Fluid Dynamics NorthHolland
Research
2 (1987) 1523
15
A new expression of force on a body in viscous vortex flow and asymptotic pressure field T. KAMBE of Physics, University of Tokyo, Hongo, Bunkyoku,
Department Received
17 January
Tokyo Il3, Japan
1987
Abstract. Unsteady vortex flow around a fixed solid body in a viscous incompressible fluid is investigated for the case where the velocity field is assumed to vanish at infinity. Consideration of the asymptotic pressure field far from the body leads to a new formula for the force acting on the body, which is given by a volume integral whose integrand is linear with respect to the vorticity and does not include the velocity. This is facilitated by using a renormalized Green’s function introduced by Howe. The formula offers an interesting interpretation for the force in the case of inviscid vortex rings moving near the body: that is, the force is proportional to the rate of change of volume flux through the rings of an imaginary potential flow around the body. The relation of the present subject to the excitation of acoustic waves by vortex motion moving near a compact body is considered.
1. Introduction In steady flow of a viscous fluid past a solid body, the force acting on the body is related to the difference in momentum flux between that entering through a plane far in front of the body and that leaving a plane far behind it. The net difference is reduced to the total loss of momentum flux in the wake far behind the body. In the present study we suppose that the fluid is at rest at infinity, and seek a similar relation between the force on a solid body and the asymptotic field far from it. In the theory of aerodynamic sound, the presence of a solid body in the vicinity of a vortex motion is known to be a source of wave fields characterized by the dipole radiation (Obermeier 1980, Kambe 1986 for an inviscid fluid). The dipolar emission of sound waves is closely related to the force exerted on the body (Curle 1955 for a viscous fluid). Curle attributed to this mechanism the excitation of the Aeolian tones emitted by wires in wind. Miyazaki and Kambe (1986) showed in an axisymmetric problem that, when a solid body is placed in an incompressible inviscid flow of a moving circular vortex ring in axisymmetric manner, the body will experience a timedependent force F in the axial direction, which is related to the coefficient A of a dipole term in the velocity potential by F=
p&l(r),
0.1)
where p is the fluid density and the velocity potential $I has an asymptotic expression: +  (A(t)/4~)xJx~ (xi: Cartesian coordinate, x: magnitude of the vector (x,)). We consider here a localized flow around a fixed solid body in a viscous incompressible fluid. Hence it is assumed that there is no uniform flow at infinity and that the velocity decays at distances far from the body. In this circumstance the flow around the body is rotational intrinsically. A typical example is a viscous vortex ring moving near a solid body. Associated 01695983/87/$3.25
0 1987, The Japan
Society of Fluid Mechanics
T. Kambe / A new expression of force on a body
16
with a net force Fj exerted by the viscous flow, the pressure field takes the asymptotic form of a dipole potential. It will be shown below for a general localized viscous flow that the dipole strength is proportional to the force Fj. Therefore, if one knows the dipolar asymptotic field, one readily finds the force on the body. Basically the force exerted on a solid body is defined by the surface integral of the mechanical stress distributed over the body surface. Once an incompressible velocity field U, is known, the pressure satisfies the Poisson equation with the source term  pa2u,uj/axJxj. Then the pressure is represented in an integral form by using a Green’s function, based on the Green theorem. In Section 3 we will present two Green’s functions, which lead to two different, but equivalent, expressions of the pressure. Their asymptotic forms at large distances are of dipole type. The equivalence of the two expressions yields two representations of the force on the body in a viscous flow: one is in the form of a surface integral, as mentioned above, and the other in the form of a volume integral, which is new. In deriving the second expression we use a renormalized Green’s function developed by Howe (1975a) and Obermeier (1980). The volume integral does not require the representation of the velocity field and is evaluated once we know the timedependent vorticity field. In Section 4, it is verified directly without recourse to the asymptotic analysis that the second force expression reduces to the first one. The expression of the volume integral offers an interesting interpretation for the force, especially in the case of vortex rings moving near a solid body in an inviscid fluid. It is shown in Section 5 that the force on the body is proportional to the rate of change of the volume flux (through the rings) of an imaginary potential flow around the body. Applying the last result to the problem of aerodynamic sound, in which it is known that the acoustic pressure of the dipole wave is proportional to the rate of change of the force on the body, we find that the wave pressure should be proportional to the second time derivatives of the volume flux. From the experimental observation of the acoustic waves generated by a vortex ring moving near a circular cylinder (Kambe, Minota and Ikushima 1986, Kambe 1986, Minota and Kambe 1987), it has been shown that the observed wave profile is reproduced by the above theoretical law with reasonable accuracy.
2. Formulation We consider an incompressible flow of a viscous fluid of uniform density p around a fixed solid body which is assumed to have a length scale 1. The body surface is denoted by S. The governing equations are the continuity equation and momentum equation
aul
axi
(2.1)
‘3
(2.2) where U, is the velocity component, assumed. The tensor Ti is given by qj = pvivj  a,,, where ajj is the viscous
p
is the pressure,
and
the summation
convention
is
(2.3)
stress tensor, (2.4)
T. Kambe /A new expression of force on a body
~1 being obtain
the shear viscosity
constant.
Taking
the divergence
11
of eq. (2.2) and
using
(2.1)
we
(2.5) Note that a27;j/axi8xj = pa2(uiuJ)/axiaxj due to eq. (2.1). However, the use of 2;/ is more useful in deriving the final formula (3.9), as will be seen in the next section. Based on the assumption that there is no uniform flow at infinity and on the wellknown property of the incompressible flow at rest at infinity (without any simple source of fluid), the magnitude of velocity decays as xP3 when the distance x = 1x ( from the origin, which is taken inside the body, becomes sufficiently large (Batchelor 1970). The boundary conditions are (I)
(a)
u=O
(2.6a)
on S,
(or, in an inviscid [0(x,
(II)
t) 1 = 0(xP3)
where n is a unit relation on S,
outward
ap Or,
(2.6b)
as x + co,
normal
ax+&Tj=O, I
(b) n. u = 0 on S)
fluid,
(2.7)
to S (from
the fluid).
The first condition
(I) yields
the
(2.8a)
atxonS,
J
ap ax,
ni
i
+
$T.’ J
=
0,
i
i
(2.8b)
at x on S , 1
in view of eq. (2.2). The following analyses can be applied to an inviscid fluid as well. Therefore the boundary conditions for the inviscid case are given in parentheses. In the following section, integral expression of the pressure p is derived by applying Green’s theorem to eq. (2.5) and using an appropriate Green’s function. Two kinds of Green’s function will be presented below, which yield two equivalent expressions of the pressure.
3. Expressions of pressure Green’s
formula p(x,
p
for the pressure /G(x,
t> = 
and a function
y) vzp(y,
G is written
t) d’Y+L[G
in the form
VPP&G]
V
where Green’s
function
v2G(x,
*n ds,
(3.1)
G satisfies
y) = 6(xy),
(3.2)
the first volume integral with respect to the variable y is taken over unbounded space V outside the surface S, and V, is the gradient operator with respect to the vector y. To the factor v ‘p in the volume integral, the expression (2.5) is substituted. (A) First we take the free space Green’s function,
G,(x,
u>= 4n,;_y,
(3.3)
7
and substitute it into (3.1). By using the property surface integral in (3.1) is rewritten as G
*ni
/s %
dS+
$JG,pni I s
dS.
aG,/ayi
=  aG,/ax,
for this form of G,, the
(3.4)
T. Kambe / A new expression of force on a body
18
The volume
integral
in (3.1) together
with (2.5) is transformed
to
7;, d3_v jv6v2p d3.v=jv(+&,

= a;. J
js
G,%;
dS+&
aYj
jGi$j I
d3y,
v
(3.5)
J
where the second equality is obtained by carrying out partial integration and eliminating surface integral over a large sphere at infinity by the condition (2.7). Further integration part for the second volume integral of (3.5) leads to
aT. /s Introducing
G+ni
a?/
a2 dS + ax+,
dS + ? jGi7;i”i ax, s
j v GA,
d3y.
the by
(3.6)
(3.4) and (3.6) into (3.1) we obtain
P(X, t) = S,G1 ay (~+~jnidS+~~G,(ps.,+~,)n,dS a2 + axiaxJ / vGIT~ Due to the boundary vanishes. In addition,
(3.7)
condition (2.8a) or ((2.8b) for an inviscid it is noted that at x on S we have
( pSij + q.j)nj The first equality expression is just the body) by the Thus, finally, we T,(x, t) and the
d3y.
dS = (p&,
 a,,)nj
dS,
case) on S, the first integral
=f, dS.
(3.8)
is simply the consequence of the condition (2.6a) (or (2.6b)). The middle the stress exerted on the body surface dS with the normal nj (pointing into surrounding fluid, which is denoted by f,dS (a;, = 0 for the inviscid case). obtain a first expression of the pressure p at x and t in terms of the tensor surface force f. on S: d3y.
(3.9)
(Similar analysis for the wave equation of aerodynamic sound instead of the Poisson equation (2.5) will be found in Curle (1955).) The first term represents a dipolar pressure field concerned with the surface distribution of the stress f,, whereas the second term represents a quadrupolar pressure field generated by a rotational flow around the solid body. Note that aT,,/ax, is expressed in terms of the vorticity w = v x 2) as aT,J/axj = p(w x u)~ + a(ipu2)/ax, + ~(0 x w),. The pressure field far from the body (x X 1) is particularly interesting. When the distance is large enough, the Green’s function is expanded in the following asymptotic series: 1 1  y,g; + 0(xj =41T ( x , 4. this into (3.9) and retaining only the leading
1 4,rrlxyl Substituting
p(x,
t) =fJ(t)D;(x)
+o(x3),
x
(3.10) term, we have (3.11)
where
4(t) = Lids=
J,(Pa,,ulj)n, ds,
(3.12) (3.13)
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T. Kambe / A new expression of force on a body
20
It is now shown that the expression (3.16) is valid up to the terms of 0(x‘). First, the function G2 satisfies the boundary conditon on S. Differentiating (3.16) with respect to y,, V G = (xiY
2
Y)V,Y
4alx
Yl3
.
Since v Y denotes the velocity of a potential flow around the body, n .v y = 0 is satisfied on S. Therefore the condition (3.14) is satisfied. Second, although the function G, is considered at y near the body in the following analysis, one wouold expect G2 to show such a singular behavior when y tends to x as required for the Green’s function. The function satisfying (3.2) must have a singular behavior G  1/[41rI x  y I] as 1xy 1 + 0. When y/I becomes large, q(y) tends to y, since ai( y) = O(ye2) for a compact body. Thus, G,(x,
1 Y)= L 4lT lxy+o(y2)
(
for y/l B 1. The location of the singularity in the y space is displaced only by the relative order x 3 from the correct position y =x. It is found that the function G, has a correct singular behavior at a displaced position. This displacement is not important because we are interested in G, only at y near the body. Third, the Green’s function G2 satisfies the equation v 2G = 0 up to the terms of O(xe2). In order to show this, we develop (3.16) for x/l z+ 1 as (3.19) The property just mentioned is evident in this form since y(y) is a harmonic function of y. Note that the 0(xe3)term is (1/8a)Y( y)q( y)(i32/ax,i3x,)xP’, which is not harmonic in general with respect to y. Thus it has been found that the function G, defined by (3.16) and (3.17) satisfies the required properties of the Green’s function up to the 0(xe2) terms. This is sufficient for our purpose since the formula considered below depends only on these terms. The second pressure formula (3.15) requires the representation of v,G,, which is given the following form:
vyG,= oi(x)V,Y(
y) + O(X~),
(3.20)
from (3.19), where 0, is given in (3.13). With restriction to the 0(xP2)term, v,G, is proportional to the velocity v Y,( y) at a fixed (observation) point x (therefore Oi( x) being a constant vector). Evidently the imaginary flow represented by V, = v Y is incompressible, i.e., div V, = 0, for each i = 1, 2, 3. Therefore one may introduce a vector potential !Pi by the relation vY=v
XT,
(= I$
(3.21)
with the condition div ‘k; = 0. The function qL is a vector for each i (= 1, 2, 3). In cases of two dimensions or axissymmetry, the vector potential !$ is related to the stream function. Since V x (v x ei)=  v '!Pj = 0,each component of !I’, is a harmonic function in the unbounded space outside S, and may be chosen such that !Pj = 0 on S. (The function !Pi may be divided into a part !Pil representing the uniform velocity ej, where ei is a unit vector in the y,direction, and a part !Pi2 representing the perturbation due to the body. The part qil is given by &ej x y. The second part is a harmonic function decaying at infinity and satisfying the boundary value  :eiXy at y on S.) Substituting (3.20) into (3.15) one obtains
Z’(x, t> =
4/voY(
y) . ( P$
+ VP) d3y + O(x3).
T. Kambe / A new expression offorce on u body
21
Use of the relation (3.21) transforms this into the following concise form: P(X, t) = Gj~~)~i(~~ + O(L3),
(3.22)
where
G;(t) = 
lI’v x ‘&).
(p$+vp)d3Y
(3.23)
(3.24) where w = D X o. Again it is found that the pressure (3.22) is represented asymptotica~y.
by a dipole term
4. Equivalence of two expressions of force The two pressure expressions, (3.11) and (3.22), must be identical. This leads to I;J.(t> =Gi(t), where Gj is given by (3.24). Thus we have 6(f)=
~;j$(ybo(p.
t)d3y.
w
This describes the fact that the ith component of the force F is related to the time derivative of the volume integral of the inner product of o and !$ where !Pi is the vector potential of an imaginary potential flow around the body with a unit velocity in the Y,?direction at infinity. The function ‘k; is independent of the time t. Therefore the force depends on the dynamics of the vorticity w(x, t). In summary, when there exists a vortex motion near a solid body, it will experience a net force given by the formula (4.1). We have obtained formula (4.1) from the equivalence of two coefficients of the as~ptotic pressure. However, the equivalence of the two expressions I;I and G, should be derived directly without recourse to the asymptotic form of the pressure, It is now shown that the expression (3.23) reduces to (3.12). Reminding relation (3.21) with r; of (3.17), we obtain
Substituting this into (3.23) yields
where eq. (2.2) is used together with the properties that au,,JaY, = Cland v'$ volume integrals are transformed to the surface integrals:
= 0. All three
T. Kambe /A
22
new expression of force on a body
The second term vanishes by the use of Condition (I) of Section 2, while the integrands of the first and third integrals take the forms uljnj (since ui = 0 (or n,uJ = 0) on S) and p6,, (by the relation (3.18)) respectively. Thus we find G,=
Js
(pSijui,)n,
dS.
This is just the expression of F,.
5. Vortex motions Formula (4.1) is valid for an inviscid fluid as well as a viscous fluid, as noted in the parentheses in the course of derivation. The vorticity w(x, t) in the inviscid fluid is governed by the equation ~w+vx(wxl;)=o.
(5.1)
Suppose that the vorficity consists of n thin closed vortex tubes, and that the flow is irrotational outside the tubes. The k th tube is represented by the closed curve C, of the center line and its crosssection area uk. With the notation ds(k) for an infinitesimal line element, the integrand wd3y(k) is given by the “mean value expression” I w lo(~,cds(k)
= r, d+),
r!f=
IwIo(J/c,
where 1w lo is a representative magnitude of the vorticity over the crosssection strength of the k th vortex which is invariant along the tube. Thus we have jv?P;u
d3y=
;
[email protected]
k=l
and I’, is the
‘k,. ds(k) ck y=v
x
*;,
(5.2)
by the Stokes theorem, where Sk is an open surface bounded by the curve C, and n is a unit normal to S,. The vector y is the velocity of an imaginary potential flow around the body. Here one may define the volume flux q.(k) of the flow through the closed curve C,: J,(k,
t) = LLQ
dS.
The flux J, depends on the position of the k th vortex relative to the body, and therefore depends on the time t because the vortex position changes, although V, is timeindependent. Using (5.2) in (4.1) we finally obtain r;l=
p$ i: k=l
TJ(k,
t).
(5.3)
Thus it is found that, when a body is located near an interacting system of n vortex rings, the force on the body is equal to the time derivative of the sum of the volume flux J,( k, t) (through the k th vortex) of a potential flow around the body, multiplied by  p. This may be regarded as a generalization of the aeroacoustic result of Howe (1975b), who showed that, in a twodimensional problem of sound generation by a line vortex moving near a rigid halfplane, the sound intensity is determined by the rate at which the vortex traverses the streamlines of a potential flow about a sharp edge.
T. Kambe / A new expression
of force on a body
23
In a viscous fluid, the solid body interacting with vortices may shed another vortex, which will cause a reaction force on the body. This effect is also evaluated with the above formula insofar as the vorticity field is known.
6. Summary and discussion
When the velocity of an incompressible flow around a fixed solid body in unbounded space vanishes at infinity, the pressure asymptotes a dipole form 4D, at distances far from the body, where 4 is a component of the force acting on the body (i = 1, 2, 3) and given by a surface integral of the stress distributed over the body surface. Using a renormahzed Green’s function, we have obtained a second expression for the dipole coefficient 4 in the form of a volume integral. Equivalence of the two coefficients yields the force expression in the form of a volume integral taken over the entire flow field. It has been verified directly that this second expression of the force reduces to the first one. Further, introducing a vector potential for an imaginary potential flow around the body with a unit velocity at infinity, we have found that the second expression is transformed into the time derivative of the volume integral of the inner product of the vorticity and the vector potential, multiplied by  p. The integrand has a linear dependence on the vorticity. In a study of the influence of solid bodies on aerodynamic sound, Obermeier (1980) obtained a similar formula by introducing a vector Green’s function. The new formula of force is valid even in an inviscid fluid, in which the vorticity is often localized in space and therefore the volume integral is conveniently restricted to the regions of nonzero vorticity. When applied to a system of n interacting vortex rings in the presence of a solid body, the force acting on it is proportional to the time derivative of the sum of the volume flux J,(k, t) through the k th vortex of the imaginary potential flow. Reaction force from the vortex shedding in a viscous fluid is also included in the formulation. The present problem is motivated in the investigation of generation of acoustic waves by a vortex ring moving near a body (Kambe 1986, Kambe, Minota and Ikushima 1986). The vortex motion in an incompressible fluid induces a timedependent dipolar pressure field at large distances, a solution of the Laplace equation. This disturbance drives a dipolar acoustic wave, a solution of the sound wave equation. The wave amplitude thus generated is proportional to the second time derivative of the flux T(k, l), which is compared with the force on the body proportional to the first derivative, reflecting the fact that the rate of change of the force generates a dipolar sound wave.
References
Batchelor, G.K. (1970) An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge) Section 2.9. Curie, N. (1955) Proc. Roy. Sot. London A 231, 505514. Howe, M.S. (1975a) J. Fluid Mech. 67, 597610. Howe, MS. (1975b) J. Fluid Mech. 71, 625673. Kambe, T. (1986) J. Fluid Mech. 173, 643666. Kambe, T., Minota, T. and Ikushima, Y. (1986) in: G. ComteBellot and J.E. Ffowcs Williams, eds., Proc. ZUTAM Symp. on Aero and HydroAcoustics, Lyon 1985 (Springer, Berlin). Minota, T. and Kambe, T. (1987), J. Sound Vib.Z19(3), (December). Miyazaki, T. and Kambe, T. (1986) Phys. Fluids 29, 40064015. Obermeier, F. (1980) J. Sound Vib. 72, 3949.