Added mass

Friedrich Wilhelm Bessel proposed the concept of added mass in 1828 to describe the motion of a pendulum in a fluid. The period of such a pendulum increased relative to its period in a vacuum (even after accounting for buoyancy effects), indicating that the surrounding fluid increased the effective mass of the system.[2]

The concept of added mass is arguably the first example of renormalization in physics.[3][4][5] The concept can also be thought of as a classical physics analogue of the quantum mechanical concept of quasiparticles. It is, however, not to be confused with relativistic mass increase.

It is often erroneously stated that the added mass is determined by the momentum of the fluid. That this is not the case, it becomes clear when considering the case of the fluid in a large box, where the fluid momentum is exactly zero at every moment of time. The added mass is actually determined by the quasi-momentum: the added mass times the body acceleration is equal to the time derivative of the fluid quasi-momentum.[4]

Unsteady forces due to a change of the relative velocity of a body submerged in a fluid can be divided into two parts: the virtual mass effect and the Basset force.

The origin of the force is that the fluid will gain kinetic energy at the expense of the work done by an accelerating submerged body.

It can be shown that the virtual mass force, for a spherical particle submerged in an inviscid, incompressible fluid is[6]

${\displaystyle \mathbf {F} ={\frac {\rho _{\mathrm {c} }V_{\mathrm {p} }}{2}}\left({\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}-{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\right),}$

where bold symbols denote vectors, ${\displaystyle \mathbf {u} }$ is the fluid flow velocity, ${\displaystyle \mathbf {v} }$ is the spherical particle velocity, ${\displaystyle \rho _{\mathrm {c} }}$ is the mass density of the fluid (continuous phase), ${\displaystyle V_{\mathrm {p} }}$ is the volume of the particle, and D/Dt denotes the material derivative.

The origin of the notion "virtual mass" becomes evident when we take a look at the momentum equation for the particle.

${\displaystyle m_{\mathrm {p} }{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=\sum \mathbf {F} +{\frac {\rho _{\mathrm {c} }V_{\mathrm {p} }}{2}}\left({\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}-{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\right),}$

where ${\displaystyle \sum \mathbf {F} }$ is the sum of all other force terms on the particle, such as gravity, pressure gradient, drag, lift, Basset force, etc.

Moving the derivative of the particle velocity from the right hand side of the equation to the left we get

${\displaystyle \left(m_{\mathrm {p} }+{\frac {\rho _{\mathrm {c} }V_{\mathrm {p} }}{2}}\right){\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=\sum \mathbf {F} +{\frac {\rho _{\mathrm {c} }V_{\mathrm {p} }}{2}}{\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}},}$

so the particle is accelerated as if it had an added mass of half the fluid it displaces, and there is also an additional force contribution on the right hand side due to acceleration of the fluid.

The added mass can be incorporated into most physics equations by considering an effective mass as the sum of the mass and added mass. This sum is commonly known as the "virtual mass".

A simple formulation of the added mass for a spherical body permits Newton's classical second law to be written in the form

${\displaystyle F=m\,a}$   becomes   ${\displaystyle F=(m+m_{\text{added}})\,a.}$

One can show that the added mass for a sphere (of radius ${\displaystyle r}$) is ${\displaystyle {\tfrac {2}{3}}\pi r^{3}\rho _{\text{fluid}}}$, which is half the volume of the sphere times the density of the fluid. For a general body, the added mass becomes a tensor (referred to as the induced mass tensor), with components depending on the direction of motion of the body. Not all elements in the added mass tensor will have dimension mass, some will be mass × length and some will be mass × length².

All bodies accelerating in a fluid will be affected by added mass, but since the added mass is dependent on the density of the fluid, the effect is often neglected for dense bodies falling in much less dense fluids. For situations where the density of the fluid is comparable to or greater than the density of the body, the added mass can often be greater than the mass of the body and neglecting it can introduce significant errors into a calculation.

For example, a spherical air bubble rising in water has a mass of ${\displaystyle {\tfrac {4}{3}}\pi r^{3}\rho _{\text{air}}}$ but an added mass of ${\displaystyle {\tfrac {2}{3}}\pi r^{3}\rho _{\text{water}}.}$ Since water is approximately 800 times denser than air (at RTP), the added mass in this case is approximately 400 times the mass of the bubble.

• Basset force for describing the effect of the body's relative motion history on the viscous forces in a Stokes flow
• Basset–Boussinesq–Oseen equation for the description of the motion of – and forces on – a particle moving in an unsteady flow at low Reynolds numbers
• Darwin drift for the relation between added mass and the Darwin drift volume
• Keulegan–Carpenter number for a dimensionless parameter giving the relative importance of the drag force to inertia in wave loading
• Morison equation for an empirical force model in wave loading, involving added mass and drag
• Response Amplitude Operator for the use of added mass in ship design

• MIT OpenCourse Ware
• Naval Civil Engineering Laboratory
• Det Norske Veritas DNV-RP-H103 Modelling And Analysis Of Marine Operations Archived 2016-03-04 at the Wayback Machine