Added mass is the additional mass that an object appears to have when it is accelerated relative to a surrounding fluid. For example, it adds to the effective inertia of boats, buoys, swimmers’ limbs, airplanes and bubbles.
While added mass effects occur in all real fluids, they are most clearly defined in ideal irrotational incompressible inviscid flow that has a “potential”, φ . For a nonrotating object of volume V moving at velocity U in such a fluid which is at rest at infinity, the impulse is:
and the kinetic energy of the surrounding fluid of density r is:
where is a symmetric tensor, called the “coefficient of added mass, that depends on the shape of the object. If referred to principal axes, the tensor is diagonal with components C_{xx}, C_{yy} and C_{zz}. Typical values for ellipsoids of revolution about the x-axis and with ratio of x- to y-axes of a/b are (Lamb, 1932):
In order to accelerate the object, both it and the surrounding fluid must be set in motion. If the volume and the added mass coefficient are constant, the required force is:
where ρ_{0} is the density of the object and is the unit tensor or Kronecker delta.
In a more general motion, the volume and added mass coefficient may change. For example, a cavitation bubble that is collapsing will accelerate.
If the object is moving steadily in a direction that is not parallel to a principal axis, it will experience a torque [Lamb (1932); Wallis (1994)] from the fluid equal to:
If it is moving steadily parallel to a principal axis, say the x-direction, and is not subject to internal (balancing) forces, the mean bulk stress (pressure) in the object is:
where p_{∞} is the pressure at infinity. For a more general steady motion, the mean bulk stress is:
as long as the restraining torque is applied in a way that does not contribute to the bulk stress (e.g., consisting of two equal and opposite forces acting at points joined by a line perpendicular to the forces).
The added mass coefficient is related to the polarization, D or net dipole moment of internal sources or dipoles that could represent the object moving at velocity U in a potential flow, by:
where is a tensor “polarizability” [Wallis (1993, 1994)].
Let the object be at rest in a potential flow with velocity U that varies slightly so that changes in U are small over the scale of the object. Then the force on an internal source of strength m at location r is −ρm(U + r · ΔU) and the net force on the object is approximately, by summation, since ∑ m = 0,
The first component is due to the overall pressure gradient while the second is the “added mass force” deduced by Taylor (1928). Effects due to the change in because of ΔU are of second order.
Where there is circulation in the flow, the object cannot be represented by flux sources alone. This may introduce a “lift” force which Auton (1987) and Drew and Lahey (1987) show, for a sphere, to be:
where U_{s} is the velocity of the sphere, which is small compared with lengths over which the velocity varies in the fluid.
Similar results may be derived for more complex objects, such as a porous sphere made up of an array of much smaller spheres [Wallis (1991)].
Added mass effects also occur in two-phase continua. For example, in one-dimensional uniform acceleration of a dispersed array in the x-direction, the equations of motion are [Wallis (1994a); Zhang and Prosperetti (1994)]:
where α_{1} and α_{2} are volume fractions of the phases; p is the macroscopic pressure; g_{1} and g_{2} are body force fields; and f_{2} is an external force per unit volume of the dispersed phase, 2, acting on phase 2. C is the component of an “internal coupling” coefficient in the duration of motion that is closely approximated by 1/2 for isotropic arrays of spheres. The combination Cα_{2} is Wallis’ (1989) “exertia” while Cα_{1}α_{2} is Geurst’s (1985) “added mass coefficient”.
Similar equations are valid for three-dimensional motion in terms of vector velocities and forces, the pressure gradient, and a tensor form of C. More general forms have been proposed when there are gradients in velocities and volume fractions [e.g., Zhang and Prosperetti (1994); Drew (1992)].
While C could be called an “added mass” coefficient, other definitions are possible. For example, if motion is due entirely to f_{2}(g_{1} = g_{2} = dp/dx = 0), the first equation becomes:
and the factor in parentheses is Wallis’ (1989, 1994a) “added mass coefficient”, C_{w}, that, in the same sense of the “drift” discussed by Darwin (1952), represents the ratio of the volumetric flux of fluid to the volumetric flux of particles induced by motion starting from rest. The second equation of motion becomes:
resembling the equivalent form for a single particle.
In an alternative frame of reference in which there is no net flux, or “bulk velocity”:
the equations of motion, without body forces, may be combined to give:
which resembles the single-phase case with an “added mass coefficient” equal to (C + α_{2})/α_{1}, as used by Zuber (1964).
Geurst (1985, 1986) derived general equations of motion for two-phase inviscid flow based on the kinetic energy contributed by relative motion that was proportional to his “added mass coefficient”, m. His results are valid if m is isotropic and depends only on the volume fraction of the dispersed phase. His equations are “ill-posed” and it is likely that a realistic model should include a representation of changes in the structure of the dispersion in response to overall strain [Wallis (1994b)].
REFERENCES
Auton, T. R. (1987) The Lift Force on a Spherical Body in a Rotational Flow, J. Fluid Mech., 183, 199-218.
Darwin, C. (1952) Note on Hydrodynamics, Proc. Camb. Phil. Soc., 49, 342.
Drew, D. A. (1992) Analytical Modeling of Multiphase Flows, in Boiling Heat Transfer, R. T. Lahey, Jr. (ed.) Elsevier Science Publishers B. V.
Drew, D. A. and Lahey, R. T., Jr. (1987) The Virtual Mass and Lift Force on a Sphere in Rotating and Straining Inviscid Flow, Int. J. Multiphase Flow, 13, 113-121. DOI: 10.1016/0301-9322(87)90011-5
Geurst, J. A. (1985) Virtual Mass in Two-Phase Bubbly Flow, Physica, 129A, 233-261. DOI: 10.1016/0378-4371(85)90168-2
Geurst, J. A. (1986) Variational Principles and Two-Fluid Hydrodynamics of Bubbly Liquid/Gas Mixtures, Physica, 135A, 455-486. DOI: 10.1016/0378-4371(86)90154-8
Lamb. H. (1932) Hydrodynamics, 6th ed., Cambridge University Press.
Taylor, G. I. (1928a) The Forces on a Body Placed in a Curved or Converging Stream of Fluid, Proc. Roy. Soc., 120A, 260-283.
Taylor, G. I. (1928b) The Energy of a Body Moving in an Infinite Fluid with an Application to Airships, Proc. Roy. Soc., 120A, 13.
Wallis, G. B. (1989) Inertial Coupling in Two-Phase Flow: Macroscopic Properties of Suspensions in an Inviscid Fluid, Multiphase Science and Technology, 5, 239-361.
Wallis, G. B. (1991) The Averaged Bernoulli Equation and Macroscopic Equations of Motion for the Potential Flow of a Dispersion, Int. J. Multiphase Flow, 17(6), 683-695. DOI: 10.1016/0301-9322(91)90050-D
Wallis, G. B. (1993) The Concept of Polarization in Dispersed Two-Phase Potential Flow, Nuclear Engineering and Design, 141, 329-342. DOI: 10.1016/0029-5493(93)90110-U
Wallis, G. B. (1994a) Added Mass Coefficients for Uniform Arrays, Int. J. Multiphase Flow, 20(4), 799-803. DOI: 10.1016/0301-9322(94)90047-7
Wallis, G. B. (1994b) The Particle Pressure of Arrays in a Potential Flow, Nuclear Engineering and Design, 151, 1-14. DOI: 10.1016/0029-5493(94)90030-2
Zhang D. Z. and Prosperetti, A. (1994) Averaged Equations for Inviscid Disperse Two-Phase Flow, J. Fluid. Mech., 267, 185-219.
Zuber, N. (1964) On the Dispersed Two-Phase Flow in the Laminar Flow Regime, Chem. Eng. Sci., 19, 897-917. DOI: 10.1016/0009-2509(64)85067-3
Les références
- Auton, T. R. (1987) The Lift Force on a Spherical Body in a Rotational Flow, J. Fluid Mech., 183, 199-218. DOI: 10.1017/S002211208700260X
- Darwin, C. (1952) Note on Hydrodynamics, Proc. Camb. Phil. Soc., 49, 342.
- Drew, D. A. (1992) Analytical Modeling of Multiphase Flows, in Boiling Heat Transfer, R. T. Lahey, Jr. (ed.) Elsevier Science Publishers B. V.
- Drew, D. A. and Lahey, R. T., Jr. (1987) The Virtual Mass and Lift Force on a Sphere in Rotating and Straining Inviscid Flow, Int. J. Multiphase Flow, 13, 113-121. DOI: 10.1016/0301-9322(87)90011-5
- Geurst, J. A. (1985) Virtual Mass in Two-Phase Bubbly Flow, Physica, 129A, 233-261. DOI: 10.1016/0378-4371(85)90168-2
- Geurst, J. A. (1986) Variational Principles and Two-Fluid Hydrodynamics of Bubbly Liquid/Gas Mixtures, Physica, 135A, 455-486. DOI: 10.1016/0378-4371(86)90154-8
- Lamb. H. (1932) Hydrodynamics, 6th ed., Cambridge University Press.
- Taylor, G. I. (1928a) The Forces on a Body Placed in a Curved or Converging Stream of Fluid, Proc. Roy. Soc., 120A, 260-283. DOI: 10.1098/rspa.1928.0148
- Taylor, G. I. (1928b) The Energy of a Body Moving in an Infinite Fluid with an Application to Airships, Proc. Roy. Soc., 120A, 13. DOI: 10.1098/rspa.1928.0131
- Wallis, G. B. (1989) Inertial Coupling in Two-Phase Flow: Macroscopic Properties of Suspensions in an Inviscid Fluid, Multiphase Science and Technology, 5, 239-361.
- Wallis, G. B. (1991) The Averaged Bernoulli Equation and Macroscopic Equations of Motion for the Potential Flow of a Dispersion, Int. J. Multiphase Flow, 17(6), 683-695. DOI: 10.1016/0301-9322(91)90050-D
- Wallis, G. B. (1993) The Concept of Polarization in Dispersed Two-Phase Potential Flow, Nuclear Engineering and Design, 141, 329-342. DOI: 10.1016/0029-5493(93)90110-U
- Wallis, G. B. (1994a) Added Mass Coefficients for Uniform Arrays, Int. J. Multiphase Flow, 20(4), 799-803. DOI: 10.1016/0301-9322(94)90047-7
- Wallis, G. B. (1994b) The Particle Pressure of Arrays in a Potential Flow, Nuclear Engineering and Design, 151, 1-14. DOI: 10.1016/0029-5493(94)90030-2
- Zhang D. Z. and Prosperetti, A. (1994) Averaged Equations for Inviscid Disperse Two-Phase Flow, J. Fluid. Mech., 267, 185-219. DOI: 10.1017/S0022112094001151
- Zuber, N. (1964) On the Dispersed Two-Phase Flow in the Laminar Flow Regime, Chem. Eng. Sci., 19, 897-917. DOI: 10.1016/0009-2509(64)85067-3