Additional mass

Additional mass (English: added mass ) or virtual mass is a physical term from fluid mechanics . It describes the additional inertia of a system that arises because an accelerated or decelerated body has to move or deflect part of the volume of the fluid surrounding it. Additional mass appears almost always, since a body and the fluid surrounding it cannot fill the same space at the same time. To simplify matters, it can be assumed that part of the fluid moves with the body. However, this is only a simplification, since in reality the entire fluid is accelerated to different degrees.

The dimensionless quantity additional mass coefficient (English: added mass coefficient ) is the additional mass divided by the mass of the fluid displaced by the body - i.e. the density of the fluid times the volume of the body. In general, the additional mass is a second order tensor which describes the relationship between the vector of the acceleration of the fluid and the force vector on the body.

Basics

The concept of additional mass was proposed by Friedrich Wilhelm Bessel in 1828 to describe the movement of a pendulum in a fluid. The period of a pendulum is increased compared to a movement in a vacuum (even after taking into account buoyancy forces), this suggests that the surrounding fluid increases the effective mass of the system.

The concept of additional mass is probably the first example of renormalization in physics. The concept can also be understood as a classic counterpart to the quantum mechanical concept of quasiparticles . It should not be confused with the increase in mass in special relativity.

It is often erroneously claimed that the additional mass is determined by the momentum of the surrounding fluid. That this is not the case becomes clear when the medium is in a large vessel, where its momentum is exactly zero at any point in time. The additional mass is determined by the quasi-impulse: The additional mass times the acceleration of the body is equal to the time derivative of the quasi-impulse of the fluid.

Virtual mass and basset power

The unsteady forces due to a change in the relative speed of a body in a fluid can be divided into two parts: the contribution of the virtual masses and the Basset force.

The cause of the force is that the work done by the accelerated body is converted into kinetic energy of the fluid.

It can be shown that the force due to the virtual mass for a spherical particle in a frictionless, incompressible fluid is given by

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

bold symbols stand for vectors, is the velocity vector of the flow field, is the velocity of the spherical particle, is the density of the fluid (continuous phase), is the volume of the particle and denotes the substantial derivative . ${\displaystyle \mathbf {u} }$ ${\displaystyle \mathbf {v} }$ ${\displaystyle \rho _{\mathrm {c} }}$ ${\displaystyle V_{\mathrm {p} }}$ ${\displaystyle {\tfrac {\mathrm {D} }{\mathrm {D} t}}}$

The reason for the term "virtual mass" becomes clear when the equation of motion of the body is considered.

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

where is the sum of all other forces on the body, such as As gravity , pressure gradient, flow resistance , dynamic lift , Basset force , etc. ${\displaystyle \sum \mathbf {F} }$

If the derivative according to the speed of the body is now shifted to the left side of the equation, the result is

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

the body is thus accelerated as if it had an additional mass as large as half the mass of the displaced fluid. In addition, there is an additional force on the right side due to the acceleration of the fluid.

Applications

The additional mass can be included in most physical equations by considering the effective mass as the sum of the mass and the additional mass. The sum is commonly referred to as the "virtual ground".

The simple expression of the additional mass for a spherical body allows Newton's 2nd law to be written in the following form:

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

The relationship between force and acceleration is more complex for any body; acceleration in one direction can also lead to forces in other directions. It then makes sense to represent the force and acceleration as a six-dimensional vector. The first three components contain force and acceleration, the other three torque and angular acceleration. The additional mass then becomes a tensor (the "induced mass tensor"); its components depend on the direction of movement of the body. Not all elements of this tensor have the dimensions of a mass, some have the dimension mass × length and others have mass × length ² .

All bodies that are accelerated in a fluid are subject to the additional mass. Since the additional mass depends on the density of the fluid, this effect is mostly neglected if bodies fall in a fluid with a significantly lower density. In those cases where the density of the fluid is comparable to or greater than the density of the body, the additional mass can be greater than the mass of the body. If this is now neglected, it can lead to significant errors in calculations.

It can be shown that the additional mass of a sphere (with radius ) in the case of potential flow is given by . A spherical air bubble that rises in water therefore has a mass of but an additional mass of . Since water is approximately 800 times denser than air (under standard conditions ), the additional mass in this case is approximately 400 times greater than the mass of the bubble. ${\displaystyle r}$ ${\displaystyle {\tfrac {2}{3}}\pi r^{3}\rho _{\text{fluid}}}$ ${\displaystyle {\tfrac {4}{3}}\pi r^{3}\rho _{\text{air}}}$ ${\displaystyle {\tfrac {2}{3}}\pi r^{3}\rho _{\text{water}}}$

shipbuilding

These principles also apply to ships, submarines and oil rigs. When designing ships, it is necessary to include the energy needed to accelerate the extra mass. In the case of ships, the additional mass can quickly amount to ¼ or ⅓ of the mass of the ship. It therefore has an essential part in indolence ; in addition there is the flow resistance due to friction and wave resistance.

In aircraft (with the exception of aircraft that are lighter than air such as balloons and airships), the additional mass is usually not taken into account, as the density of the air is negligibly low.

See also

• Basset force describes the influence of the history of the movement of the body on the viscosity in a Stokes flow
• Bassett Boussinesq Oseen equation describes the motion - and the force on - a particle which is located in a non-steady flow at low Reynolds numbers is
• Darwin drift for the relationship between the additional mass and the Darwin drift volume
• Response Amplitude Operator as an example of the use of the concept of additional mass in the construction of ships
• Keulegan-Carpenter number as a dimensionless number which describes the relationship between flow resistance and inertial forces under wave loads
• Morison's equation as an empirical model for the forces under shaft loading taking into account additional mass and flow resistance

Web links

• MIT OpenCourse Ware
• Naval Civil Engineering Laboratory

Individual evidence

1. ↑ L. Prandtl , K. Oswatitsch, K. Wieghardt: Guide through the flow theory . Vieweg + Teubner Verlag, 2013, ISBN 978-3-322-99491-2 .
2. ^ John Nicholas Newman : Marine hydrodynamics . MIT Press , Cambridge, Massachusetts 1977, ISBN 0-262-14026-8 , §4.13, p. 139.
3. ^ GG Stokes : On the effect of the internal friction of fluids on the motion of pendulums . In: Transactions of the Cambridge Philosophical Society . 9, 1851, pp. 8-106. bibcode : 1851TCaPS ... 9 .... 8S .
4. ^ José González, Miguel A. Martín-Delgado, Germán Sierra, Angeles H. Vozmediano: Quantum electron liquids and high-T _c superconductivity . Springer, 1995, ISBN 978-3-540-60503-4 , p. 32.
5. ↑ ^{a b} Gregory Falkovich: Fluid Mechanics, a short course for physicists . Cambridge University Press, 2011, ISBN 978-1-107-00575-4 , Section 1.3.
6. ↑ A. Biesheuvel, S. Spoelstra: The added mass coefficient of a dispersion of spherical gas bubbles in liquid . In: International Journal of Multiphase Flow . 15, No. 6, 1989, pp. 911-924. doi : 10.1016 / 0301-9322 (89) 90020-7 .
7. ^ Clayton T. Crowe, Martin Sommerfeld, Yutaka Tsuji: Multiphase flows with droplets and particles . CRC Press, 1998, ISBN 0-8493-9469-4 , p. 81.
8. ^ ^{A b} A Review of Added Mass and Fluid Inertial Forces Report CR 82.010 NAVAL CIVIL ENGINEERING LABORATORY, Port Hueneme, California, January 1982