Momentum transfer

In fluid mechanics, added mass or virtual mass is the inertia added to a system because an accelerating or decelerating body must move (or deflect) some volume of surrounding fluid as it moves through it. Added mass is a common issue because the object and surrounding fluid cannot occupy the same physical space simultaneously. For simplicity this can be modeled as some volume of fluid moving with the object, though in reality "all" the fluid will be accelerated, to various degrees.

The dimensionless added mass coefficient is the added mass divided by the displaced fluid mass – i.e. divided by the fluid density times the volume of the body. In general, the added mass is a second-order tensor, relating the fluid acceleration vector to the resulting force vector on the body.

Background

Friedrich 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.

The concept of added mass is arguably the first example of renormalization in physics. 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 it is not so is clear from 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.

Virtual mass force

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

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

where bold symbols denote vectors, $\mathbf{u}$ is the fluid flow velocity, $\mathbf{v}$ is the spherical particle velocity, ${\rho }_{\mathrm{c}}$ is the mass density of the fluid (continuous phase), $V_{\mathrm{p}}$ 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.

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

where $\sum \mathbf{F}$ 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

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

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.

Applications

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

$F=ma$ Template:Pad becomes Template:Pad $F=(m+m_{\text{added}})a.$

One can show that the added mass for a sphere (of radius $r$) is $\frac{2}{3}\pi r^3{\rho }_{\text{fluid}}.$ 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 $\frac{4}{3}\pi r^3{\rho }_{\text{air}}$ but an added mass of $\frac{2}{3}\pi r^3{\rho }_{\text{water}}.$ 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.

Naval architecture

These principles also apply to ships, submarines, and offshore platforms. In ship design, the energy required to accelerate the added mass must be taken into account when performing a sea keeping analysis. For ships, the added mass can easily reach ¼ or ⅓ of the mass of the ship and therefore represents a significant inertia, in addition to frictional and wavemaking drag forces.

In aircraft (other than lighter-than-air balloons and blimps), the added mass is not usually taken into account because the density of the air is so small.

See also

• 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
• Response Amplitude Operator for the use of added mass in ship design
• 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

References

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External links

• MIT OpenCourse Ware
• Naval Civil Engineering Laboratory