Hydrostatics Tutorial

This tutorial covers hydrostatic calculations for ship hulls, including displacement, buoyancy, and metacentric heights.

Introduction

Hydrostatics is the study of fluids at rest. For ships, hydrostatic calculations determine the static equilibrium conditions and initial stability characteristics.

Key hydrostatic properties include:

  • Displacement: The weight of water displaced by the hull

  • Center of Buoyancy (B): The centroid of the displaced water volume

  • Waterplane Area: The area of the hull at the waterline

  • Center of Flotation (F): The centroid of the waterplane

  • Metacentric Height (GM): A measure of initial stability

Loading a Hull

First, load your hull geometry from an STL or VTK file:

from pynavaltoolbox import Hull

# Load the hull
hull = Hull("path/to/hull.stl")

# View the hull dimensions
bounds = hull.get_bounds()
length = bounds[1] - bounds[0]
beam = bounds[3] - bounds[2]
depth = bounds[5] - bounds[4]

print(f"Length: {length:.2f} m")
print(f"Beam: {beam:.2f} m")
print(f"Depth: {depth:.2f} m")

Calculating Hydrostatics at a Draft

The HydrostaticsCalculator computes hydrostatic properties at a given draft, trim, and heel:

from pynavaltoolbox.hydrostatics import HydrostaticsCalculator

calc = HydrostaticsCalculator(hull, water_density=1025.0)

# Calculate at 5m draft, level trim, no heel
state = calc.calculate_at_draft(draft=5.0, trim=0.0, heel=0.0)

# Display results
print("=== Hydrostatic Properties ===")
print(f"Draft: {state.draft:.2f} m")
print(f"Displacement: {state.displacement_mass:.0f} kg")
print(f"Displacement Volume: {state.displacement_volume:.2f} m³")
print()
print("=== Centers ===")
print(f"LCB (from AP): {state.lcb:.2f} m")
print(f"TCB: {state.tcb:.2f} m")
print(f"KB (VCB): {state.vcb:.2f} m")
print(f"LCF (from AP): {state.lcf:.2f} m")
print()
print("=== Metacentric Heights ===")
print(f"KMt (transverse): {state.kmt:.2f} m")
print(f"KMl (longitudinal): {state.kml:.2f} m")
print(f"BMt: {state.bmt:.2f} m")
print(f"BMl: {state.bml:.2f} m")
print()
print("=== Waterplane Properties ===")
print(f"Waterplane Area: {state.waterplane_area:.2f} m²")
print(f"LWL: {state.lwl:.2f} m")
print(f"BWL: {state.bwl:.2f} m")

Understanding the Results

The HydrostaticState contains many properties:

Hydrostatic Properties

Property

Unit

Description

displacement_volume

m³

Volume of water displaced by the submerged hull

displacement_mass

kg

Mass of water displaced (= volume × density)

lcb

m

Longitudinal Center of Buoyancy from AP

tcb

m

Transverse Center of Buoyancy (positive = starboard)

vcb (KB)

m

Vertical Center of Buoyancy from keel

lcf

m

Longitudinal Center of Flotation from AP

kmt

m

Height of transverse metacenter above keel

kml

m

Height of longitudinal metacenter above keel

bmt

m

Transverse metacentric radius (BM)

bml

m

Longitudinal metacentric radius (BM)

waterplane_area

m²

Area of the waterplane

wetted_surface_area

m²

Wetted surface area of the hull

Finding Equilibrium

To find the equilibrium draft and trim for a given loading condition:

# Define loading condition
displacement = 5000000.0  # kg
cog = (70.0, 0.0, 10.0)  # LCG, TCG, VCG in meters

# Find equilibrium
equilibrium = calc.find_equilibrium(
    displacement_mass=displacement,
    center_of_gravity=cog,
    initial_draft=5.0  # Optional starting guess
)

print(f"Equilibrium Draft: {equilibrium.draft:.2f} m")
print(f"Equilibrium Trim: {equilibrium.trim:.2f}°")
print(f"Draft at AP: {equilibrium.draft_ap:.2f} m")
print(f"Draft at FP: {equilibrium.draft_fp:.2f} m")

The solver adjusts draft and trim until:

  1. The displacement matches the target weight

  2. LCB aligns with LCG (longitudinal moment = 0)

Generating Hydrostatic Tables

You can generate hydrostatic tables by calculating at multiple drafts:

import numpy as np

drafts = np.arange(3.0, 8.0, 0.5)

print("Draft\tDispl(t)\tLCB\tKB\tKMt\tAw")
print("-" * 60)

for draft in drafts:
    state = calc.calculate_at_draft(draft=draft)
    print(f"{draft:.1f}\t{state.displacement_mass/1000:.0f}\t\t"
          f"{state.lcb:.2f}\t{state.vcb:.2f}\t"
          f"{state.kmt:.2f}\t{state.waterplane_area:.0f}")

Including Shell Thickness

For more accurate calculations, you can account for shell thickness:

# Calculator with 12mm shell thickness
calc = HydrostaticsCalculator(
    hull,
    water_density=1025.0,
    shell_thickness=0.012
)

This adds volume to account for water displaced by the shell plating.

Next Steps