow: • F 2 , the reference Froude number (ratio material/acoustic velocity), • R u , the quasi-1D parameter (ratio transverse/longitudinal velocity), Finally, the non-dimensional form of the 2D shallow water system is: • δ , the shallow water parameter (ratio height/reference length), • I 0 and J 0 , the reference topography and friction slopes. h t + ( hu ) x + ( hv ) y = 0 , � � u 2 + R 2 � � � u t + uu x + vu y + 1 h + φ x = − J 0 − I 0 b x , δ F 2 u 2 + R 2 � � � v t + uv x + vv y + h + φ y = − J 0 . δ F 2
1. Governing equations 2. Asymptotic expansions 3. Transverse averaging 4. A zeroth-order model 5. Numerical treatment of real data 6. Numerical validation of the model on an academic test case 7. Conclusion and perspectives
Asymptotic expansions setup In the regime under consideration, we have understand the weak dependency of the solution in y . Goal : Perform asymptotic expansions in this regime, to better C 2 h p v J 0 b x J 0 C 2 h p J 0 8/31 J 0 • ε := δ F 2 ≪ 1 (in practice, F 2 ≪ 1, δ ≪ 1, J 0 ≪ 1 and J 0 ∼ δ ), • R u ≪ 1 (quasi-unidimensional setting), and R u = O ( ε ) . Highlighting the dominant terms in the system, we get: h t + ( hu ) x + ( hv ) y = 0 , √ � � u 2 + ε 2 v 2 δ u t + uu x + vu y + 1 ( h + φ ) x = 1 − u − I 0 , ε ε √ u 2 + ε 2 v 2 δ v t + uv x + vv y + 1 ( h + φ ) y = − 1 . ε 3 ε
Free surface expansion C 2 h p J 0 terms, we get We consider the third equation: C 2 h p J 0 which we rewrite as follows to highlight the dominant term: 9/31 J 0 v √ u 2 + ε 2 v 2 δ v t + uv x + vv y + 1 ( h + φ ) y = − 1 , ε 3 ε √ u 2 + ε 2 v 2 δ ( h + φ ) y = ε 2 v + ε 3 ( v t + uv x + vv y ) . � ε 2 � Neglecting the O δ � ε 2 � ( h + φ ) y = O , � ε 2 � O and there exists H = H ( x , t ) such that H ( x ) h ( x , y ) φ ( x , y ) � ε 2 � H ( x , t ) = h ( x , y , t ) + φ ( x , y ) + O . � ε 2 � � the free surface h + φ is almost fmat in the y -direction, up to O
Longitudinal velocity expansion C 2 h p Next step : Build a 1D model consistent with these expansions. H x . J 0 J 0 , straightforward computations yield: Highlighting the dominant terms, the second equation reads: b x J 0 10/31 J 0 √ � � u 2 + ε 2 v 2 δ u t + uu x + vu y + 1 ( h + φ ) x = 1 − u − I 0 . ε ε To perform the asymptotic expansion of u with respect to ε , we write u ( x , y , t ) = u ( 0 ) 2 D ( x , y , t ) + O ( ε ) . � ε 2 � Since h + φ = H + O Λ � p / 2 , u ( 0 ) � 2 D = C H − φ � | Λ | b x − δ where we have defjned the corrected slope Λ ( x , t ) = − I 0
1. Governing equations 2. Asymptotic expansions 3. Transverse averaging 4. A zeroth-order model 5. Numerical treatment of real data 6. Numerical validation of the model on an academic test case 7. Conclusion and perspectives
The river cross-section x 0 To obtain a 1D model, we start by averaging the 2D equations: y z 11/31 z below, we display the cross-section of the river, with respect to x . z y � ε 2 � O H ( x ) � y + S ( x ) = h ( x , y ) dy y − h ( x , y ) � H ( x ) � ε 2 � = L ( x , z ) dz + O L ( x , z ) φ ( x , y ) z = 0 ⊙ y − y +
Averaging the 2D system over the river width hu dy . u dy J 0 J 0 h 1. The original mass conservation equation reads: 2. Arguing the mass conservation and integrating the second equation x 12/31 h t + ( hu ) x + ( hv ) y = 0 . Therefore, since v ( y − ) = v ( y + ) = 0, we get: � y + � y + h t dy + ( hu ) x dy = 0 ⇒ S t + Q x = 0 , = y − y − � y + where the averaged discharge Q is given by Q = y − (times h ) between y − and y + yields: � � y + � � � � y + b x − δ Q t + = 1 − I 0 ( h + φ ) x hu 2 dy ε y − y − √ u 2 + ε 2 v 2 � y + − 1 dy . C 2 h p − 1 ε y −
Averaging the 2D system x Next step : From the averaged system, build a truly 1D model that is 13/31 : � ε 2 � Finally, the averaged system reads as follows, up to O S t + Q x = 0 , � � y + � � � � y + u | u | Q t + = 1 Λ S − + O ( ε ) . hu 2 dy ε C 2 h p − 1 dy y − y − zeroth-order accurate (up to O ( ε ) ). That is to say, the new model needs to ensure Q = Q ( 0 ) 2 D + O ( ε ) , where � y + Q ( 0 ) hu ( 0 ) 2 D = 2 D dy y − � y + � C ( H − φ ) 1 + p / 2 dy . = | Λ | sgn ( Λ ) y −
1. Governing equations 2. Asymptotic expansions 3. Transverse averaging 4. A zeroth-order model 5. Numerical treatment of real data 6. Numerical validation of the model on an academic test case 7. Conclusion and perspectives
Setting up the model x friction term. the unknown u , which depends on y . We cannot directly use this equation in a 1D model, since it contains equation is neglected, and we get: The integrated discharge equation, highlighting the dominant terms 14/31 and multiplying by ε , is u | u | � � � y + � � � y + � ε 2 � Λ S − C 2 h p − 1 dy = ε Q t + + O . hu 2 dy y − y − At the zeroth order, i.e. up to O ( ε ) , the right-hand side of this � y + u | u | Λ S − C 2 h p − 1 dy = O ( ε ) . y − Instead, we approximate the integral, up to O ( ε ) , with a new 1D
The friction model C 2 5 The coeffjcient C 2 S 2 2 D 2 D C 2 of the friction coeffjcient: First, we choose this 1D friction term as a usual hydraulic C 2 15/31 C 2 engineering model. Thus, we impose the following formula: Q | Q | � y + u | u | 1 D S = C 2 h p − 1 dy + O ( ε ) . y − It contains a 1D friction coeffjcient 5 C 1 D , to be determined. According to the discharge equation, we get, up to O ( ε ) : Q | Q | 1 D = Q | Q | 1 D S = Λ S + O ( ε ) ⇒ Λ S 2 + O ( ε ) . = Second, we impose Q = Q ( 0 ) 2 D + O ( ε ) , to get the following expression 1 D = Q ( 0 ) � Q ( 0 ) � � � � y + � 2 � C ( H − φ ) 1 + p / 2 dy = 1 . Λ S 2 y − 1 D usually contains the hydraulic radius, the Chézy coeffjcient, …
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