Added Mass

In the fluid mechanics referred Added mass or virtual mass, the additional inertia of a system which arises because an accelerated or decelerated body needs to move or deflect a portion of the volume of the surrounding fluid. Added Mass, there is almost always in evidence, as a body and the fluid surrounding it can not fill the same space at the same time. For simplicity may be assumed to be moved along a portion of the fluid to the body. But this is only a simplification, since in reality all of the fluid is accelerated to different extents.

The dimensionless added mass coefficient is the added mass divided by the mass of the displaced fluid from the body - that is divided by the density of the fluid times the volume of the body. In general, the added mass is a second order, which describes the relationship between the vector of the acceleration of the fluid and the force vector of the body tensor.

Basics

The concept of added mass was proposed in 1828 by Friedrich Wilhelm Bessel, to describe the motion of a pendulum in a fluid. The period of a pendulum with respect to a movement in the vacuum is increased ( after the consideration of buoyancy forces ), this suggests that the surrounding fluid increases the effective mass of the system.

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

It is often stated erroneously that the added mass will determined by the momentum of the surrounding fluid. That this is not so, it is clear when the medium is located in a large vessel, in which the pulse at any given time is exactly zero. The added mass is determined by the quasi- pulse: The added mass times the acceleration of the body is equal to the time derivative of the quasi- impulse of the fluid.

Virtual mass and Basset force

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

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

It can be shown that the force is provided due to virtual ground for a spherical particle in inviscid, incompressible fluid through

The focus bold symbols for vectors, the speed vector of the flow field, the speed of the spherical particle, the density of the fluid (continuous phase), the volume of the particle and d / dt denotes the substantial dissipation.

The cause of the term " virtual ground " will be apparent when the motion equation of the body is considered.

It is the sum of all other forces acting on the body, such as Gravity, pressure gradient, flow resistance and dynamic lift, Basset force, etc.

If now the derivative is shifted to the left side of the equation for the velocity of the body, resulting

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

Applications

The added mass can be added to most phyisikalischen equations by the effective mass is considered as the sum of the mass and the added mass. The sum is commonly referred to as " virtual ground ".

The simple expression of the added mass for a spherical body allows the second Newton's law in the following form to write:

For any body, the relationship between force and acceleration is more complex, an acceleration in a direction can also lead to forces in other directions. It is useful to represent force and acceleration as six-dimensional vector. The first three components include power and acceleration, the other three torque and angular acceleration. Mass is then added to a tensor (the " induced mass tensor " ), its Kompenenten are dependent on the direction of movement of the body. Not all elements of this tensor have the dimensions of a mass, some have the dimension of mass × length and other mass × length2.

All bodies, which are accelerated in a fluid, subject to the added mass. However, because the added mass is a function of the density of the fluid, this effect is usually negligible when the body fall within a fluid with significantly lower density. In the cases where the density of the fluid is comparable to or greater than the density of the body, the added mass may be greater than the mass of the body. Will this now neglected, may lead to significant errors in calculations.

It can be shown that the added mass of a sphere (with radius ) at given potential flow through. A spherical bubble, which rises in the water, therefore has a mass but a measure of added. Since water is about 800 times denser than air ( at standard conditions ), the added mass is about 400 times greater than the mass of the bubble in this case.

Shipbuilding

This Gesätzmäßigkeiten also apply to ships, submarines and oil rigs. In the planning of ships, it is necessary to include the energy required to accelerate the added mass. For vessels can be quickly ¼ or ⅓ of the mass of the ship, the added mass. It therefore has a significant share of inertia; For this, the flow resistance is due to friction and wave drag.

In aircraft (other than aircraft that are lighter than air like balloons and airships ), the added mass is usually not considered, since the density of the air is negligible.