Accelerate a body through a fluid and it must move the surrounding fluid too, so it behaves as if it were heavier. This tool computes that "added (virtual) mass" from the shape, size, fluid density and acceleration. See the added-mass coefficient Ca, the effective mass and the force needed to accelerate a sphere, cylinder, plate or square prism update in real time.
Parameters
Body shape
The added-mass coefficient Ca is set by the shape
Characteristic dimension (diameter or side)
m
Diameter for sphere/cylinder, side length for plate/prism
Span length L
m
For cylinder/plate/prism (unused for the sphere)
Fluid density ρ
kg/m³
Seawater ≈ 1025, fresh water ≈ 1000, air ≈ 1.2
Body mass m_body
kg
Acceleration a
m/s²
Results
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Added-mass coefficient Ca
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Reference volume V (m³)
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Added mass m_a (kg)
—
Effective (virtual) mass (kg)
—
Force to accelerate F (N)
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Added-mass influence (%)
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Acceleration in a fluid and the entrained fluid
As the body accelerates left and right in the fluid, the surrounding fluid (the pale-blue halo) moves with it. The extent of the entrained fluid suggests the size of the added mass. The arrow shows the direction of acceleration.
Added mass vs characteristic dimension
Force needed vs acceleration
Theory & Key Formulas
$$m_a=C_a\,\rho\,V,\qquad m_{eff}=m_{body}+m_a$$
Added mass m_a and effective (virtual) mass m_eff. Ca: added-mass coefficient, ρ: fluid density, V: reference displaced volume, m_body: body mass.
$$F=(m_{body}+m_a)\,a$$
Force F needed to accelerate the body at acceleration a in the fluid. Because the entrained fluid is accelerated too, F = m_eff·a acts on the effective mass.
A sphere has Ca = 0.5; a cylinder accelerated transverse to its axis has Ca = 1.0. The blunter and more angular the shape, the larger Ca becomes.
What is Added Mass?
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When I try to move a ball quickly underwater it feels much heavier than in air. Is that just the water's drag?
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Half right, half not. The "heaviness" you feel at constant speed really is drag. But the heavy feeling you get the instant you jolt it from rest into motion has a different cause — that is the "added mass". When you accelerate a body, you have to push and move the surrounding water too. The inertia of accelerating that water makes the body feel as if it became heavier itself.
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Wait, I'm moving the surrounding water too? It looks like only the body is moving.
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Yes — the water right next to the body is carried along with it, and water a bit farther away is pushed aside and moves too. Look at the animation top-right: there is a pale-blue halo around the body. That is a picture of the "entrained fluid". Every time the body accelerates left and right, the water in the halo sloshes with it. The amount of water moved is the added mass, written m_a = Ca·ρ·V — Ca is a shape coefficient, ρ the water density, V the reference displaced volume.
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So the coefficient changes with shape. Are a sphere and a cylinder really that different?
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Quite different. A sphere has Ca = 0.5 — it carries "half" the mass of the water it displaces. A cylinder accelerating sideways to its axis has Ca = 1.0, carrying water equal to its full displaced volume. A square prism is heavier still at Ca ≈ 1.19. The blunter and more angular the shape, the worse it lets the water flow past, so the more it drags along. Switch the "Body shape" on the left and watch how the Ca number changes.
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I see. But how is this different from drag? Both are "heavy because of the water", aren't they?
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Good question. The decisive difference is "when it acts". Drag acts even at constant speed and is a function of velocity. Added mass acts only while you accelerate and is proportional to acceleration. And added mass appears even in an ideal, zero-viscosity fluid — drag does not. That is why in the equation it shows up as an inertial term, F = (m_body + m_a)·a, multiplying the acceleration. At constant speed the added-mass force is zero; it only piles on during the instant of acceleration.
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Does this actually cause problems in real design?
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Very much so. Take a structure vibrating underwater. Its natural frequency scales as √(stiffness/mass), but submerge it and the effective mass rises by the added mass, so the natural frequency drops noticeably below the in-air value. Ship starting, stopping and turning, and offshore platforms or risers swaying in waves, simply do not match prediction without the added mass. "Things move sluggishly underwater" is exactly this inertia of the entrained water.
Frequently Asked Questions
When a body accelerates through a fluid, it must also push and accelerate the surrounding fluid. The entrained fluid gains kinetic energy, so the body behaves as if it had extra mass. This is the added (virtual) mass. It is written m_a = Ca·ρ·V, where Ca is the added-mass coefficient set by the shape, ρ is the fluid density and V the reference displaced volume. Added mass is not drag — it appears even in an ideal, inviscid fluid.
Ca is a dimensionless number set by the body shape. A sphere has Ca=0.5 (it carries half the mass of the fluid it displaces); a cylinder accelerating perpendicular to its axis has Ca=1.0 (it carries fluid equal to the displaced volume); a flat plate accelerating broadside also gives Ca=1.0 referenced to its circumscribing-cylinder volume; a square prism accelerating normal to a face has Ca≈1.19. Slender, streamlined shapes have a small Ca; blunt, angular shapes a larger one.
Drag is a force that exists even at constant velocity; it comes from viscosity and the pressure difference of the wake and is a function of speed. Added mass is an inertial effect that appears only while the body accelerates, and is proportional to acceleration. It is the inertia of feeding kinetic energy to the fluid, and exists even in an inviscid (zero-viscosity) fluid. At constant velocity the added-mass force is zero; during acceleration it adds on top of the drag, independently of it.
It is decisive for the manoeuvrability of ships and submarines (the response to starting, stopping and turning), for offshore platforms and risers moving in waves, and for the natural frequencies of structures vibrating in water. Because a natural frequency scales as √(stiffness/mass), submerging a structure raises its effective mass by the added mass and noticeably lowers the natural frequency. It is also essential for the dynamics of pump impellers and underwater robots.
Real-World Applications
Manoeuvrability of ships and submarines: When a ship or submarine starts, stops or turns, it must accelerate and decelerate not only the hull but the surrounding seawater. In sway motion the added mass can equal or exceed the hull mass itself, and ignoring it badly distorts the predicted manoeuvring response. Manoeuvring simulations and autopilot control design routinely build added mass (and added moment of inertia) into the equations of motion.
Offshore structures and risers: The floating foundations of offshore wind turbines, oil platforms and the risers running to the seabed are accelerated cyclically by waves. The Morison equation for wave force is the sum of a "drag term" and an "inertia term (which includes added mass)", and for slender members the inertia term dominates. A wrong added-mass estimate leads to wrong predictions of structural motion and fatigue life.
Vibration and natural frequencies of underwater structures: Sluice gates, submerged pipework, the cold-water pipe of ocean-thermal plants and the arms of underwater robots all gain effective mass from the added mass when immersed, lowering the natural frequency below the in-air value. A lower natural frequency is directly tied to vortex-induced vibration and resonance with waves, so underwater vibration analysis always adds the added mass to the mass matrix.
Use in CAE and fluid analysis: As a pre-study before a coupled fluid-structure (FSI) analysis, an analytical estimate like this tool gives the order of magnitude of the added mass, which helps judge whether the mesh and time step are adequate. Added mass from ideal-flow potential theory is also used as a sanity check on CFD results, and as an input for a quick correction of low-frequency structural response.
Common Misconceptions and Pitfalls
A common misconception is that "added mass is just a kind of fluid drag". Added mass and drag are entirely different physics. Drag comes from viscosity and the wake (pressure difference), exists at constant velocity and is a function of speed. Added mass, by contrast, appears only during acceleration, is proportional to acceleration and exists even in an inviscid ideal fluid — in potential-flow theory the drag is zero yet the added mass remains intact. Confusing the two leads you to mix up the acceleration term and the velocity term when writing the equation of motion.
Next, the assumption that "the added-mass coefficient Ca is always a fixed constant". The Ca values here are representative figures for an ideal fluid, an infinitely large fluid domain, a rigid body and one specific direction of acceleration. In reality Ca changes greatly near a free surface (shallow draft) or near a wall or seabed. It also depends on the direction of acceleration — even a cylinder accelerated along its axis has a much smaller Ca. At high wave frequencies a frequency dependence appears. Design requires a Ca chosen for the actual operating conditions.
Finally, the mistake that "added mass is a density effect like buoyancy, so there is no need to distinguish them". Buoyancy is a static force that always acts to balance gravity, even when the body is at rest. Added mass is a dynamic inertial effect that produces no force unless the body is accelerating. Supporting a body underwater is a buoyancy calculation; accelerating or vibrating it underwater is an added-mass calculation, and the two must be treated separately. In natural-frequency work, remember it is not that "buoyancy makes it lighter" but that "added mass raises the effective mass".
How to Use
Enter the body's characteristic dimension (diameter for cylinders, length for plates) in meters using dimDNum
Specify the body length or span perpendicular to flow direction using lengthLNum
Input fluid density (e.g., seawater 1025 kg/m³, fresh water 1000 kg/m³) in rhoFluidNum
Enter the body's structural mass in kg using bodyMassNum
Run the simulator to obtain added-mass coefficient, reference volume, and effective virtual mass
Review the force required and added-mass influence percentage to assess acceleration requirements
Worked Example
A 0.5 m diameter cylindrical pile (steel, mass 800 kg) is accelerated horizontally in seawater (density 1025 kg/m³) over a 6 m length. Reference volume V = 1.178 m³. With added-mass coefficient Ca = 1.0 for a cylinder, added mass m_a = 1207 kg. Effective virtual mass becomes 800 + 1207 = 2007 kg. To accelerate this pile at 2 m/s², the required force F = 2007 × 2 = 4014 N. Added-mass influence = 1207/(800+1207) × 100 = 60.1%, showing that fluid inertia dominates the acceleration response in offshore wave loading scenarios.
Practical Notes
Added-mass coefficients vary with geometry: cylinders perpendicular to flow use Ca ≈ 1.0; elliptical sections may reach Ca = 2.0 depending on aspect ratio
For submarine hulls and underwater vehicles, added mass can exceed structural mass by 300%, critically affecting maneuverability calculations
Oscillating bodies in confined spaces (harbors, pipelines) show increased Ca due to wall proximity; apply blockage corrections for confined geometries
Added-mass effects dominate only at high accelerations; for quasi-static (creeping) flows, inertial forces become negligible