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 φ = 0 1 (b) Bernoulli for P-Flow (steady or unsteady) p = − ρ φ t + 2 + gy + C ( t ) 2 | φ | 3.7.2 Boundary Conditions for P-Flow Types of Boundary Conditions: ∂φ (c) Kinematic Boundary Conditions- specify the flow velocity v at boundaries. = U n ∂n (d) Dynamic Boundary Conditions- specify force F or pressure p at flow boundary. 1 2 p = − ρ φ t + ( φ ) + 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 flux condition) • v n ˆ = U n ˆ = U n = Given · · fluid velocity boundary velocity nornal boundary velocity v = φ φ n ˆ = U n · ⇒ ∂ ∂ ∂ ( n 1 ∂x 1 + n 2 ∂x 2 + n 3 ∂x 3 ) φ = U n ⇒ ∂φ ∂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 = − ρ φ t + ( φ ) + gy + C ( t ) = Given 2 2 ( 29 ) ( ) 2 1 ( 2 2 t C gy p t + + ∇ +- = = ∇ φ φ ρ φ = + + ∇ +-- GIVEN ) ( ) ) ( 2 1 ( : DBC 19) (Lecture : KBC surface Free linear non 2 t C gy t 3 2 1 φ φ ρ GIVEN U n n = = ∂ ∂ φ : 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 flow boundary value problem is linear . Potential function φ . • B on f U n n = = ∂ φ ∂ V in 2 = φ ∇ Stream function ψ . • V in 2 = ψ ∇ ψ =g on B Linear Superposition: if φ 1 ,φ 2 ,... are harmonic functions, i.e., 2 φ i = 0, then φ = α i φ i , where α i are constants, are also harmonic, and is the solution for the boundary value problem provided the kinematic boundary conditions are satisfied, i.e., ∂φ ∂ = ( α 1 φ 1 + α 2 φ 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 ψ . The K.B.C specify the value of ψ on the boundaries. 4 = 3.8.1 Example x denote a unit-source flow with source at x i , i.e., 1 ln Let φ i φ i x ≡ φ source x, x i (in 2D) x − x i 2 π − 1 (in 3D), = − 4 π x − x i then find m i such that φ = i m i φ i ( x ) satisfies KBC on B Caution: φ must be regular for x ∈ V , so it is required that x / ∈ V ....
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.20 taught by Professor Dickk.p.yue during the Spring '05 term at MIT.

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

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