lecture10 - Lecture 10- Marine Hydrodynamics Lecture 10 3.7...

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Unformatted text preview: Lecture 10- Marine Hydrodynamics Lecture 10 3.7 Governing Equations and Boundary Conditions for P-Flow 3.7.1 Governing Equations for P-Flow (a) Continuity 2 = 0 1 (b) Bernoulli for P-Flow (steady or unsteady) p = t + 2 + gy + C ( t ) 2 | | 3.7.2 Boundary Conditions for P-Flow Types of Boundary Conditions: (c) Kinematic Boundary Conditions- specify the ow velocity v at boundaries. = U n n (d) Dynamic Boundary Conditions- specify force F or pressure p at ow boundary. 1 2 p = t + ( ) + gy + C ( t ) (prescribed) 2 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20 The boundary conditions in more detail: Kinematic Boundary Condition on an impermeable boundary (no ux condition) v n = U n = U n = Given uid velocity boundary velocity nornal boundary velocity v = n = U n ( n 1 x 1 + n 2 x 2 + n 3 x 3 ) = U n n = U n ( 29 3 2 1 n , n , n n = v U v Dynamic Boundary Condition: In general, pressure is prescribed 1 2 p = t + ( ) + gy + C ( t ) = Given 2 2 ( 29 ) ( ) 2 1 ( 2 2 t C gy p t + + +- = = = + + +-- GIVEN ) ( ) ) ( 2 1 ( : DBC 19) (Lecture : KBC surface Free linear non 2 t C gy t 3 2 1 GIVEN U n n = = : KBC boundary Solid 3.7.3 Summary: Boundary Value Problem for P-Flow The aforementioned governing equations with the boundary conditions formulate the Boundary Value Problem (BVP) for P-Flow. The general BVP for P-Flow is sketched in the following figure. It must be pointed out that this BVP is satisfied instantaneously . 3 3.8 Linear Superposition for Potential Flow In the absence of dynamic boundary conditions , the potential ow boundary value problem is linear . Potential function . B on f U n n = = V in 2 = Stream function . V in 2 = =g on B Linear Superposition: if 1 , 2 ,... are harmonic functions, i.e., 2 i = 0, then = i i , where i are constants, are also harmonic, and is the solution for the boundary value problem provided the kinematic boundary conditions are satisfied, i.e., = ( 1 1 + 2 2 + ... ) = U n on B . n n The key is to combine known solution of the Laplace equation in such a way as to satisfy the kinematic boundary conditions (KBC). The same is true for the stream function . The K.B.C specify the value of on the boundaries. 4 = 3.8.1 Example x denote a unit-source ow with source at x i , i.e., 1 ln Let i i x source x, x i (in 2D) x x i 2 1 (in 3D), = 4 x x i then find m i such that = i m i i ( x ) satisfies KBC on B Caution: must be regular for x V , so it is required that x / V ....
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lecture10 - Lecture 10- Marine Hydrodynamics Lecture 10 3.7...

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