Offshore Hydromechanics

Transcription

1 Offshore Hydromechanics Module 1 : Hydrostatics Constant Flows Surface Waves OE4620 Offshore Hydromechanics Ir. W.E. de Vries Offshore Engineering

2 Today First hour: Schedule for remainder of hydromechanics module 1 Assignment stability Basic flow properties Potential flow concepts Second hour: Potential flow elements Superposition of flow elements Cylinder in uniform flow

3 Schedule OE4620 Module 1 13 November Constant potential flow phenomena 20 November Constant real flow phenomena 1 27 November Constant real flow phenomena 2 4 December Ocean surface waves 1 (!) 11 December Ocean surface waves 2 18 December Extra/Examples

4 Assignment: Stability Make the assignment for next lecture Next lecture the assignment will be worked out briefly Questions regarding the assignment or the subjects of module 1? Contact: Room 2.78 Tel

5 CONSTANT POTENTIAL FLOW PHENOMENA Flows in this chapter: Obey simplified laws of fluid mechanics Are subject to limitations Give qualitative impressions of flow phenomena

6 Basic Flow Properties Ideal fluid : Non viscous Incompressible Continuous Homogeneous

7 Continuity Condition Increase of mass per unit time: m t = t ( ρ dxdydz) min = ρ u dydzdt

8 Continuity Condition Mass through a plane dxdz during a unit of time: min = ρ v dxdzdt min = ρ u dydzdt

9 Continuity Condition Mass through a plane dxdz during a unit of time: min = ρ v dxdzdt min = ρ u dydzdt ( ρv) mout = ρv + dy dxdzdt y

10 Continuity Condition Mass through a plane dxdz during a unit of time: min = ρ v dxdzdt min = ρ u dydzdt ( ρv) mout = ρv + dy dxdzdt y ( ρv) m = ρv + dy dxdz ρudxdz t y

11 Continuity Condition Mass through a plane dxdz during a unit of time: min = ρ v dxdzdt min = ρ u dydzdt ( ρv) mout = ρv + dy dxdzdt y ( ρv) ( ρu) m = ρv + dy dxdz ρudxdz = dxdydz t y y

12 Continuity Condition Combining for x, y and z direction yields the Continuity Equation ( ρu) ( ρv) ( ρw) m ρ = dxdydz = + + t t x y z dxdydz

13 Continuity Condition Combining for x, y and z direction yields the Continuity Equation ( ρu) ( ρv) ( ρw) m ρ = dxdydz = + + t t x y z dxdydz

14 Continuity Condition Combining for x, y and z direction yields the Continuity Equation ( ρu) ( ρv) ( ρw) m ρ = dxdydz = + + t t x y z dxdydz

15 Continuity Condition Combining for x, y and z direction yields the Continuity Equation ( ρu) ( ρv) ( ρw) ρ = 0 t x y z

16 Continuity Condition For an ideal, incompressible fluid: u v w + + = x y z 0

17 Continuity Condition For an ideal, incompressible fluid: u v w + + = x y z 0 Or, with = + + : x y z uv V = 0

18 Deformation of flow particles Deformation v = tan & α & α x u = tan & β & β y

19 Deformation of flow particles Deformation v = tan & α & α x u = tan & β & β y Deformation velocity (dilatation): & α + & β 1 v u = x y

20 Deformation of flow particles Deformation v = tan & α & α x u = tan & β & β y Deformation velocity (dilatation): & α + & β 1 v u = x y Rotation (in 2 D) : 1 v & ϕ = 2 x u y

21 Velocity Potential A scalar function, Φ The velocity in any point of the fluid and in any direction is the derivative to that direction of the potential u Φ = v x Φ = w y Φ = z

22 Properties of Velocity Potentials Combining the continuity condition with the velocity potentials results in the Laplace equation : Φ Φ Φ x y z = or 2 Φ = 0 Potential flow is rotation-free by definition : v x u = y 0 w y v = z 0 u z w = x 0

23 Properties of Velocity Potentials Combining the continuity condition with the velocity potentials results in the Laplace equation : Φ Φ Φ x y z = or 2 Φ = 0

24 Euler and Bernoulli Euler: Apply Newton s 2 nd law to non viscous and incompressible fluid Du Du p dm = ρ dxdydz = dx dydz Dt Dt x

25 Euler and Bernoulli Euler: Apply Newton s 2 nd law to non viscous and incompressible fluid Du Du p dm = ρ dxdydz = dx dydz Dt Dt x

26 Euler and Bernoulli Euler: Apply Newton s 2 nd law to non viscous and incompressible fluid Du Du p dm = ρ dxdydz = dx dydz Dt Dt x Mass* acceleration = pressure*area

27 Euler and Bernoulli Euler Equations: u u u u 1 p = u + v + w = t x y z ρ x With 2 u Φ Φ 1 Φ 2 2 u = = x x x x x 2 u Φ Φ 1 v Φ = = y y x y 2 x y 2 u Φ Φ 1 Φ 2 w = = z z x z x z 2 2 2

28 Euler and Bernoulli Differentiate w.r.t. x, y or z gives 0: Φ 1 Φ Φ Φ p = 0 x t 2 x y z ρ (similar for y, z)

29 Euler and Bernoulli Differentiate w.r.t. x, y or z gives 0: Φ 1 Φ Φ Φ p = 0 x t 2 x y z ρ (similar for y, z) This expression is a function of time only, providing the Bernoulli Equation: Φ 1 p = t 2 ρ ( ) 2 V gz C t

30 Euler and Bernoulli For a stationary flow along a streamline : 1 2 p ρ 2 V + + gz = C This can be written in two alternative ways : 2 V 2g p + + z = ρg C ρ ρ V + p + gz = C

31 Stream Function Ψ Ψ= constant along a streamline Ψ Definition : u = and v y Ψ = x Stream lines and equipotential lines are orthogonal No cross flow, impervious boundary of stream tube

32 Potential Flow Elements Uniform flow Source Sinks Circulation Superposition is allowed

33 Uniform Flow Φ = + U x Φ = U x Ψ = + U y Ψ = U y

34 Source & Sink Q Q Φ = ln r Φ = + ln r 2π 2π Ψ = + Q θ 2π Ψ = Q θ 2π

35 Circulation (Vortex) Counter clockwise flow: Φ = + Γ θ 2π Γ Ψ = ln r 2π Clockwise flow Φ = Γ θ 2π Ψ = + Γ ln r 2π

36 Superposition - principle

37 Superposition - principle

38 Superposition - principle

39 Superposition - principle

40 Uniform flow + sink Q y Ψ = arctan U 2π x y

41 Separated Source + sink s Q 2π 2ys x + y + s Ψ = arctan 2 2 2

42 Uniform flow + source + sink No flow through boundary (= streamline) Rankine ship forms Q 2ys Ψ = arctan U π x + y s y

43 Doublet or Dipole A source and sink pair placed very close together (s -> 0) Ψ = µ x y + y 2 2 Φ = µ x x + y 2 2

44 Dipole + uniform flow Model for cylinder X-axis is a streamline Boundary at R = µ U is also a streamline

45 Uniform flow around a cylinder v θ Ψ µ sinθ = = U r sinθ r r r r= R r= R v θ = 2U sinθ V θ = 0 at stagnation points V θ = 2*U at sides of cylinder

46 Uniform flow + circulation Pressure asymmetry creates a lift force

47 Continuity Condition Continuity Equation, general case Ideal fluid, incompressible ( ρu) ( ρv) ( ρw) ρ = 0 t x y z u v w + + = x y z 0

48 Deformation of flow particles Deformation velocity or dilatation: & α + & β 1 v u = x y Rotation (in 2 D) : 1 v & ϕ = 2 x u y

49 Velocity Potential A scalar function, Φ The velocity in any point of the fluid and in any direction is the derivative to that direction of the potential u = Φ/ x etc

50 Properties of Velocity Potentials The continuity condition results in the Laplace equation : Potential flow is rotation-free by definition :

51 Euler and Bernoulli Force = mass * acceleration : Euler non viscous and incompressible fluid Energy conservation along a flowline : Bernoulli non viscous and incompressible fluid stationary flow :

52 Stream Function Ψ Companion of the Velocity Potential Definition : Stream lines and equipotential lines are orthogonal No cross flow, impervious boundary of stream tube

53 Potential Flow Elements Uniform flow Sources and sinks Circulating flows Superposition is allowed

54 Polar Coordinate Description Basic definitions : Source : (Sink = - source) Vortex :

55 Superposition - principle

56 Uniform flow + sink

57 Source + sink

58 Uniform flow + source + sink

59 Dipole Strength µ = Qs/π

60 Dipole + uniform flow

61 Uniform flow around a cylinder Stagnation points on the X axis V max = 2 U at Y axis

62 Uniform flow + circulation Pressure asymmetry creates a lift force