(1)
Huang Shehua Li Wei
Department of River Engineering
Wuhan University, 430072, P. R. China
E-mail: hshh2@263.net
Chung-Hwan CHUN
Dept. of Mechanical Engineering
Pohang University of Science & Technology
San 31, Hyoja-dong, Pohang, Republic of Korea
Abstract: In this paper, a concept of virtual mass force in viscous flows is introduced through rigorous analysis of a particle transient motion at very low particle Reynolds number. For the particle transient motion at high Reynolds number in viscous flows, the corresponding virtual mass force is researched by method of vorticity analysis. The result is shown that the virtual mass force acting on a particle accelerating at high Reynolds in viscous flows has the same formulation as that in nonviscous flows.
Keywords: virtual mass force, particle Reynolds number, particle transient motion
In two phase flow or multiphase flow, small particle’s transient motion is very complicated. For the transient motion of one particle, the complexity mainly lies in interactions between the particle and fluid, that is, the unsteady drag of flow around the particle. For high particle Reynolds number, our current knowledge of transient forces on particles are still limited. It is difficult to present a universal mathematical model for particle unsteady motion in fluids theoretically.[1-4] It is well known that object will be subject to virtual mass force when it translates in ideal and potential flow unsteadily, and magnitude of the virtual mass is equal to one half of the fluid mass which it expels out. However, in real fluid due to the viscosity forces acting on moving objects become more complicated. So far there has not been an obvious definition and complete research on virtual mass force of an object in viscous flow. In this paper we first analyze the spherical particle’s transient motion at very low Reynolds number, with a definition of virtual mass force on a particle in viscous flow being introduced. Then based on the vorticity method, the virtual mass force on a particle translating in infinite viscous flow field unsteadily at high Reynolds is explored theoretically.
Assume that a particle moves unsteadily along a beeline in state and unbounded flow field where its velocity relative to fluids is equal to its absolute velocity. When particle Reynolds number << 1, the fluid flow due to the particle motion can be described approximately by Navior-Stokes equation with convection term being neglected. Equations of stream function and vorticity hence have the following simplified forms for time-dependent axisymmetric flow in spherical reference frame: [5]
(1)
(2)
where is Stokes operator and defined by
(3)
Substitute equation (2) into (1) to eliminate vorticity, and we obtain
(4)
Equation (4) can be resolved with following given boundary conditions:
(5)
According to the characteristics that a general solution to linear differential equation can be got by combining of some special solution, particle velocity is expressed as the corresponding Fourier series:
(6)
For any Fourier component of particle velocity
in Fourier space
, particle motion is completely similar to that of a small sphere vibrating
along beeline in unbounded flow field. The forces on small oscillating sphere
are as follows: [7,8]
(7)
Integrated at interval (-¥，+¥) with respect to
, equation (7) becomes the general drag on a particle moving at arbitrary
velocity
, that is,
(8)
Denote the three terms
on the right-hand side of equation (8) with
、
and
respectively, then we get
The first term:
(9)
the second terms：
(10)
Differentiate on both sides of equation (6) with respect to time t：
(11)
Substitute equation (11) into equation (10) and obtain
(12)
The third one:
(13)
It can be proved that the real part of integral in equation (13) is an even function because of
when
; and
when
.
Therefore, equation (13) becomes
(14)
For equation (11) a
reverse Fourier transformation is made[9]：
(15)
On both sides of
equation (15)
being multiplied and consequential
result substutided into equation (14)， the equation is
further reorganized and its order of integral is exchanged to yield
(16)
The real part of above equation having been separated and some simplifications made it, one at last get following result:
(17)
By considering equation (9), equation (12) and equation (17) together, the total drag on a particle is
According to the
assumption of beginning, in above equation
is particle velocity in quiecent
flow field. Hence, for a uniform flow of movement
should be regarded as particle
velocity related to fluid. Subsituting
for
in above equation, it becomes
(18)
It is shown from equation (18) that the general drag for particle transient motion at low Reynolds number comprises three terms. The first term is referred as to quasi-steady Stokes drag because it has the same formulation as that on a particle moving steadily in Stokes creeping flow. The second term takes its form of the virtual mass force on a particle accelerating in idea non-rotational flow. The physical meaning of virtual mass force in idea non-rotational flow is that the particle pushes part of fluids to move at the same acceleration as it accelerates in flow field, and the volume of the fluid part which seems to be attached to the particle is just equal to a half of the particle volume. The third one is so-call history integral force or Basset force, reflecting an extra contribution of fluid viscosity to the drag of particle transient motion.
If three components of general drag on a particle are classified by their relationship to particle acceleration, the general drag is composed of two parts: quasi-unsteady force and unsteady forces. If the three components of general drag are classified according to their relationship to fluid viscosity, the general drag can be divided into another two parts, that is, viscous forces and non-viscous force. For last classification, non-viscous part of general drag is equivalent to virtual mass force on a particle accelerating in idea non-rotational flow. Based on above analysis, here we present a general definition of virtual mass force in both viscous and ideal fluids. An virtual mass force is a kind of fluid force on an object accelerating in fluid which has nothing to do with the fluid viscosity and can be specified by the object and its acceleration in relation to fluid.
In a infinite incompressible flow field around rigid particles, take fluids and particles as a united fluid- solid coupled kinematic system. Particle velocity can be written as
where
is rate of rotation of particle motion. It follows that a
common kinematic constraint is satisfied by velocities of both particles and
fluids, that is,
For a particle transient motion in unbounded 3D flow field, the vorticity field due to particle motion will focus on finite areas of vicinity to particle position and its wakes. Because vortex line can not be ended on particle surface, which implies that all vortex lines will be closed circles in the united kinematic system of particle and fluid, following conservation equation is valid:
(19)
It is illustrated from equation (19) that general vorticities are zero in united kinematic system of particle and fluid. Based on this conservative property of general vorticities, the relation of vorticity integral in fluid domain to that in solid area is deduced as follows,
where
、
are volumes of fluid and particle respectively. It
can be found from above relation that total vorticites are finite value in
unbounded flow field.
Taking curl on both sides of Navior – Stokes equation of incompressible flow, one gets a following vorticity transport equation:
(20)
In order to get the
asymptotic property of vorticity distribution in far field from particle center,
we ignore convection term in equation (20) and assume that at a space position r and time
the vorticity due to particle
motion can be described with a transient source of space. For any time t>
, a basic solution of vorticity distribution in
unbounded flow field is
If the vorticity
due to particle motion in in
unbounded flow field R at tome
is given, for any time t, total vorticity in the space is calculated by following integral:
(21)
where subscript “0”
refers to
space in which the integration is made. It is shown from
equation (21) that at any following time t the vorticity at position of flow
field decays and trends to zero exponentially. The region of no-zero vorticity
is finite. For situations that the convection term of vortex transport equation
is not negligible, vorticity formulation of flow field will be different from
equation (21), while the above asymptotic property of vorticity field is not
changed by its convection because convection velocity of the fluid is a finite
value. By means of Biot-Savart formulation, we can further get asymptotic
velocity in far field of vorticity as follows:[11]
(22)
where is
the impulse of unbounded flow
field.
For particles motion along a beeline in
unbounded flow field which is quiescent in infinite region, as discussed above,
the vorticity of flow field exits in the vicinity of particles and their wakes,
which shows that .it is reasonable to assume that the field of vorticity due to
particle motion is within a finite space region, with vorticity intensity
decreasing along distance r apart from particle center and trending to zero at infinite.
Therefore, we can draw a spherical surface
which is big enough to compass the
particle and associated vorticity field, as depicted in Fig. 1. In terms of
momentum equation, we attain
Fig.1 A
sketch of particle motion in infinite flow field
Where
is total forces of fluid on a
particle;
is velocity vector of fluid flow
due to partical motion; p is pressure on surface
; and
is fluid volume embodied by
spherical surface. In above equation by plusing and substracting a term
，it yields
(23)
When sphere radium is
large enough, on spherical surface
unsteady Beroulli equation is
applied to determine pressure p as follows:
According to previous
assumption that the velocity of flow trends to be zero in the infinite, that is,
～
, we deduce that
is negligible. From equation (22)
velocity potential function of flow field can be expressed as
Then a surface integral
of pressure is calculated as follows
Noting the relations of
and
on the spherical surface
, we get
Since I is a constant in the
integral domain, there is
and further we obtain
(24)
When the sphere space is
full of fluids, momentum of fluids within sphere is defined by following
expression[12]
(25)
Substituting equation
(24) and (25) into (23) yields
(26)
Equation (26) represents the total forces on a particle translating arbitrarily along a beeline.
It is shown from equation (26) that the total
forces on a particle consist of two parts: one part is a derivative of the space
integral of vortcity distributed in flow field with respect to time, and the
second part is a derivative of momentum of fluids excluded by the particle with
respect to time, while the fluids has the same volume as the particle. Because
the excluded fluids and their volume is virtual, which is easy to understand
from above discussion, and their velocity field is unknown in space
, it is impossible to calculate the second part of total forces directly from
equation (26). However, when we think of particle motion in irrotational flow,
as
is true at any position, equation
(26) becomes
According to theories of moving objects in irrotational flow (Lam, [13]), it is well known that
(27)
where Fm is virtual mass force on a moving object. It is shown that the second term on the right hand side of equation (26) is an equivalence of the virtual mass force exerted on a particle accelerating in irrotational flows.
For the first term on the right hand side of equation (26), we consider if it contains any component independent of fluid viscosity and determined only by relative acceleration of fluids. Based upon Helmholtz equation of incompressible flow [which is the equation (20) with viscous dissipation term being neglected], the first term on the right hand side of equation (26) can be written
From previous assumption that when r®¥ there is w =0, in above equation the surface integral is equal to zero. While for volume integral of the second term on right hand side of above equation, because of
we get
Similarly, due to the fact that when r®∞ there is v =0, the above two surface integrals are equal to zero. Therefore, the first term on the right hand side of equation (26) becomes
(28)
Substituting equation (27) and equation (28) into equation (26) and writing it in form of component of tensor, at last we obtain the following result:
(29)
Since equation (26) represents the total forces on a particle accelerating rectilinearly in flows, it is concluded from equation (29) that the virtual mass force acting on a particle moving unsteadily in viscous flows has the same formulation as that in nonviscous flows.
References
[1] E. Michaelides, Reviews—The transient equation of motion of particles, bubbles and droplets, J. of Fluid Eng., Vol. 119, 1997.
[2] Mei, R., Flow due to an oscillating sphere and an expression for unsteady drag on on the sphere at finite Reynolds number, J. of Fluid Mechanics, Vol. 270, 1994, pp133-174.
[3] Maxey, M. R. and Riley, J. J., Equation of motion for a small rigid sphere in a non-uniform flow, Phys. Fluids 26(4),1983.
[4] Rizk, M. A. and Elghobashi, S. E., The motion of a spherical Particle suspended in a turbulent flow near a plane wall, Phys. Fluids 28(3),1985.
[5] White, F. M., Viscous Fluid Flow, McGraw-Hill Book Company, 1982.12.
[6] Fan Yinchuan, Higher Mathematics, Higher Education Press, 1978.
[7] Yi Jiaxun, Fluid Mechanics, Higher Education Press, 1983.12.(in Chinese)
[8] Landau, L.D. and Lifshitz, E.M., Fluid mechanics, Translated from the Russian by Peng Xulin, Higher Education Press, 1978.(in Chinese)
[9] Wang Huzheng et al，Handbook of modern engineering mathematics，Vol. 1, Press of Huazhong Engineering College, 1985.
[10] Wu, J. C., Theory of for Aerodynamic force and moment in viscous flows, AIAA Journal Vol. 19 No.4, 1980.
[11] Tong Binggang，Theory of Vorticity motion, Press of China University of Science and Technology, 1994.
[12] Bachelor, G. K., An introduction to fluid dynamics, Cambridge University Press, 1994.
[13] Lamb, H., Hydrodynamics, 6th ed. Cambridge University Press, 1932.