## Abstract

The fundamental understanding of the added mass phenomenon associated with the motion of a solid body relative to a fluid is revisited. This paper focuses on the two-dimensional flow around a circular cylinder oscillating transversely in a free stream. A virtual experiment reveals that the classical approach to this problem leads to a paradox. The inertial force is derived afresh based on analysis of the motion in a frame of reference attached to the cylinder centroid, which overcomes the paradox in the classical formulation. It is shown that the inertial force depends not only on the acceleration of the cylinder *per se*, but also on the relative motion between body and fluid embodied in a parameter called alpha, *α*, which represents the ratio of the maximum transverse velocity of the cylinder to the free-stream velocity; the induced inertial force is directionally varying and non-harmonic in time depended on the alpha parameter. It is further shown that the component of the inertial force in the transverse direction is negligible for *α*<0.1, increases quadratically for *α*<0.5, and tends asymptotically to the classical result as , i.e. in still fluid.

## 1. Introduction

Whenever acceleration is imposed on a fluid by acceleration of a body moving through the fluid, or acceleration of the fluid relative to the body, additional inertial forces act on the body due to the ‘added mass’ effect. The calculation of these inertial forces has been of fundamental interest in fluid mechanics for more than a century. It can also be of considerable significance in some engineering applications involving lightweight structures submerged in dense fluids. One such example is the ‘vortex-induced vibration’ of offshore structures encountered in exploration and extraction of petroleum from sea beds; marine cables, pipelines, risers, platforms, etc., can be excited into significant vibration by sea currents and waves and dealing with its consequences has provided impetus and funding for concentrated research on the problem over the last few decades. Some relatively recent fundamental studies have highlighted the importance of added mass in vortex-induced vibration and increased interest in this topic [1–3].

The scope of this work is to revisit the fundamental understanding of the added mass effect with particular attention to the ‘deceivingly’ simple case of a circular cylinder oscillating transversely in a free stream. A review of the literature in §2 reveals that some fundamental aspects of the added mass notion may have been ignored in the classical formulation of this problem. The inconsistency of the classical formulation with the physical world is demonstrated through a virtual experiment in §3. A new formulation of the added mass in a frame of reference moving with the cylinder is given in §4 to derive the inertial force acting on a cylinder oscillating transversely to a free stream from potential flow theory. However, a purely mathematical treatise of the flow potential is beyond the objectives of the present work. The paper concludes with some remarks from the present work in §5.

## 2. Literature review

The notion of *added mass*, also referred to as *hydrodynamic mass* or *virtual mass* in the literature, can be found in textbooks of classical hydrodynamics [4,5], but according to some sources, it was first introduced by Dubua back in 1776 [6, ch. 1]. A survey of the literature shows that added mass is customarily—but not uniquely—derived from the kinetic energy associated with the motion of the fluid. The formulation below is adapted from Brennen [7].

The kinetic energy of the fluid, *E*_{K}, can be expressed in tensor notation as
2.1where is the entire volume of fluid, *u*_{i} (*i*=1,2,3) represent the spatial components of the fluid velocity in this volume, and repeated indices imply a summation. It is assumed throughout that the fluid is incompressible with density *ρ*. Any increase in the kinetic energy of the fluid implies that additional work must be done on the fluid to supply the extra energy, and vice versa.

Consider the motion of a non-deformable body through an unbounded fluid otherwise at rest is a rectilinear translation at velocity, *U*. The kinetic energy can be conveniently expressed as
2.2If the velocity of the body is steady, the kinetic energy of the fluid remains constant in time even for unsteady and/or turbulent flows (in a statistical sense). If the body is accelerated the velocity at each point in fluid relative to the body varies in direct proportion to the change in *U*. The rate of additional work required is equal to the rate of change of the kinetic energy with time. This comes to the effect that an additional inertial force, *F*_{I}, is experienced by the body such that the work done by the body −*F*_{I}*U* is equal to d*E*_{K}/d*t*, i.e.
2.3The inertial force has the same form and sign as that required to accelerate the mass, *m*, of the body itself, namely *m* (d*U*/d*t*). Consequently, it is rational to consider the mass of fluid given by *ρI* as an *added mass*, denoted *m*_{A}, of fluid being accelerated ‘by’ the body—but not ‘with’ the body. This subtle distinction has some important ramifications which are addressed further below. As pointed out by Brennen [7], p. 3, ‘there is no such identifiable fluid mass; rather all of the fluid is accelerating to some degree such that the total kinetic energy of the fluid is increasing’.

The above statement reveals another subtlety in the analytical description of incompressible flows: in theory, a change in pressure on the body surface must be instantly radiated throughout the entire fluid domain. This can be problematic when it comes to the computation of volume integrals over the entire fluid domain such as that defining the kinetic energy in (2.1). In this respect, it may worth drawing the attention to the indeterminacy of the integral defining Darwin's *drift mass* which, coincidentally, is always equal to the added mass for incompressible flows [8]. The point is made to illustrate that the definition of the added mass needs be modified if the fluid far from the body is not at rest as assumed hitherto.

Now consider the motion of a body relative to an unbounded fluid stream (‘free stream’) of steady velocity, , far from the body. In this case, the kinetic energy within an infinitely large domain becomes infinite and the above formulation of the added mass is meaningless. The appropriate fluid velocities, *u*_{i}, to be employed in defining the kinetic energy in (2.1) are those relative to the free-steam velocity,
2.4Then, the integral defining *E*_{K} becomes determinable. It can be shown that, following the same procedure as above, this Galilean transformation leads to no alteration in the inertial force since
2.5which is the same result as (2.3). It is assumed above that *U* and are aligned but even if the two velocities are not collinear, it can readily be verified that the same result is obtained by employing vectors. This is a deceivingly simple result which is very often employed as the starting point to define added mass in the literature, rather than the opposite. In fact, this has been the root of much confusion regarding the added mass as its meaning is often misinterpreted as
irrespectively of the fluid motion relatively to the body. The above expression is incorrect unless the fluid far from the body is at rest. For all other cases, it is not in accord with the physical world as is demonstrated through a virtual experiment in §3. It is worth noting here the words of Sarpkaya [2]: ‘the added mass is not a concentrated mass attached to the centroid of the body. It is distributed throughout the fluid set in motion by the body. Thus, its magnitude and centroid change with time as the intensity and distribution of the kinetic energy of the fluid change with time.’

For completeness, it is useful to generalize the definition of the added mass. For a body of arbitrary shape moving within a fluid, there are in general three translational accelerations, and three rotational accelerations , where the overdot denotes differentiation with respect to time. The added mass expresses in a unified way the linear forces and moments, *F*_{i}(*i*=1,2,…,6), imposed on the body due to all six possible components of acceleration,
2.6For the general case of any body in arbitrary motion through a fluid, there are 36 different added-mass components, *m*_{ij}(*i*,*j*=1,2…6). Under specific conditions, some of these components can be related, e.g. due to simplifications introduced by geometric symmetries. For a body with three planes of symmetry, the number of independent added mass components reduces to 12. Further simplifications arise for potential flow in which case the added mass matrix is symmetric ([9] cited in [7]). The general expression in (2.6) is the appropriate starting point for the study of the dynamic behaviour of a body moving within a fluid, rather than (2.3) as there is a risk of overlooking terms that may be significant [10].

The above formulation to derive the added mass from the kinetic energy of the fluid makes no assumption about the nature of the flow (rotational or irrotational, laminar or turbulent, etc.). It follows from the foregoing that all characteristic flow structures and processes generated by the motion of a body within a viscous fluid, such as boundary layers, free shear layers, wakes, formation and shedding of concentrated vorticity in the wake, contribute to the kinetic energy of the fluid and their evolution in time is what determines the inertial force due to the added mass. Hence, knowledge of the entire flow field over time is required to evaluate the kinetic energy when a body moves within a real fluid; this is one of the most important and challenging problems in fluid mechanics [11]. Particularly in the case of unsteady separated turbulent flows, the problem is even more complicated.

Evaluation of the inertial forces through the general expression in (2.6) require as a precondition that the flow induced by each component of the body motion or velocity in a fluid are linearly superposable. This is generally not true because the equations of fluid motion (e.g. Navier–Stokes) are nonlinear. However, there are two models of fluid flow which do satisfy the superposability condition, namely the potential flow for infinitely large Reynolds numbers and the Stokes flow for asymptotically small Reynolds numbers [7]. Quoting from the latter source (p. 9): ‘Only in these limiting cases can the added mass matrix be regarded as an exact representation of the relation between fluid inertial force and body acceleration. In other types of flow it could however be regarded as a reasonable approximation.’

In the framework of inviscid potential flow, analytical expressions have been developed to evaluate added mass components for specific body geometries in different kinds of motion, which can be found in the literature [7,10]. The problem becomes much more difficult in viscous flows because of the effect of vorticity generated on the body surface and preexisting within the fluid. However, a number of studies have demonstrated that added-mass effects are independent of the nature of the flow. First and foremost, Basset [12] showed that the general equations of impulsive motion of a viscous fluid are the same as those of a perfect fluid based on a very simple but strict reasoning. Chang [13] showed in a rigorous manner that the effect due to the body's acceleration can be separated from the contribution from vorticity on the force acting on the body. Chang [13] also showed that the potential flow has a *geometric influence* on the contribution of vorticity attached on the body surface and within the flow to the drag and lift forces, i.e. viscous forces are not free from inertial effects. Leonard & Roshko [1] argued that added mass definitions obtained for irrotational flow are more generally applicable to incompressible flow because an incremental change in velocity instantaneously generates a potential velocity field proportional to that velocity change which is superimposed on the existing velocity field. Mougin & Magnaudet [14], based on the mathematical treatment of the equations of motion by Howe [15], showed that the linear and angular momentum balances for a body moving within a viscous rotational flow at rest at infinity can be cast in a form which comprises exactly the added-mass components as for the corresponding motion within an inviscid fluid; the contributions from vorticity appear as separate terms in the equations. Mougin & Magnaudet [14] further argued that added-mass effects are independent of the body surface, i.e. interchanging slip and no-slip boundary conditions makes no difference to the inertial forces. Wakaba & Balachandar [16] firmly established through direct numerical simulations the independence of the added mass coefficient to the acceleration number and to the state of flow prior to acceleration for flows around a rigid sphere in relative motion to a viscous fluid; the explanation provided by the latter authors is that the potential flow part establishes instantaneously as in the case of an ideal fluid (no memory effect).

A critical review of previous studies discussed in the preceding paragraphs reveals three important aspects: (i) added mass can be properly defined in both ideal inviscid and real viscous flows but exact evaluations can be made only in the former case, (ii) inviscid flow added-mass effects are inherent in viscous flows for the same kind of body motion relative to the fluid independently of the nature of the flow (i.e. for instantaneous Reynolds numbers from zero to infinity), and (iii) the correct way to evaluate added masses is in a frame of reference fixed with the body moving within a fluid otherwise at rest far from the body. The last point appears not to be understood clearly. The required forces in a fixed frame of reference have to be obtained by ‘projecting’ the absolute velocity field from the ‘moving’ to the fixed frame of reference [13,14].

## 3. The added mass paradox: a virtual experiment

This work is concerned with two-dimensional flow around a circular cylinder oscillating transversely in a free stream. This is one of the conceptually most simple configurations to study the added mass effect, but not as simple as it may originally seem. Surprisingly, no solutions specific to this flow configuration can be found in textbooks or other probable sources in the literature. The derivation of the inertial force is postponed to §4.

The following example is adapted to the case of a circular cylinder from one with a prolate spheroid given by Imlay [10]. Consider a circular cylinder in rectilinear motion through an ideal inviscid fluid in the direction of the *x*-axis, and experiencing acceleration, . The potential flow would have streamlines of the sort shown in figure 1*a*. Consider now the cylinder is at the same position having the same acceleration as before but due to orbital motion at a fixed angle and rate of drift. The nature of a real flow is to align with the instantaneous direction of the orbiting cylinder; this can be clearly seen in flow visualization photographs of the wake behind an orbiting cylinder (see [17], fig. 3). The streamlines for the corresponding potential flow would be those depicted in figure 1*b*. Although the actual mass of the body would react in the same way in both cases, the added mass effect, i.e. the inertial force in the *x*-direction, would be different because of the difference in flow patterns attending the two different motions. Recalling that a given flow pattern requires a unique velocity potential to produce it; that the kinetic energy is derivable from the velocity potential; and that the added mass effects are the result of changes in the kinetic energy of the fluid; it reasonable to expect different added mass effects in the two different flow patterns [10]. The above example illustrates that the added mass does not depend on the acceleration of the body *per se*, so the added mass effect may differ under two sets of circumstances where the acceleration is the same. It is clear that, the change in the magnitude of the linear velocity would cause an inertial force in the *x*-direction in the first case, whereas the same acceleration in the second case would not produce any inertial force since the magnitude of the linear velocity of the cylinder and the total kinetic energy of the fluid remain constant in time.

Now, consider the flow around a circular cylinder oscillating transverse to a free stream. It follows from the foregoing that it is very unlikely that the added mass effect is the same as if the cylinder is oscillating in still fluid. However, that is invariably what is assumed in the classical approach to this problem in the literature. Let the cylinder undergo sinusoidal oscillations transverse to the free stream, its motion being described as a function of time by , where *A* is the amplitude of oscillation and *ω* is the cyclic frequency. According to the classical formulation, the potential-flow added mass depends only on the shape of body and the direction of motion [18], p. 567. Hence, the only component of the inertial force due to the added-mass effect acts in the transverse *y*-direction,
3.1where *m*_{A} is the added mass. For two-dimensional potential flow around a body with geometry having three planes of symmetry, the only independent components in the added mass matrix are the diagonal terms. For the circular cylinder, in particular, the two components (in Cartesian coordinates) associated with translational acceleration are identical and equal to the fluid mass displaced by the cylinder. Suppose one carries out experiments where *A* and *ω* can be varied at will, and the force acting on the cylinder is measured. A set of parameters is employed for which the amplitude is kept negligibly small but finite, and the frequency is varied from zero to a very large value. It is obvious that the flow pattern around the cylinder in all experiments would be similar to that around a stationary cylinder since the amplitude is negligibly small; hence, the kinetic energy of the fluid would also remain unaffected, and there would be no induced inertial force acting on the cylinder. However, the measured force is expected to vary in direct proportion to *ω*^{2} according to the classical formulation in (3.1). This is obviously impossible and is called here the added-mass paradox.

## 4. Derivation of the added-mass force on an oscillating cylinder

Consider a circular cylinder oscillating transversely in a free stream of steady uniform velocity, . The motion of the cylinder induces a time-dependent relative velocity where ** i** and

**are the orthonormal basis in a fixed frame of reference in Cartesian coordinates {**

*j**x*,

*y*},

*Y*(

*t*) describes the cylinder's motion and the overdot denotes a derivative with respect to time. The relative velocity vector can also be expressed in a frame of reference moving with the cylinder as

**(**

*U**t*)=

*U*(

*t*)

*e*_{η}where

*e*_{η}is the basis vector aligned with the instantaneous direction of the cylinder's velocity relative to the fluid in polar coordinates {

*η*,

*θ*} as shown in figure 2. In essence, the cylinder motion

*vis à vis*the free stream can be decomposed into an equivalent translation plus a rotation of the cylinder about its centre. It should be noted that the cylinder does not literally rotate in the fixed frame of reference. The acceleration can be obtained by differentiation of the relative velocity with respect to time in the frame of reference attached to the cylinder, 4.1where

*θ*is the angle defined in figure 2, is the angular acceleration of the frame of reference, and

*e*_{θ}the corresponding basis vector.

Each of the two terms on the right-hand-side of (4.1) is associated with independent components of the added mass matrix due to translational and rotational accelerations and corresponding inertial forces acting on the body in the general case of bodies with arbitrary geometry in two dimensions. For a circular cylinder, the potential-flow added mass for translation equals the fluid mass displaced by the cylinder, whereas the added mass for rotation about its centre is zero. Hence, only the first term contributes to the inertial force induced by the added mass effect, i.e.
4.2where *m*_{A} is the added mass of a circular cylinder associated with translation (the only non-zero term). Equation (4.2) shows that the induced inertial force is directionally varying (aligned with the instantaneous relative flow direction) and its magnitude varies in time according to the derivative of the relative velocity magnitude, whatever that may be. Hence, the result in (4.2) while specific to the circle is generally applicable to all kinds of oscillation, e.g. sinusoidal or non-sinusoidal, with a single frequency or multiple components, and even more broadly to all kinds of accelerations, e.g. impulsive or random.

If the cylinder oscillation is assumed for simplicity to be sinusoidal in time, , the magnitude of the relative velocity vector is given by
4.3where is the ratio of maximum transverse velocity of the cylinder to the free-stream velocity, called the alpha parameter hereafter. The inertial force in the moving frame of reference can be obtained by differentiation of (4.3) and substitution of the result in (4.2). Projection of the inertia force back into the fixed frame of reference can be accomplished *via* the transformation . After some mathematics, the following result is obtained for the inertial force in the *y*-direction:
4.4where *G*(*t*;*α*) is a dimensionless function of time which depends solely on the alpha parameter. This result demonstrates that the inertial force due to added mass is not only a function of the body geometry and the direction of motion, but also of the relative motion, embodied in the alpha parameter, in contrast to the classical formulation. The new result in (4.4) differs from the classic one in (3.1) only in the term *G*(*t*;*α*). However, this is not a simple correction. The analytical form of the *G* function is
4.5Figure 3 shows the temporal variation of the *G* function for different values of *α*. The *G* function is positively defined varying between minimum and maximum values of zero and (1+*α*^{−2})^{−1}, respectively. *G*(*t*) is a non-harmonic function of time with a fundamental frequency at twice the cylinder's oscillation frequency. Hence, *the added mass even for the simple case of potential flow around a circular cylinder undergoing sinusoidal oscillation transverse to a free stream gives rise to a complex non-harmonic inertial force in the same direction.*

As a side note, the inertial force induced in the transverse direction, *F*_{Iy}(*t*), comprises main components at the fundamental and the third harmonic of the cylinder's oscillation frequency since it approximately varies according to the following trigonometric relationship .

The contribution of the added mass to the transverse force for an oscillating cylinder can be appreciated from the variation of as a function of the alpha parameter, which is shown in figure 4. For *α*<0.1, the inertial force in the transverse direction is practically negligible since . For 0.1<*α*<10, increases rapidly tending asymptotically to unity as . For , the inertial force increases quadratically with the alpha parameter since .

## 5. Concluding remarks

In this paper, a new formulation is employed to derive the inertial force on a body oscillating transversely to a free stream due to the added-mass effect. The formulation is based on analysis of the body motion in a frame of reference fixed to the centroid of the cylinder moving through still fluid as this is a prerequisite for the proper definition of the added mass from the kinetic energy of the fluid. It is shown that the inertial force depends not only on the body geometry and the direction of its motion, but also on the flow pattern induced by the motion of the body relative to the free stream. This result stands in sharp contrast to the classical assumption that the inviscid potential-flow force is simply equal to the product of a shape- and motion-dependent constant added mass times the acceleration of the body. The latter assumption is frequently employed in the literature but is not physically consistent as demonstrated by a virtual experiment. This is so because the added mass resides outside the surface of the body, whereas in the classical formulation, it is replaced by an equivalent mass attached to the centroid of the body.

For a circular cylinder in sinusoidal oscillation transverse to a free stream, the dependency of the inertia force on the relative motion between body and fluid can be embodied in a single parameter called alpha, . Even in this simple case, the added-mass for potential flow gives rise to a complex directionally varying non-harmonic in time inertial force. Projection of the inertial force to the transverse direction yields a component which is negligible for *α*<0.1 but its contribution increases quadratically for *α*<0.5. It can be readily verified that the new result in (4.4) recovers the classical result in still fluid and dispels the added mass paradox in the presence of a free stream; i.e. the paradox is not a shortcoming of potential flow *per se* but of the limitation of an inviscid fluid to meet the boundary conditions of a viscous fluid.

A key finding of the present study is that a better way to conceptualize the inertia force due to the added mass is
For the specific configuration considered here, the added-mass acceleration is equal to the acceleration of the oscillating body *d*^{2}*Y* /d*t*^{2} times the dimensionless function *G*(*t*;*α*), whose analytical form can be worked out. For geometries other than a circle, the analytical calculation of the inertial force becomes more complicated because it comprises, in general, the contributions due to both translational and rotational accelerations and thereby different added mass components.

Based on the outcome of the present study, the physical meaning of the added mass is the following: as a body moves through a fluid, it displaces the fluid particles permanently in the direction of its motion; if the body accelerates part of the fluid close to the body will also accelerate while being pushed aside by the body. Thence, an additional inertial force is required to accelerate this part of the fluid. Even though the part of the fluid being accelerated by the body is infinitely widely spread in theory, this does not matter to the physical world; it has been demonstrated in a few studies that contributions of flow structures to the forces acting on the body decay rapidly with distance from the body [13,19]. This development is consistent with the equivalence of the added mass and the drift mass. The interested reader may wish to consult the resolution of the indeterminacy of the drift mass integral provided by Yih [8]. By virtue of its definition and its physical meaning, added mass is a positively defined quantity. The total added mass cannot be negative even though the drift mass may exhibit a local reflux. However, the component of the induced inertial force in the direction of body motion can become negligible under certain conditions as is shown in the present work.

Added mass is a generic property in fluid mechanics and its fundamental understanding has far-reaching implications in problems involving flow–structure interactions. In particular, added mass has a significant effect on the vortex-excited oscillation of cylindrical structures when the density of the structure is comparable with the fluid density, which has been the subject of much debate in the literature [2]. It is expected that the present study will shed new light on this outstanding issue.

- Received February 25, 2013.
- Accepted May 9, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.