ABSTRACT
We consider the propagation of a high-Reynolds-number gravity current at the bottom of a horizontal channel along the horizontal coordinate x. The bottom and top of the channel are at https://www.w3.org/1998/Math/MathML"> z = 0 , H , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq529.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , and the cross-section is given by the general https://www.w3.org/1998/Math/MathML"> − f 1 ( z ) ≤ y ≤ f 2 ( z ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq530.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for 0 ≤ z ≤ H. We use a one-layer, Boussinesq, Shallow-Water (SW) formulation to solve the time-dependent motion produced by release from rest of a fixed volume of fluid from a lock. The dependent variables are the position of the horizontal interface, https://www.w3.org/1998/Math/MathML"> h ( x , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq531.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , and the speed (averaged over the area of the current), https://www.w3.org/1998/Math/MathML"> u ( x , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq532.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . The non-rectangular cross-section geometry enters the formulation via f(h) and integrals of f(z) and https://www.w3.org/1998/Math/MathML"> z f ( z ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq533.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where https://www.w3.org/1998/Math/MathML"> f ( z ) = f 1 ( z ) + f 2 ( z ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq534.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is the width of the channel. For a given geometry https://www.w3.org/1998/Math/MathML"> f ( z ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq535.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , the only input parameter in the lock-release problem is the height ratio https://www.w3.org/1998/Math/MathML"> H / h 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq536.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of ambient to lock. In general, the solution is obtained by a finite-difference numerical code. Analytical results are derived for the initial dam-break slumping motion, and for the long-time self-similar phase. The model is illustrated for various cross-section shapes: power-law ( https://www.w3.org/1998/Math/MathML"> f ( z ) = b z α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227202/7e950746-38f4-4035-8ec6-0dcc19f8564d/content/eq537.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where b, α are positive constants), trapezoidal, V-shaped valley, and circle-segment. When the cross-section of the channel expands upwards, the speed of propagation is larger, and the decay of speed after the slumping stage is weaker, than in the classical rectangular counterpart. The theoretical results are in good agreement with previously-published experimental data, but a sharp comparison is not feasible because the experiments were performed in full-depth-lock configurations where the one-layer model is not expected to be accurate.
