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General Modelling and Scaling Laws - IV - NTNU

TMR7 Experimental Methods in Marine Hydrodynamics lecture in week 34. General Modelling and Scaling laws Dimensionless numbers Similarity requirements Derivation of dimensionless numbers used in model testing Froude Scaling Hydroelasticity Cavitation number 1 Chapter 2 in the lecture notes Dimensionless numbers Without dimensionless numbers, experimental progress in fluid mechanics would have been almost nil;. It would have been swamped by masses of accumulated data (R. Olson). Example: Due to the beauty of dimensionless numbers, Cf of a flat, smooth plate is a function of Re only (not function of temperature, pressure or type of fluid1). 1As long as the fluid is Newtonian, which means that it has a linear stress/strain rate, with zero stress for zero strain 2. 3. Types of similarity Geometrical similarity Kinematic similarity Dynamic similarity What are the similarity requirements for a model test?

1 General Modelling and Scaling Laws • Dimensionless numbers • Similarity requirements • Derivation of dimensionless numbers used in model testing

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Transcription of General Modelling and Scaling Laws - IV - NTNU

1 TMR7 Experimental Methods in Marine Hydrodynamics lecture in week 34. General Modelling and Scaling laws Dimensionless numbers Similarity requirements Derivation of dimensionless numbers used in model testing Froude Scaling Hydroelasticity Cavitation number 1 Chapter 2 in the lecture notes Dimensionless numbers Without dimensionless numbers, experimental progress in fluid mechanics would have been almost nil;. It would have been swamped by masses of accumulated data (R. Olson). Example: Due to the beauty of dimensionless numbers, Cf of a flat, smooth plate is a function of Re only (not function of temperature, pressure or type of fluid1). 1As long as the fluid is Newtonian, which means that it has a linear stress/strain rate, with zero stress for zero strain 2. 3. Types of similarity Geometrical similarity Kinematic similarity Dynamic similarity What are the similarity requirements for a model test?

2 4. Geometrical Similarity The model and full scale structures must have the same shape All linear dimensions must = LF LM. have the same scale ratio: This applies also to: The environment surrounding the model and ship Elastic deformations of the model and ship 5. Kinematic Similarity Similarity of velocities: The flow and model(s) will have geometrically similar motions in model and full scale Examples: - Velocities in x and y direction must have the same ratio, so that a circular motion in full scale must be a circular motion also in model scale - The ratio between propeller tip speed and advance speed must be the same in model and full scale: VF VM VF VM. = or = J= JM. nF (2 RF ) nM (2 RM ). F. nF DF nM DM. 6. Dynamic Similarity Geometric similarity and Similarity of forces Ratios between different forces in full scale must be the same in model scale If you have geometric and dynamic similarity, you'll also have kinematic similarity The following force contributions are of importance: Inertia Forces, Fi Viscous forces, Fv Gravitational forces, Fg Pressure forces, Fp Elastic forces in the fluid (compressibility), Fe.

3 Surface forces, Fs. 7. Inertia Forces (mass forces). dU 3 dU dx 3. Fi L = L U 2 L2. dt dx dt is fluid density U is a characteristic velocity t is time L is a characteristic length (linear dimension). 8. Gravitational Forces Fg gL 3. Just mass times acceleration g is acceleration of gravity 9. Viscous Forces dU 2. Fv L UL. dx is dynamic viscosity [kg/m s]. - a function of temperature and type of fluid 10. Pressure Forces F p pL 2. Force equals pressure times area p is pressure 11. Elastic Fluid Forces Fe v E v L 2. v is compression ratio Ev is the volume elasticity (or compressibility). v Ev=elasticity modulus K [kg/m s2]. 12. Surface Forces Fs L. is the surface tension [kg/s2]. 13. Froude number Fn The ratio between inertia and gravity: Inertia force Fi U 2 L2 U 2. = =. Gravity force Fg gL 3. gL. Dynamic similarity requirement between model and full scale: U 2 U 2.

4 M. = F. gLM gLF. UM UF. = = Fn gLM gLF. Equality in Fn in model and full scale will ensure that gravity forces are correctly scaled Surface waves are gravity-driven equality in Fn will ensure that wave resistance and other wave forces are 14 correctly scaled Reynolds number Re Equal ratio between inertia and viscous forces: Inertia forces Fi U 2 L2 UL UL. = = == Re Viscous forces Fv UL .. is the kinematic viscosity, = [m2/s].. Equality in Re will ensure that viscous forces are correctly scaled 15. Kinematic viscosity of fluids (from White: Fluid Mechanics). To obtain equality of both Fn and Rn for a ship model in scale 1:10: m= 16. Mach number Mn Equal ratio between inertia and elastic fluid forces: Fi U 2 L2.. Fe v E v L2. By requiring v to be equal in model and full scale: U 2 L2 U 2 L2 . 2 . = 2 . v E v L M v Ev L F.

5 UM UF. = = Mn Ev , M Ev , F.. Ev is the speed of sound Fluid elasticity is very small in water, so usually Mach number similarity is not required It is only when Mach numbers get close to 1 that it is important to 17. consider compressibility effects. When Mach< , incompressible flow is assumed Weber number Wn The ratio between inertia and surface tension forces: Inertia forces Fi U 2 L2 U 2 L. = =. Surface tension forces Fs L . Similarity requirement for model and full scale forces: U 2 L U 2 L . = = at 20 C. M F. UM UF. = = Wn M F. ( L) M ( L) F. When Wn>180, we assume that a further increase in Wn doesn't influence the fluid forces 18. Scaling ratios used in testing of ships and offshore structures Symbol Dimensionless Number Force Ratio Definition Re Reynolds Number Inertia/Viscous UL.. Fn Froude Number Inertia/Gravity U.

6 GL. Mn Mach's Number Inertia/Elasticity U. EV . Wn Weber's Number Inertia/Surface tension U. L. St Strouhall number - fv D. U. KC Keulegan-Carpenter Number Drag/Inertia U AT. D. 19. Froude Scaling UM UF LF. = UF = UM = UM . gLM gLF LM. Using the geometrical similarity requirement: = LF LM. If you remember this, most of the other Scaling relations can be easily derived just from the physical units 20. Froude Scaling table Physical Parameter Unit Multiplication factor Length [m] . Structural mass: [kg] 3 F M. Force: [N] 3 F M. Moment: [Nm] 4 F M. Acceleration: [m/s2] a F = aM. Time: [s] . Pressure: [Pa=N/m2] F M. 21. Hydroelasticity Additional requirements to the elastic model Correctly scaled global stiffness Structural damping must be similar to full scale The mass distribution must be similar Typical applications: Springing and whipping of ships Dynamic behaviour of marine risers and mooring lines 22.

7 Scaling of elasticity F. FL3.. EI. Hydrodynamic force: F C U 2 L2. F M. Geometric similarity requirement: = F = M. LF LM. Requirement to structural rigidity: U 2 L4 U 2 L4 . = ( EI )=. F. ( EI )M 5. EI F EI M. 23. Scaling of elasticity geometrically similar models Geometrically similar model implies: IF = IM 4. Must change the elasticity of material: E F = EM . Elastic propellers must be made geometrically similar, using a very soft material: EM = EF . Elastic hull models are made geometrically similar only on the outside. Thus, E is not scaled and I M = I F 5. 24. Cavitation Dynamic similarity requires that cavitation is modelled Cavitation is correctly modelled by equality in cavitation number: ( gh + p0 ) pv =. 1/ 2 U 2. To obtain equality in cavitation number, atmospheric pressure p0 might be scaled pv is vapour pressure and gh is hydrostatic pressure Different definitions of the velocity U is used 25.

8 General Modelling and Scaling laws Dimensionless numbers Similarity requirements Derivation of dimensionless numbers used in model testing Froude Scaling Hydroelasticity Cavitation number 26.


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