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Unformatted text preview: Lecture 10 Marine Hydrodynamics Lecture 10 3.7 Governing Equations and Boundary Conditions for PFlow 3.7.1 Governing Equations for PFlow (a) Continuity 2 = 0 1 (b) Bernoulli for PFlow (steady or unsteady) p = t + 2 + gy + C ( t ) 2   3.7.2 Boundary Conditions for PFlow 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 PFlow The aforementioned governing equations with the boundary conditions formulate the Boundary Value Problem (BVP) for PFlow. The general BVP for PFlow 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 unitsource 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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 Spring '05
 DickK.P.Yue

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