PHYSICS OF FLUIDS
VOLUME 16, NUMBER 4
APRIL 2004
Two-fluid jets and wakes
Andrzej Herczynski
Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467-3811
Patrick D. Weidman
Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309-0427
Georgy I. Burde
Jacob Blaustein Institute for Desert Research, Ben Gurion University, Sede-Boker Campus 84990, Israel
共Received 11 August 2003; accepted 8 January 2004; published online 8 March 2004兲
Similarity solutions for laminar two-fluid jets and wakes are derived in the boundary-layer
approximation. Planar and axisymmetric fan jets as well as classical and momentumless planar
wakes are considered. The interface between the immiscible fluids is stabilized by the action of
gravity, with the heavier fluid, taken to be a liquid, placed beneath the lighter fluid. Velocity profiles
for the jets and the classical wake depend intimately, but differently, on the parameter ␹
⫽ ␳ 1 ␮ 1 / ␳ 2 ␮ 2 , where ␳ i and ␮ i are, respectively, the density and absolute viscosity of the fluid in
the upper (i⫽1) and lower (i⫽2) fluid domains, while the momentumless wake profile depends on
the parameter ⍀⫽ ␳ 1 ␮ 32 / ␳ 2 ␮ 31 . Generally, all interfaces deflect from horizontal except the fan jet.
However, while the interface for the classical planar two-fluid wake is never flat, the interfaces for
the planar jet and the momentumless wake become flat in the particular case ␮ 1 ⫽ ␮ 2 . Velocity
profiles illustrating the strongly asymmetrical jet and wake profiles that arise in air-over-water,
oil-over-water, and air-over-oil flows are presented. © 2004 American Institute of Physics.
关DOI: 10.1063/1.1651481兴
two disparate fluids was reported by Wang10 and their oblique impingement was studied by Tilley and Weidman.11 In
the latter case, an oblique stagnation flow of upper fluid 1
asymptotically inclined to the horizontal at angle ␪ 1 impinges on a lower layer of immiscible fluid 2. The problem is
to determine the asymptotic response angle ␪ 2 of the oblique
stagnation flow that ensues in the lower fluid. These
stagnation-point flows are not strictly boundary-layer flows
since they provide exact solutions of the Navier–Stokes
equations. A practical application of planar normal two-fluid
stagnation-point flow analysis to a swept wing problem is
reported in Coward and Hall.12
Another two-fluid flow that may be analyzed using
boundary-layer techniques is the jet discharge of fluid 1 into
a quiescent immiscible fluid 2. This problem has attracted the
interest of many researchers because of its practical importance, particularly when gravitational effects are included.
Here we only cite the work of Burde,13 who, with gravity
excluded, found an explicit boundary layer solution for penetrating free jet flow in the special case ␹⫽1 with the aid of
von Mises variables. This restricted case was further exploited to determine analytical boundary-layer solutions for
the wall jet and a weakly swirling free fan jet of fluid 1, in
each case penetrating into the quiescent domain of immiscible fluid 2. Exact solutions of the Navier–Stokes equations
describing the motion of a two-dimensional two-fluid cylindrical source are reported in a recent publication by
Putkaradze.14
In the present study we investigate yet another class of
laminar boundary-layer flows referred to as two-fluid jets
I. INTRODUCTION
Two-fluid flows appear in many industrial applications.
Examples are given by Joseph and Renardy1 in their book
Fundamentals of Two-Fluid Dynamics. The problems considered in that work do not dwell on boundary-layer type flows
which is the focus of the present investigation. Boundarylayer flows involving two fluids often occur in engineering
practice as illustrated in the studies reviewed below.
The first important paper involving a boundary-layer
analysis of a two-fluid flow appears to be that of Lock2 who
investigated the spatial development of two horizontal immiscible fluid streams moving parallel to each other at speed
U 1 for the upper fluid and speed U 2 for the lower fluid. He
found that this locally driven flow is described by, in addition
to the velocity ratio ␭⫽U 1 /U 2 , the parameter ␹
⫽ ␳ 1 ␮ 1 / ␳ 2 ␮ 2 , where ␳ i and ␮ i are the densities and absolute viscosities of the two fluids. The radial counterpart of
Lock’s problem—the steady laminar shear layer between two
radial streams of incompressible and immiscible fluids—is
considered in Katoshevski et al.3 In the same spirit, Wang4
considered spatially developing boundary layers produced by
the uniform shear flow of one fluid streaming over a second,
quiescent, heavier immiscible liquid.
We will also mention in passing the works of Ting,5
Libby and Liu,6 Mills,7 Klemp and Acrivos,8 and Small,9 in
which the problem of the laminar mixing of two parallel
streams of the same fluid was discussed and debated. Since
there is no interface, those works do not bear directly on the
subject of the present investigation.
The normal impingement of stagnation-point flows of
1070-6631/2004/16(4)/1037/12/$22.00
1037
© 2004 American Institute of Physics
Downloaded 04 Jan 2005 to 136.167.58.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
1038
Phys. Fluids, Vol. 16, No. 4, April 2004
and wakes. Like the problems13,14 described above, these are
not locally driven flows; rather, they owe their existence to a
source 共jet兲 or a sink 共wake兲 of momentum upstream of the
spatially developing flow. In the quiescent state, the two immiscible fluids are separated by a horizontal flat interface,
with the heavier fluid residing below the lighter fluid to ensure hydrostatic stability. For the planar jet, imagine a narrow slot straddling the interface through which one or both
fluids are discharged parallel to the interface into the quiescent two-fluid ambient. The interface is free to deform, owing to normal viscous stresses and pressure forces acting in
concert with gravity at the material interface. Unlike the planar jet, it will be shown that the radial two-fluid jet emanating from a horizontal cylindrical slot into a quiescent twofluid ambient maintains a flat interface for all values of the
governing parameter ␹. For both planar and radial jets, the
ensuing motion far from the source does not depend on the
relative momenta discharged in each fluid, only on the total
momentum J passing through the slot. Indeed, all the momentum could be imparted by the discharge of just one fluid
for a slot with one edge coincident with the interface. The
momentum acquired asymptotically far downstream by each
fluid is a manifestation of lateral diffusion across the immiscible interface. The two-fluid jets studied here can be used to
understand velocity profiles obtained in the planar and axisymmetric intrusion experiments of Didden and
Maxworthy,15 where a thin layer of fresh water is made to
spread across the top of a deep layer of weakly stratified
saline water, in which case the second immiscible fluid is the
ambient air above the free surface. For planar wakes, two
immiscible fluids are in horizontal translation at common
uniform velocity U. The fluids tangentially stream past a
symmetric slender body, aligned with the fluid–fluid interface, and drift off the trailing edge to form a two-fluid wake.
This represents the immiscible two-fluid, equal-velocity limit
of stratified flow off a splitter plate in the experiments of
Koop and Browand.16 The momenta removed from each
fluid by the frictional drag on opposite sides of the plate are
generally not equal, so the downstream wake is expected to
be asymmetric. Indeed this is the case, but analogous to the
jet problem, the wake deficit velocity profile in each fluid
depends only on the total body drag D and not on the separate frictional drag components imparted from opposite sides
of the plate. For self-propelled slender bodies, the second
moment of momentum K is the conserved quantity and
asymmetric deficit wake profiles with zero momentum are
obtained. In all problems considered here, stable laminar
flow is assumed.
The presentation is as follows. In Sec. II the boundarylayer problems for two-fluid planar and axisymmetric fan
jets are analyzed and sample jet velocity profiles are presented. A special nonself-similar solution for the planar jet
found for ␹⫽1 is reported and illustrated with plots of twofluid velocity profiles. Both classical and momentumless planar two-fluid wakes are analyzed in Sec. III, again with
graphical examples of wake profiles. A discussion of results
and concluding remarks are given in Sec. IV.
Herczynski, Weidman, and Burde
II. PLANAR JETS AND AXISYMMETRIC FAN JETS
In the usual manner, both the Cartesian (x,y) 共index j
⫽0) and cylindrical (r,y) 共index j⫽1) two-fluid jet flows
may be handled simultaneously by identifying r⫽x for j
⫽1 in the following development. Consider an interface at
y⫽ ␾ (x) gravitationally separating upper fluid (i⫽1) from
immiscible lower fluid (i⫽2). Envision a straight ( j⫽0) or
cylindrical ( j⫽1) slot straddling the two fluids through
which upper and lower layer fluids are separately or simultaneously discharged into the surrounding quiescent twofluid ambient. At sufficiently large Reynolds number, the
two-fluid laminar flow for narrow jets are described by the
zero pressure gradient boundary-layer equations in each fluid
layer (i⫽1,2) given by
共 x j u i 兲 x ⫹ 共 x j v i 兲 y ⫽0,
共1a兲
u i共 u i 兲 x⫹ v i共 u i 兲 y ⫽ ␯ i共 u i 兲 y y ,
共1b兲
where ␯ i are the kinematic fluid viscosities, with j⫽0 for the
planar jet and j⫽1 for the fan jet. These equations are to be
solved with the far-field and interfacial conditions
u i →0,
兩 y 兩 →⬁,
共2a兲
u 1 ⫽u 2 ,
y⫽ ␾ 共 x 兲 ,
共2b兲
v 1⫽ v 2 ,
y⫽ ␾ 共 x 兲 ,
共2c兲
␮ 1共 u 1 兲 y ⫽ ␮ 2共 u 2 兲 y ,
y⫽ ␾ 共 x 兲 ,
共2d兲
⫺p 1 ⫹2 ␮ 1 共 v 1 兲 y ⫽⫺p 2 ⫹2 ␮ 2 共 v 2 兲 y ,
y⫽ ␾ 共 x 兲 .
共2e兲
Here p i are the pressures in each layer, ␾ (x) is the position
of the interface, and absolute viscosities ␮ i and fluid densities ␳ i are assumed constant. Equations 共2b兲 and 共2c兲 state
that y⫽ ␾ (x) is a streamline and Eqs. 共2d兲 and 共2e兲 enforce
continuity of tangential stress and normal stress, respectively,
at the material interface. Moreover, the total streamwise momentum flux of the system
J⫽x j
冉冕
⬁
␾共 x 兲
␳ 1 u 21 dy⫹
冕
␾共 x 兲
⫺⬁
␳ 2 u 22 dy
冊
共2f兲
is conserved. A scaling analysis of 共1兲 and 共2f兲 reveals the
characteristic coordinate and velocity scales 关 y c ,u c , v c 兴 in
each fluid layer, viz.
y c ⫽ ␣ x 共 2⫹ j 兲 /3,
共3a兲
u c ⫽ ␤ x ⫺ 共 1⫹2 j 兲 /3,
共3b兲
v c ⫽ ␥ x ⫺ 共 2⫹ j 兲 /3,
共3c兲
␳␯ 2
J
1/3
共3d兲
J2
1/3
␣⫽
␤⫽
␥⫽
冉 冊
冉 冊
,
,
␳ 2␯
冉 冊
J␯
␳
共3e兲
1/3
,
共3f兲
where ␥⫽␣␤ is introduced for convenience.
Downloaded 04 Jan 2005 to 136.167.58.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
Phys. Fluids, Vol. 16, No. 4, April 2004
Two-fluid jets and wakes
We now address the two-fluid problem using streamfunctions ␺ i (i⫽1,2) satisfying the continuity equation in
each fluid layer, namely (u i , v i )⫽ 关 x ⫺ j ( ␺ i ) y ,⫺x ⫺ j ( ␺ i ) x 兴 .
Inserting this into 共1b兲 yields the governing equations
共 ␺ i 兲 y 共 ␺ i 兲 xy ⫺ 共 ␺ i 兲 x 共 ␺ i 兲 y y ⫺ jx ⫺1 共 ␺ i 兲 2y ⫽ ␯ i x j 共 ␺ i 兲 y y y .
共4兲
The stream functions scale as ␺ c ⫽u c y c x j ⫽ ␥ x (1⫹2 j)/3. Normalizing ␺ i and y with their respective scales, a straightforward calculation reveals that the similarity form of the
streamfunctions may be taken as
␺ i⫽
␩ i⫽
冉 冊
6␥i
x 共 1⫹2 j 兲 /3 f i 共 ␩ i 兲 ,
1⫹2 j
共 y⫺ ␾ 共 x 兲兲
␣i
共 2⫹ j 兲 /3
共5b兲
.
Substituting 共5兲 into governing Eqs. 共4兲, boundary conditions
共2a兲–共2d兲 and integral constraint 共2f兲 furnishes the boundaryvalue problem
f i⵮ ⫹2 共 f i f i⬙ ⫹ f i⬘ 兲 ⫽0,
共6a兲
f i⬘ →0,
兩 ␩ i 兩 →⬁,
共6b兲
f 1⬘ 共 0 兲 ⫽ ␹ 1/3 f 2⬘ 共 0 兲 ,
共6c兲
2
冑
f 1 共 0 兲 ⫽ ␹ 1/6
␯2
f 共 0 兲,
␯1 2
共6d兲
f ⬙1 共 0 兲 ⫽ f 2⬙ 共 0 兲 ,
冕
⬁
0
f ⬘1 2 d ␩ 1 ⫹
冕
0
⫺⬁
f 2⬘ 2 d ␩ 2 ⫽
冉 冊
1⫹2 j
6
共6e兲
,
共 1⫹2 j 兲 ␹
␯1
␯2
f i共 ␩ i 兲 ⫽
2 共 1⫹ ␹ 1/2兲 1/3
␯1
␯2
冉
2 共 1⫹ ␹ 1/2兲 1/3
冉 冊 冉 冊冉 冊
冉 冊
␾共 x 兲⫽
2/3
3␳1
␳1
␮1
2
␮2
␳2
g 1⫺
␳1
␳2
冊
␩i ,
共7兲
⫺1
1
共 1⫹ ␹ 1/2兲 2/3␹ 2/3
x 4/3
.
共8兲
It is seen from 共8兲 that the sign of the interface deflection
depends on the relation between absolute viscosities 共recall
that ␳ 2 ⬎ ␳ 1 ): it is positive in the case of ␮ 1 ⬎ ␮ 2 and it is
negative in the opposite case so that the interface approaches
the plane y⫽0 as x→⬁ from the side of the fluid having the
larger absolute viscosity. Introducing the dimensionless variables
x̃⫽
冉 冊
J
g␳1
⫺1/2
x,
˜ 共 x̃ 兲 ⫽
␾
˜⫽
␾
冉 冊
␯ 21
9g
⫺1/3
␾,
␳共 ␹⫺␳ 兲
共9兲
x 2 共 2⫹ j 兲 /3
共6g兲
It is immediately seen that, in the case j⫽1 corresponding to
the fan jet, the first term in 共6g兲, representing a difference of
viscous normal stresses in the two fluids, vanishes so that the
only solution to 共6g兲 is ␾ (x)⫽0. Thus, the interface for the
two-fluid fan jet is flat. Note that, after imposing the condition ␾ (x)⫽0, the viscous normal stress becomes not simply
continuous, as it is in the free shear layer problem studied by
Lock2 共see also the discussion of this point by Jones and
Watson in Rosenhead兲,17 but zero at the interface for both
共10兲
,
where ␳ ⬅ ␳ 1 / ␳ 2 and the ratio ␮ 1 / ␮ 2 has been expressed
through ␹ and ␳.
The dimensional self-similar jet velocity profiles obtained from 共5a兲 and 共7兲 are
u i 共 x, ␩ i 兲 ⫽ ␤ i
3 共 1⫹ j 兲 /3␭ 2i
2 共 1⫹ ␹ 兲
⫻sech2
⫺1 f 1⬘ 共 0 兲
1
共 1⫺ ␳ 兲共 1⫹ ␹ 1/2兲 2/3␹ 2/3 x̃ 4/3
1/2
1/2
⫺g 共 ␳ 2 ⫺ ␳ 1 兲 ␾ 共 x 兲 ⫽0.
tanh
3 共 2 j⫺1 兲 /3␭ i
where ␭ 1 ⫽ ␹ 1/6 and ␭ 2 ⫽1.
The solution of the fan jet problem is given by Eqs. 共5兲
and 共7兲 with j⫽1 and ␾ (x)⫽0. To specify a solution for the
planar jet problem, we introduce f 1⬘ (0) evaluated from 共7兲
into 共6g兲, taken for j⫽0, to obtain ␾ (x) in the form
共6f兲
冋冉 冊 册
冉 冊
␹
3 共 2 j⫺1 兲 /3␭ i
yields
2
where ␹ ⫽ ␳ 1 ␮ 1 / ␳ 2 ␮ 2 is the same parameter that appears in
the work of Lock.2 In the integral invariant 共6f兲, use has been
made of the relation J⫽ ␳ i ␥ 2i / ␣ i obtained from 共3d兲 and 共3f兲.
The boundary-value problem 共6a兲–共6f兲 does not include
the position ␾ (x) of the interface and thus determines f i ( ␩ i )
only. The interface position is obtained from the normal
stress condition 共2e兲 wherein the pressure drop across the
interface is p 2 ⫺p 1 ⫽( ␳ 1 ⫺ ␳ 2 )g ␾ (x). After substituting 共5兲
into 共2e兲 and taking into account 共6c兲 and 共6e兲, one obtains
4 共 1⫺ j 兲共 J 2 ␯ 21 ␳ 1 兲 1/3
fluids. The interfacial position for the planar jet, however, is
determined from 共6g兲 after solving Eqs. 共6a兲–共6f兲 for f i ( ␩ i ).
Integrating 共6a兲 and using conditions 共6b兲–共6f兲, one
readily finds solutions in the form
J␯1
1
x
共5a兲
1039
1/2 2/3
冉
1
x
共 1⫹2 j 兲 /3
3 共 2 j⫺1 兲 /3␭ i
2 共 1⫹ ␹ 1/2兲 1/3
冊
␩i ,
共11兲
where the ␩ i are defined by 共5b兲 with ␾ (x)⫽0 for j⫽1 and
␾ (x) given by 共8兲 for j⫽0.
The mass fluxes in each fluid layer are readily found to
be
3 共 2⫺ j 兲 /3␭ i 共 1⫹2 j 兲 /3
ṁ i ⫽ 共 ␳ i ␮ i J 兲 1/3
x
共 1⫹ ␹ 1/2兲 1/3
共 i⫽1,2兲
共12a兲
so the total mass flux is
ṁ⫽ṁ 1 ⫹ṁ 2 ⫽3 共 2⫺ j 兲 /3共 ␳ 2 ␮ 2 J 兲 1/3共 1⫹ ␹ 1/2兲 2/3x 共 1⫹2 j 兲 /3.
共12b兲
Note that the mass flux ratio ṁ 1 /ṁ 2 ⫽ ␹ 1/2 depends only on
␹. Similarly, the momentum fluxes in each layer are
Downloaded 04 Jan 2005 to 136.167.58.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
1040
Phys. Fluids, Vol. 16, No. 4, April 2004
Herczynski, Weidman, and Burde
TABLE I. Fluid properties for air, water, and a high viscosity Dow Corning 200 silicone oil at 20 °C.
J 1⫽
Fluid
␳
共g/cm3兲
␯
共cm2/s兲
␮
共g/cm s兲
␳␮
共g2/cm4 s兲
Air
Oil
Water
0.001 205
0.9762
0.9982
0.1500
11.197
0.010 04
1.8075⫻10⫺4
10.931
0.010 022
2.178⫻10⫺7
10.6704
0.010 00
冉 冊
␹ 1/2
1⫹ ␹ 1/2
J,
J 2⫽
冉 冊
1
1⫹ ␹ 1/2
J
共13兲
and the momentum flux ratio is, therefore, J 1 /J 2 ⫽ ␹ 1/2.
Equation 共13兲 implies that the momentum flux of each
fluid layer is a separately conserved quantity. It might seem
that the problem could be solved by imposing separate momentum constraints in each fluid layer, instead of 共2f兲. However, it would be inconsistent for the following reasons.
First of all, the x-independent state of momentum J 1 in
the upper layer and J 2 in the lower layer is a feature of the
solution obtained, which, like most self-similar solutions,
should describe one of the possible asymptotic states that
may be realized far downstream from the source of two-fluid
momentum. It is evident that, closer to the source, there
should be exchange of momenta between the layers and only
the total momentum flux is conserved. Thus, independent
momentum constraints on the asymptotic state cannot be imposed from the outset, before the adopted ansatz reveals a
solution with zero tangential stress in each fluid at the interface; this, of course, implies no exchange of momentum between the layers.
Next, even being aware of this solution feature, if one
tried to impose separate momentum constraints in each fluid
layer using this solution form, it would transpire that, after
satisfying boundary conditions 共2a兲–共2f兲, the solution is attended by the necessary condition J 1 /J 2 ⫽ ␹ 1/2. This shows
that J 1 and J 2 cannot be independently conserved, and therefore the solution 共7兲 with J⫽J 1 ⫹J 2 as the single integral
constant is recovered.
The two-fluid planar jet solution derived in this section
is not self-similar in an original meaning of the term since it
cannot be reduced to a universal velocity profile by proper
choice of x-dependent scales for the velocity and the transversal coordinate. From another point of view, we do not
have the situation common for the existence of self-similar
solutions when the physical parameters in the problem fail to
provide both a fundamental length and a fundamental
velocity—here the parameter g plays a significant role in the
solution by providing a missing scale. Nevertheless, one can
treat the results found as generalized self-similar solutions
since they can be reduced to universal profiles by a combination of scaling for the velocity, a shift of the origin of the
transverse coordinate to the interface position, and a scaling
for the shifted transverse coordinate. It is also seen that while
the parameter g plays a crucial role in determining the interface position ␾ (x), it does not take part in the problem determining the functions f i which define the form of the velocity profile.
Velocity profiles for the two-fluid flow are best normalized using the interfacial velocity which is also the maximum
jet speed. From 共11兲 we see that the interfacial velocities of
each fluid
u 共 x, ␾ 共 x 兲兲 ⫽ ␤ 1
⫽␤2
3 共 1⫹ j 兲 /3␹ 1/3
2 共 1⫹ ␹ 兲
1/2 2/3
1
x
3 共 1⫹ j 兲 /3
共 1⫹2 j 兲 /3
1
2 共 1⫹ ␹ 兲
1/2 2/3
x
共 1⫹2 j 兲 /3
共14兲
are equal, as they must be, since ␤ 2 / ␤ 1 ⫽ ␹ 1/3. Thus the velocity distribution across the two-fluid planar and fan jets
may be written
u i 共 x,y 兲 ⫽u 共 x, ␾ 共 x 兲兲 sech2
冉
冊
y⫺ ␾ 共 x 兲
,
␦ i共 x 兲
共15兲
in which the ␦ i (x) provides a measure of the jet thickness in
each fluid layer, namely
␦ i共 x 兲 ⫽ ␣ i
冉
2 共 1⫹ ␹ 1/2兲 1/3
3 共 2 j⫺1 兲 /2␭ i
冊
x 共 2⫹ j 兲 /3.
共16兲
In this form, it is clear that the velocity profiles are not symmetrically disposed about the interface except when ␣ 1 / ␣ 2
⫽ ␹ 1/6, which corresponds to the condition ␹⫽1. Thus symmetric jets are possible for different fluids that possess a
special relation among densities and viscosities.
Using air, water, and a high viscosity silicone oil we
present examples of planar two-fluid jet flows. The properties of air and water at 20 °C taken from Batchelor18 and
those for the silicone oil at the same temperature taken from
Lasso and Weidman19 are given in Table I.
Since the heavier fluid must reside below the lighter
fluid, we have ␹ ⫽2.178⫻10⫺4 for air over water, ␹
⫽1.067⫻103 for oil over water, ␹ ⫽2.053⫻10⫺8 for air
over oil, and clearly ␹⫽1 for water in both layers. These
planar jet velocity profiles are displayed in Fig. 1 for the
choice J⫽1000 g/s2 at downstream position x⫽100 cm. It is
clear from Eq. 共16兲 that the jet widths at fixed station x
depend on ␣ i as well as the wide-ranging values of ␹. The
ratio of the jet widths, however, depends only on the ratio
of kinematic viscosities in the fashion ␦ 1 (x)/ ␦ 2 (x)
⫽( ␯ 1 / ␯ 2 ) 1/2.
Downloaded 04 Jan 2005 to 136.167.58.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
Phys. Fluids, Vol. 16, No. 4, April 2004
Two-fluid jets and wakes
1041
FIG. 1. Normalized planar jet velocity
profiles for combinations of air, oil,
and water, with fluid properties as
given in Table I, computed for jet momentum J⫽1000 g/s2 at downstream
position x⫽100 cm: 共a兲 air over water,
共b兲 oil over water, 共c兲 air over oil, and
共d兲 water only.
Nonself-similar solution for the planar jet when ␹Ä1
The solution of the planar free jet problem found in the
previous section and defined by Eqs. 共5兲 and 共7兲 may be
represented in the form
␺ i ⫽A 冑␯ i x 1/3 tanh共 ␩ i 兲 ,
where
A⫽
冉 冊
J2
␯ 1 ␳ 21
1/6
␩ i⫽
A 共 y⫺ ␾ 共 x 兲兲
6 冑␯ i x 2/3
3 2/3␹ 1/6
共17a兲
J 1⫽
共17b兲
共 1⫹ ␹ 1/2兲 1/3
with ␾ (x) defined by 共8兲. This solution can be generalized to
␺ i ⫽Ã 冑␯ i 关 ⫺B⫹x
␩ i⫽
à 共 y⫺ ␾ 共 x 兲兲
6 冑␯ i x 2/3
1/3
tanh共 ␩ i 兲兴 ,
⫹arctanh
冉 冊
B
x 1/3
共18兲
,
where à and B are constants. The solution defined by Eq.
共18兲 satisfies governing equations 共4兲 共taken for j⫽0) and
the conditions 共2a兲–共2c兲. However, the condition of continuity of tangential stress 共2d兲 can be satisfied only if ␹⫽1. The
difference with this solution is in the tangential stress at the
interface which, calculated on the basis of 共18兲, is proportional to B and therefore does not vanish. If ␹⫽1 is assumed,
the integral condition 共2f兲 also can be satisfied to give
Ã⫽
冉 冊
J̃ 2
␯ 1 ␳ 21
1/6
,
J̃⫽
9J
.
2
Despite the fact that the above expression for à does not
contain B, and simply coincides with what expression 共17b兲
gives for ␹⫽1, there is a significant difference between solutions 共17兲 and 共18兲. Now, as distinct from 共13兲, the momentum fluxes J 1 and J 2 in each layer are not constant but functions of x given by
共19兲
冉
冉
冊
冊
2x⫺3Bx 2/3⫹B 3
J,
4x
2x⫹3Bx 2/3⫺B 3
J
J 2⫽
4x
共20a兲
so that the momentum flux ratio is
冉
冊
2x⫺3Bx 2/3⫹B 3
J1
⫽
.
J2
2x⫹3Bx 2/3⫺B 3
共20b兲
The variation of the momentum fluxes along the stream reveals momentum exchange across the interface which is absent in solution 共17兲, since there the tangential stress in each
fluid is zero at the interface. It may be noted from 共20兲 that
asymptotically far downstream the momentum carried in
each layer tends to the equipartitioned value J/2, independent
of B.
The interface position ␾ (x), determined from continuity
of normal stress 共2e兲, is
␾共 x 兲⫽
冉 冊冉 冊
1 J␯1
g 6␳1
2/3
␳ 1 共 x 2/3⫺3B 2 兲
.
␳2
x2
共21兲
Downloaded 04 Jan 2005 to 136.167.58.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
1042
Phys. Fluids, Vol. 16, No. 4, April 2004
Herczynski, Weidman, and Burde
FIG. 2. The interface shape for the nonself-similar ␹⫽1 planar jet solution
at x⫽1 plotted for ␳⫽1/4 at different values of b.
Thus the solution of the problem is given by 共18兲 where à is
defined by 共19兲, ␾ (x) by 共21兲, and B is an arbitrary constant.
Some remarks on formula 共21兲 are needed. First, for B
⫽0 it corresponds to the solution 共8兲 in which ␾ (x) is positive since the relations ␳ 1 ⬍ ␳ 2 and ␹⫽1 lead to ␮ 1 ⬎ ␮ 2 .
Next, even though 共21兲 seems to be valid for any values of
␳ 1 and ␳ 2 , the case ␳ 1 ⫽ ␳ 2 is excluded. The point is that the
jump in viscous stress at y⫽ ␾ (x) in 共2e兲 is proportional to
( ␳ 1 ⫺ ␳ 2 ). If it is further assumed that ␳ 1 ⫽ ␳ 2 corresponding
to two identical fluids, 共2e兲 is satisfied identically so that any
␾ (x) is permitted, as may be expected.
Renormalizing 共21兲 using variables 共9兲 yields
2/3
2/3
˜ 共 x̃ 兲 ⫽ ␳ 共 x̃ ⫺b 兲 ,
␾
2 2/3x̃ 2
b⫽3B 3
␳⫽
冉 冊
3g ␳ 1
J
1/2
,
共22兲
␳1
.
␳2
The interface shape is displayed in Fig. 2 for selected values
of b.
Equations 共18兲, 共19兲, and 共21兲 do not represent a selfsimilar solution even in the generalized sense discussed previously. The longitudinal velocity profile cannot be reduced
to a universal profile, independent of the longitudinal coordinate, by a shift of the origin for the transverse coordinate to
the interface position and scalings of the coordinate and velocity. The nonself-similar nature of the solution also reveals
itself by the presence of an arbitrary constant B which provides an additional length scale: it determines, in particular,
the values of x at which the interface deflection takes its
maximum and zero values. It is convenient, while displaying
the velocity profiles, to use the scaling based on this length
scale, viz.
X⫽
x
,
x*
y
Y⫽
,
y*
U⫽
u
,
u*
␾
⌽⫽
,
y*
共23a兲
FIG. 3. The longitudinal velocity profiles for the nonself-similar ␹⫽1 planar
jet solution plotted for ␳⫽1/4 and b⫽0.2 at different values of X.
x * ⫽ 共 B 冑3 兲 ,
u *⫽
3
y *⫽
冉
␯ 21 ␳ 1 x * 2
J̃
冊
1/3
冉 冊
J̃ 2
␳ 21 ␯ 1 x *
1/3
,
共23b兲
.
Here the x coordinate is scaled by (B 冑3) 3 to assign the value
X⫽1 to the point ⌽(X)⫽0, and the scales for u and y are
chosen such that y * ⫽x * /Re1/2, where Re⫽u*x*/␯1 .
The longitudinal velocity profiles U(Y ) at selected values of X are plotted in Fig. 3. The position of the interface
for each profile manifests itself by a break of smoothness.
The profiles are not symmetric about the interface and the
velocity maximum is always situated in the lower fluid. The
position of the velocity maximum approaches the interface
as X increases.
III. PLANAR WAKES
The asymptotic structure of the laminar wake behind a
slender symmetric planar body aligned with the uniform flow
of a homogeneous fluid was originally investigated by
Goldstein20 for the classical wake and by Birkhoff and
Zarantonello21 for the momentumless wake of a symmetric
self-propelled body. A new study for the intermediate wake
region behind finite bodies reported by Tordella and Belan22
requires an analysis using matched asymptotic expansions.
Here we consider the two-fluid variation of the classical
asymptotic wake analysis. Let one fluid of density ␳ 1 and
viscosity ␮ 1 in the upper-half plane, and another immiscible
fluid of density ␳ 2 and viscosity ␮ 2 in the lower-half plane,
stream at uniform speed U over a slender body of finite
length symmetrically placed at the two-fluid interface y⫽0.
The total frictional force 共per unit span兲 on the plate wrought
by the two fluids is D. Here we need only the Cartesian ( j
⫽0) form of the zero pressure gradient boundary-layer equa-
Downloaded 04 Jan 2005 to 136.167.58.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
Phys. Fluids, Vol. 16, No. 4, April 2004
Two-fluid jets and wakes
tions 共1兲 which are to be solved with appropriate far-field and
interfacial boundary conditions along with the constraint that
the momentum deficit in the wake be equal to the body drag.
The origin x⫽0 is placed at the trailing edge of the plate. Far
downstream (x→⬁), the velocity deficit w i (x,y)⫽U
⫺u i (x,y) is small so that, to leading order, boundary layer
equations 共1兲 become
⫺ 共 w i 兲 x ⫹ 共 v i 兲 y ⫽0,
共24a兲
␳ iU 共 w i 兲 x⫽ ␮ i共 w i 兲 y y ,
共24b兲
where the neglected terms in 共1b兲 are O(x ⫺1/2) relative to
those retained. The incompressibility condition 共24a兲 admits
a streamfuction ␺ i in each fluid layer giving the velocity field
as (w i , v i )⫽ 关 ( ␺ i ) y ,( ␺ i ) x 兴 . Thus to leading order the twofluid wake flow is described by the linear diffusion equation
U ␳ i 共 ␺ i 兲 xy ⫽ ␮ i 共 ␺ i 兲 y y y .
共25兲
Defining the unknown position of the interface as y
⫽ ␾ (x), the above equation is subject to the far-field and
interfacial conditions
共 ␺ i 兲 y →0,
␺ 1⫽ ␺ 2 ,
y→⫾⬁,
共26a兲
y⫽ ␾ 共 x 兲 ,
共26b兲
共 ␺1兲y⫽共 ␺2兲y ,
y⫽ ␾ 共 x 兲 ,
␮ 1共 ␺ 1 兲 y y ⫽ ␮ 2共 ␺ 2 兲 y y ,
共26c兲
y⫽ ␾ 共 x 兲 ,
共26d兲
2 ␮ 2 共 ␺ 2 兲 xy ⫺2 ␮ 1 共 ␺ 1 兲 xy ⫽ 共 p 2 ⫺p 1 兲
⫽共 ␳ 1⫺ ␳ 2 兲g ␾共 x 兲,
y⫽ ␾ 共 x 兲 .
共26e兲
The complete formulation of the problem requires an additional constraint characterizing whether or not the wake
transports momentum. These two problems are considered
separately in the sequel.
A. Classical wakes
The boundary-value problem for the classical wake is
closed by the constraint that the momentum deficit in the
wake is equal to the body drag 共per unit span兲, viz.
D⫽U
冉冕
⬁
␾共 x 兲
␳ 1 共 ␺ 1 兲 y dy⫹
冕
␾共 x 兲
⫺⬁
冊
␳ 2 共 ␺ 2 兲 y dy .
共27兲
We generalize the coordinate expansion of Goldstein20 to
include an expansion for the interface deflection
␺ i⫽ ␣ i␤ iU
冋
f 共i 0 兲 共 ␩ i 兲 ⫹
␾ 共 x 兲 ⫽ ␾ 共 0 兲⫹
1
x
1/2
1
x
f 共 1 兲 共 ␩ i 兲 ⫹¯
1/2 i
册
,
1
1
␾ 共 1 兲 ⫹ ␾ 共 2 兲 ⫹ 3/2 ␾ 共 3 兲 ⫹¯,
x
x
共28a兲
共28b兲
where
␩ i⫽
共 y⫺ ␾ 共 x 兲兲
␣ i x 1/2
,
共28c兲
␣ i⫽
␤ i⫽
冑
2␯i
,
U
1043
共28d兲
D
3/2
2 冑␲␳ i ␯ 1/2
i U
共28e兲
.
Inserting 共28兲 into 共25兲 and 共26兲 yields the following
boundary-value problem governing the similarity flow. For
simplicity, we now drop the superscript notation for
f (0)
i ( ␩ i ) to obtain
f⵮
i ⫹␩i f ⬙
i ⫹ f i⬘ ⫽0,
共29a兲
f ⬘i →0,
共29b兲
兩 ␩ i 兩 →⬁,
f 1共 0 兲 ⫽ f 2共 0 兲 ,
共29c兲
f ⬘1 共 0 兲 ⫽ ␹ 1/2 f 2⬘ 共 0 兲 ,
共29d兲
f ⬙1 共 0 兲 ⫽ f 2⬙ 共 0 兲 ,
共29e兲
U 关 ␮ 1 ␤ 1 f ⬘1 共 0 兲 ⫺ ␮ 2 ␤ 2 f ⬘2 共 0 兲兴
冉
D⫽U 2 ␳ 1 ␣ 1 ␤ 1
冕
⬁
0
1
x 3/2
⫽共 ␳ 1⫺ ␳ 2 兲g ␾共 x 兲,
f 1⬘ d ␩ 1 ⫹ ␳ 2 ␣ 2 ␤ 2
冕
0
⫺⬁
冊
共29f兲
f ⬘2 d ␩ 2 , 共29g兲
where ␹ ⫽ ␳ 1 ␮ 1 / ␳ 2 ␮ 2 is the same parameter that appears in
the free jet problem. Use of the tangential stress boundary
condition 共29e兲 has been made to give the normal stress condition 共29f兲. The solutions for f i ( ␩ i ) are
f i 共 ␩ i 兲 ⫽ 冑2 ␲
␭i
erf共 ␩ i / 冑2 兲 ,
共 1⫹ ␹ 1/2兲
共30兲
where ␭ 1 ⫽ ␹ 1/2, ␭ 2 ⫽1, and erf(x) is the error function of
argument x. Then from 共29f兲 it is clear that ␾ (k) ⫽0 for k
⫽0, 1, 2 so that the leading order interface deflection in
共28b兲 is given by ␾ (3) /x 3/2. Evaluation of f i⬘ (0) in 共29f兲
using 共30兲 yields
␾ 共 x 兲 ⫽2U ␯ 1 ␤ 1
冉 冊
␹ 1/2
1⫹ ␹
1/2
1
x 3/2
,
共31兲
showing that the two-fluid interface is always deflected upward, regardless of the viscosity contrast. The wake deficit
velocity profiles in each fluid domain are
w i 共 x, ␩ i 兲 ⫽
␤ iU
2␭ i
1/2
共 1⫹ ␹
x
1/2
2
兲
e ⫺ ␩ i /2.
共32兲
The deficit mass flux in each fluid layer is then
ṁ i ⫽
冉 冊
␭i
1⫹ ␹
1/2
D
,
U
共33兲
so that ṁ 1 /ṁ 2 ⫽ ␹ 1/2 and the total deficit mass flux is ṁ
⫽ṁ 1 ⫹ṁ 2 ⫽D/U. Similar results are found for the momentum fluxes J i since the linearization used to obtain 共25兲 leads
to the approximation J i ⫽Uṁ i .
Downloaded 04 Jan 2005 to 136.167.58.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
1044
Phys. Fluids, Vol. 16, No. 4, April 2004
Herczynski, Weidman, and Burde
FIG. 4. Normalized planar wake deficit velocity profiles for combinations
of air, oil, and water, with fluid properties as given in Table I, computed for
freestream velocity U⫽100 cm/s at
downstream position x⫽100 cm: 共a兲
air over water, 共b兲 oil over water, 共c兲
air over oil, and 共d兲 water only.
The common wake deficit velocity at the interface,
which is also the maximum deficit velocity in the wake, is
given by
w 共 x, ␾ 共 x 兲兲 ⫽2U ␤ 1
␹ 1/2
1⫹ ␹
1/2
1
x
1/2
⫽2U ␤ 2
1
1⫹ ␹
1
1/2
x 1/2
.
共34兲
The two expressions are 共necessarily兲 equal since ␤ 2 / ␤ 1
⫽ ␹ 1/2. Thus the deficit velocity distribution across the wake
takes the form
冋冉
y⫺ ␾ 共 x 兲
w i 共 x,y 兲 ⫽w 共 x, ␾ 共 x 兲兲 exp ⫺
␦ i共 x 兲
冊册
2
,
共35兲
Birkhoff and Zarantonello21 show that the conserved quantity
for this flow is obtained by multiplying the boundary layer
form of the momentum equation by y 2 and integrating over
the fluid domain. The resulting conserved integral for the
wake flow behind a symmetric self-propelled body is the
second moment of axial deficit momentum.
The integral constraint for the two-fluid system is obtained by multiplying Eq. 共24b兲 by y 2 and integrating over
the entire two-fluid domain, which yields
冋 冕
U ␳2
共36兲
It is clear that symmetry about the interface is obtained only
when ␣ 1 ⫽ ␣ 2 , corresponding to ␯ 1 ⫽ ␯ 2 , independent of the
value of ␹. The ratio of wake thicknesses is given by
␦ 1 (x)/ ␦ 2 (x)⫽( ␯ 1 / ␯ 2 ) 1/2. Three two-fluid wake velocity
profiles, and one homogeneous wake profile for comparison,
are presented in Fig. 4 for the choice D⫽200 dynes/cm, U
⫽100 cm/s, and x⫽100 cm.
B. Momentumless wakes
For a self-propelled symmetric slender planar body moving along the interface of a two-fluid stream, the drag on the
body is zero. Hence Eq. 共27兲 gives no information on the
structure of the similarity flow. For a homogeneous fluid,
⫺⬁
⫽␮2
in which ␦ i (x) is the e-folding measure of jet thickness in a
given layer, namely
␦ i 共 x 兲 ⫽ 冑2 ␣ i x 1/2.
␾共 x 兲
冕
y 2 共 w 2 兲 x dy⫹ ␳ 1
␾共 x 兲
⫺⬁
冕
⬁
␾共 x 兲
y 2 共 w 2 兲 y y dy⫹ ␮ 1
y 2 共 w 1 兲 x dy
冕
⬁
␾共 x 兲
册
y 2 共 w 1 兲 y y dy.
共37兲
Two integrations by parts of the right-hand side, using farfield boundary conditions of zero velocity and slope, give
U
冋 冕
冋 冕
⳵
␳
⳵x 2
␾共 x 兲
⫺⬁
⫽2 ␮ 2
冕
␮ 冕
⬁
y 2 w 2 dy⫹ ␳ 1
␾共 x 兲
⫺⬁
w 2 dy⫹
␾共 x 兲
1
y 2 w 1 dy
册
⬁
␾共 x 兲
册
共38兲
w 1 dy .
For a momentumless wake, the integrals on the right-hand
side are zero. Consequently, the constraint for the two-fluid
momentumless wake posed here is
冋 冕
K⫽U ␳ 2
␾共 x 兲
⫺⬁
y 2 w 2 dy⫹ ␳ 1
冕
⬁
␾共 x 兲
册
y 2 w 1 dy ,
共39兲
Downloaded 04 Jan 2005 to 136.167.58.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
Phys. Fluids, Vol. 16, No. 4, April 2004
Two-fluid jets and wakes
where K is a constant representing the second moment of
deficit momentum of the two-fluid wake flow.
A scaling analysis of (y,w, v ) using Eqs. 共24兲 and constraint 共39兲 yields
y c⫽
␯ 1/2
U 1/2
w c⫽
x 1/2,
KU 1/2 1
␳␯ 3/2 x 3/2
K 1
v c⫽
␳␯ x 2
,
␩ i⫽
共 y⫺ ␾ 共 x 兲兲
␣ i x 1/2
w 共 x, ␾ 共 x 兲兲 ⫽⫺ ␤ 1
共40a兲
⫽⫺ ␤ 2
共40b兲
,
where
␣ i⫽
␤ i⫽
冑
4␯i
,
U
共40c兲
KU 1/2
共40d兲
,
3/2
with ␾ (x) expanded as in 共28b兲. Inserting solution form 共40兲
into governing equation 共25兲, boundary conditions 共26兲, and
integral constraint 共39兲, gives
f⵮
i ⫹2 ␩ i f ⬙
i ⫹6 f ⬘
i ⫽0,
共41a兲
f ⬘i →0,
共41b兲
兩 ␩ i 兩 →⬁,
f 1共 0 兲 ⫽ f 2共 0 兲 ,
共41c兲
⍀ 1/2 f 1⬘ 共 0 兲 ⫽ f 2⬘ 共 0 兲 ,
共41d兲
f ⬙1 共 0 兲 ⫽ f 2⬙ 共 0 兲 ,
共41e兲
3 关 ␮ 1 ␤ 1 f 1⬘ 共 0 兲 ⫺ ␮ 2 ␤ 2 f 2⬘ 共 0 兲兴 ⫽ 共 ␳ 1 ⫺ ␳ 2 兲 g ␾ 共 x 兲 ,
共41f兲
K⫽U ␳ 1 ␣ 31 ␤ 1
冕␩
⬁
0
2
3
1f⬘
1d ␩ 1⫹ ␳ 2␣ 2␤ 2
冕
⫺⬁
0
冊
␩ 22 f ⬘2 d ␩ 2 ,
共41g兲
where ⍀⫽ ␳ 1 ␮ 32 / ␳ 2 ␮ 31 . Thus a new ratio of fluid properties,
different from ␹, appears in the two-fluid momentumless
wake problem. Solution of 共41兲 for f i ( ␩ i ) is readily obtained
in the form
f i 共 ␩ i 兲 ⫽⫺
4␭ i
共 1⫹⍀ 1/2兲
␩ ie
2
⫺␩i
共42兲
,
where ␭ 1 ⫽1 and ␭ 2 ⫽⍀ 1/2. The normal stress condition in
共41f兲 shows that ␾ (k) ⫽0 for k⫽0, 1, 2, 3, 4 and evaluation
of f ⬘i (0) using 共42兲 gives
␾ 共 x 兲 ⫽12␤ 1
冉
冉
y⫺ ␾ 共 x 兲
␦ i共 x 兲
y⫺ ␾ 共 x 兲
␦ i共 x 兲
4
1⫹⍀
1/2
4⍀ 1/2
1⫹⍀
1/2
冊
冊
冊册
共 ␮ 1⫺ ␮ 2 兲
共 1⫹⍀
1/2
1
兲共 ␳ 2 ⫺ ␳ 1 兲 x 5/2
.
Thus the interface deflection is upward for ␮ 1 ⬎ ␮ 2 and
downward for ␮ 1 ⬍ ␮ 2 . Analogous to the two-fluid planar
jet, the interface approaches the plane y⫽0 as x→⬁ from
the side of larger absolute viscosity.
The wake deficit velocity profiles in each fluid domain
are
冊册
2
2
,
共43兲
1
x 3/2
1
x 3/2
共44兲
are necessarily equal since ␤ 1 / ␤ 2 ⫽⍀ 1/2. It is easy to verify
that the integrals on the right-hand side of Eq. 共38兲 are indeed zero. Symmetric profiles are obtained only for ␯ 1
⫽ ␯ 2 . Also, the e-folding measure of lateral wake penetration
into each fluid is
␦ i 共 x 兲 ⫽ ␣ i x 1/2
16冑␲␳ i ␯ i
冉
冋冉
⫻exp ⫺
and the interfacial velocities
which motivates the similarity solution ansatz
␣ i␤ i
␺ i 共 x,y 兲 ⫽
f 共 ␩ 兲,
x i i
冋 冉
w i 共 x,y 兲 ⫽w 共 x, ␾ 共 x 兲兲 1⫺2
1045
共45兲
so again one finds the wake thickness ratio ␦ 1 (x)/ ␦ 2 (x)
⫽( ␯ 1 / ␯ 2 ) 1/2.
Three two-fluid deficit velocity profiles and one homogeneous deficit velocity profile for a momentumless wake
are presented in Fig. 5 using realistic laboratory scale parameter values K⫽500 dyne cm, U⫽100 cm/s, and x⫽100 cm.
IV. DISCUSSION AND CONCLUSION
The analytical solutions presented in this paper are valid
only in the boundary-layer approximation. In particular, for
the two-fluid jets, this requires the local jet Reynolds numbers to be large and the interface slope for the planar jet,
determined from differentiation of Eq. 共8兲, be small. Thus the
limits of our solution for laminar two-fluid jets are that
Jx 1⫺ j
␳␯ 2
冏
Ⰷ1,
冉 冊
J␯1
␳1
2/3
␳ 2 共 ␮ ⫺1 兲
冏
1
g 共 1⫺ ␳ 兲共 1⫹ ␹ 1/2兲 2/3␹ 2/3 x 7/3
Ⰶ1,
where the first inequality applies to both fluids in the planar
( j⫽0) and radial ( j⫽1) cases, and in the second inequality
␳ ⫽ ␳ 1 / ␳ 2 and ␮ ⫽ ␮ 1 / ␮ 2 .
The interface deflection due to the presence of gravity
occurs in different manners for different flows. For the parallel streaming of two fluids studied by Lock2 and for the fan
jet reported here, there is no interface deflection. For the
planar jet and the momentumless wake, the deflection is upward for ␮ 1 ⫺ ␮ 2 ⬎0 and downward in the opposite case.
Classical two-fluid wakes, on the other hand, deflect upward
regardless of the relative magnitude of their absolute fluid
viscosities. Although solutions for the two-fluid planar and
fan jets mimic their classical single-fluid counterparts, there
is an interesting distinction in the two-fluid case. For the
two-fluid fan jet the viscous normal stress is continuous at
the interface; this provides continuity of the pressure which
Downloaded 04 Jan 2005 to 136.167.58.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
1046
Phys. Fluids, Vol. 16, No. 4, April 2004
Herczynski, Weidman, and Burde
FIG. 5. Normalized momentumless
wake deficit velocity profiles for combinations of air, oil, and water, with
fluid properties as given in Table I,
computed for freestream velocity U
⫽100 cm/s at downstream position x
⫽100 cm: 共a兲 air over water, 共b兲 oil
over water, 共c兲 air over oil, and 共d兲 water only.
implies a flat interface. For the two-fluid planar jet, on the
other hand, there is a jump in the viscous normal stress
which is compensated for by the pressure difference due to
the interface deflection.
The solution of the two-fluid planar jet problem defined
in Sec. II represents a similarity solution in a generalized
sense. Although the velocity profile cannot be reduced to a
universal profile by x-dependent scalings of the velocity and
the transverse coordinate, it can be done if the scalings are
combined with a shift of the transverse coordinate to the
interface position. A common self-similar ansatz cannot provide a correct solution of the two-fluid jet problem. Should
such a solution form be used to describe flows with interfaces or free boundaries, the interface position y⫽ ␾ (x) can
be taken only proportional to the y scale, which is ␣ x 2/3 in
the case of the planar jet, to provide a constant value of the
similarity variable in boundary conditions at the interface.
This immediately would lead to a contradiction since imposing the condition of continuity of normal stress yields y
⫽ ␾ (x)⬃x ⫺4/3.
Solutions presented in Sec. II, like common self-similar
solutions, exhibit singularities at x⫽0 and therefore must be
regarded as only asymptotic solutions valid at large distances
from the jet source. For common liquid-into-liquid jets, it
was found by Andrade and Tsien23 that the accuracy of the
description of the far field by the self-similar solutions can
be improved by displacing the location of the point source
from the jet exit, as allowed by the translational invariance of
the governing equations. A theoretical justification for this
empirical finding has been presented by Revuelta et al.24
who performed a perturbation analysis of the self-similar
round and planar jet solutions 共following the comparable
analysis for the fan jet solution by Riley兲,25 using the properly scaled distance from the jet exit as an asymptotically
large quantity. The analysis revealed that the first order correction is indeed equivalent to a displaced origin for the
point source of momentum.
In the two-fluid planar jet problem, a trivial shift of the
origin, x→x⫺x 0 , cannot significantly improve the description of the far field as is evident from the solution provided
in Sec. II. There, the interface position ␾ (x) defined by Eq.
共8兲 monotonically approaches the plane y⫽0 as x→⬁. This
implies that the solution is applicable only far from the
source—closer to the source the function ␾ (x) passes
through a maximum 共at least once兲 to return to the original
interface position ␾ (x)⫽0 near the source. Such behavior
cannot be achieved by a shift of the location of the jet source.
At the same time, the ␹⫽1 solution for a planar jet found in
Sec. II provides this flow feature; see Eq. 共23兲 and Fig. 2.
Even though for smaller x 共roughly speaking, smaller than
B 3 ) the solution also becomes unrealistic, it is evident that
an improved description of the far field has been achieved.
This solution provides a nontrivial example of the exact
solution of a jet flow that contains a nonself-similar correction to a self-similar solution. It includes the arbitrary constant B which, in practice, could be found by a comparison of
the corrected solution either with experimental data 共as in
Andrade and Tsien23兲 or with the results of numerical integration of the boundary-layer equations 共as in Revuelta
et al.24兲.
Our study gives additional grounds for the conclusion
that two-fluid flows are, in general, characterized by the pa-
Downloaded 04 Jan 2005 to 136.167.58.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
Phys. Fluids, Vol. 16, No. 4, April 2004
Two-fluid jets and wakes
1047
TABLE II. Interfacial jet velocities u(x, ␾ (x)), classical wake deficit velocities w c (x, ␾ (x)), and momentumless wake deficit velocities w m (x, ␾ (x)) for the
profiles, respectively, plotted in Figs. 1, 4, 5, and the ratio of boundary layer thicknesses. For the planar jets J⫽1000 g/s2 and x⫽100 cm; for the planar wakes,
D⫽200 dynes/cm, U⫽100 cm/s, and x⫽100 cm; for the momentumless wakes, K⫽500 dynes cm, U⫽100 cm/s, and x⫽100 cm.
Fluids
Air over water
Oil over water
Air over oil
Water only
␹
⍀
⫺5
2.1772⫻10
1.0666⫻103
2.0412⫻10⫺8
1.00
2
2.0577⫻10
7.5380⫻10⫺10
2.729 82⫻1011
1.00
rameter ␹ ⫽ ␳ 1 ␮ 1 / ␳ 2 ␮ 2 . These include the planar and radial
jets and classical momentum wakes studied here, the planar
two-fluid mixing layer,2 the radial two-fluid mixing layer,3
uniform shear flow over a quiescent fluid,4 and normal and
oblique stagnation-point flows of one fluid impinging upon
another.10,11 To date, the only exception appears to be the
two-fluid momentumless wake shown here to be governed
by the parameter ⍀⫽ ␳ 1 ␮ 32 / ␳ 2 ␮ 31 . This is not surprising
since the constraint for self-propelled bodies involves the
second moment of momentum and not the momentum itself.
Solutions for two-fluid planar and fan jets may be
thought of as parallel free jets differently penetrating each
fluid domain. The interesting and important result for our
solution ansatz is that, regardless of conditions at the jet
source, the asymptotic flow achieves a determined partitioning, J 1 /J 2 ⫽ ␹ 1/2, of jet momentum. The interface velocity is
a complicated function of ␹ and ␤ i for one fluid, with ␤ i
defined in 共3e兲. Likewise, the respective jet thicknesses are a
complicated function of ␹ and ␣ i , with ␣ i defined in 共3d兲,
although the ratio of upper-to-lower fluid jet thicknesses is
quite simply ( ␯ 1 / ␯ 2 ) 1/2.
Solutions for the two-fluid classical wake represent
asymptotic Goldstein20 wakes penetrating differently into
each fluid domain with interface deflection proportional to
x ⫺3/2. The interface deficit velocity is a function of ␹ and ␤ i
for one fluid, with ␤ i now defined by 共28e兲. The penetration
depths for the two-fluid wake, in contrast with the two-fluid
jet, are independent of ␹ and depend only on ␣ i as defined in
共28d兲. This simplification of parameter dependence may be
traced to the linearity of the governing partial differential
equation describing the wake flow. The upper-to-lower fluid
wake thickness ratio is ( ␯ 1 / ␯ 2 ) 1/2.
For the momentumless wake, the interface deficit velocity depends on both ⍀ and ␤ i , here defined by 共40d兲, with
interface deflection proportional to x ⫺5/2. The penetration
depths of the two-fluid momentumless wake are independent
of ⍀ and depend only on ␣ i as defined in 共40c兲. Like the
classical wake, the ratio of upper-to-lower wake thicknesses
is again ( ␯ 1 / ␯ 2 ) 1/2.
All two-fluid jet and wake similarity solutions developed
here have velocity profiles smooth at the interface. For planar and fan jets, our results are in agreement with the zerostress interfacial velocity profiles reported in Fig. 2共a兲 of
Didden and Maxworthy15 in their air-over-water experiments. By contrast, asymptotic solutions for spatially driven
two-fluid flows, like the immiscible parallel streams studied
by Lock2 or the uniform shear flow over a quiescent fluid
u(x, ␾ (x))
共cm/s兲
w c (x, ␾ (x))
共cm/s兲
w m (x, ␾ (x))
共cm/s兲
␦1 /␦2
71.88
6.916
7.056
45.42
11.23
0.033 52
0.3454
5.641
⫺656.5
⫺0.019 28
⫺0.019 28
⫺351.15
3.865
33.40
0.1157
1.00
reported by Wang,10 have continuous profiles with a kink at
the interface. This suggests that asymptotic boundary layer
similarity solutions for freely evolving two-fluid flows are
smooth, whereas those for locally driven two-fluid flows exhibit kinks. Mathematical verification of this generality, however, would have to be confirmed by an investigation of the
full Navier–Stokes equations.
The interfacial velocities used to scale the profiles in
Figs. 1, 4, and 5 vary considerably with the fluid combinations considered. These dimensional velocities are listed in
Table II along with the penetration thickness ratios
␦ 1 (x)/ ␦ 2 (x).
In closing, it is understood that jets and wakes are stable
at moderately small Reynolds numbers. Indeed, the choice of
fluids and values of J for the two-fluid jet and of U and D for
the planar two-fluid wake used to generate the profiles presented in Figs. 1 and 4 was predicated on what is expected
for moderate Reynolds number laboratory experiments. The
two-fluid jet and wake solutions presented here provide base
flow solutions on which a stability analyses of these flows
may be performed.
1
D. D. Joseph and Y. Y. Renardy, Fundamentals of Two-Fluid Dynamics,
Part I and Part II 共Springer, New York, 1992兲, pp. 443 and 445.
2
R. C. Lock, ‘‘The velocity distribution in the laminar boundary layer between parallel streams,’’ Q. J. Mech. Appl. Math. 4, 42 共1951兲.
3
D. Katoshevski, I. Frankel, and D. Weihs, ‘‘Viscous interaction between
parallel radial streams,’’ Fluid Dyn. Res. 12, 153 共1993兲.
4
C. Y. Wang, ‘‘The boundary layer due to shear flow over a still fluid,’’
Phys. Fluids A 4, 1304 共1992兲.
5
L. Ting, ‘‘On the mixing of two parallel streams,’’ J. Math. Phys. 38, 153
共1959兲.
6
P. A. Libby and T. M. Liu, ‘‘Further solutions of the Falkner–Skan equation between parallel streams,’’ AIAA J. 5, 1040 共1967兲.
7
R. Mills, ‘‘Numerical and experimental investigations of the shear layer
between two parallel streams,’’ J. Fluid Mech. 33, 591 共1968兲.
8
J. B. Klemp and A. Acrivos, ‘‘A note on the laminar mixing of two uniform parallel semi-infinite streams,’’ J. Fluid Mech. 55, 25 共1972兲.
9
R. D. Small, ‘‘Two-stream mixing: A variational solution,’’ AIAA J. 16, 83
共1978兲.
10
C. Y. Wang, ‘‘Stagnation flow on the surface of a quiescent fluid—An
exact solution of the Navier–Stokes equations,’’ Q. Appl. Math. 43, 215
共1985兲.
11
B. S. Tilley and P. D. Weidman, ‘‘Oblique two-fluid stagnation-point
flow,’’ Eur. J. Mech. B/Fluids 17, 205 共1998兲.
12
A. V. Coward and P. Hall, ‘‘The stability of two-phase flow over a swept
wing,’’ J. Fluid Mech. 329, 247 共1996兲.
13
G. I. Burde, ‘‘Some exact solutions of problems of laminar jets of immiscible fluids,’’ J. Appl. Mech. Tech. Phys. 29, 400 共1988兲 关translated from
Zh. Prikl. Mekh. Teknicheskoi Fiz. 3, 94 共1988兲兴.
14
V. Putkaradze, ‘‘Radial flow of two immiscible fluids: Analytical solutions
and bifurcations,’’ J. Fluid Mech. 477, 1 共2003兲.
Downloaded 04 Jan 2005 to 136.167.58.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
1048
15
Phys. Fluids, Vol. 16, No. 4, April 2004
N. Didden and T. Maxworthy, ‘‘The viscous spreading of plane and axisymmetric gravity currents,’’ J. Fluid Mech. 121, 27 共1982兲.
16
G. C. Koop and F. K. Browand, ‘‘Instability and turbulence in a stratified
fluid with shear,’’ J. Fluid Mech. 93, 135 共1979兲.
17
L. Rosenhead, Laminar Boundary Layers 共Clarendon, Oxford, 1963兲, pp.
252–254.
18
G. K. Batchelor, An Introduction to Fluid Dynamics 共Cambridge University Press, Cambridge, 1973兲.
19
I. A. Lasso and P. D. Weidman, ‘‘Stokes drag on hollow cylinders and
conglomerates,’’ Phys. Fluids 29, 3921 共1986兲.
20
S. Goldstein, ‘‘On the two-dimensional steady flow of a viscous fluid
behind a solid body, Part I,’’ Proc. R. Soc. London, Ser. A 142, 545 共1933兲.
Herczynski, Weidman, and Burde
G. Birkhoff and E. H. Zarantonello, Jets, Wakes and Cavities 共Academic,
New York, 1957兲.
22
D. Tordella and M. Belan, ‘‘A new matched asymptotic expansion for the
intermediate and far flow behind a finite body,’’ Phys. Fluids 15, 1897
共2003兲.
23
E. N. Andrade and H. S. Tsien, ‘‘The velocity distribution in a liquid-intoliquid jet,’’ Proc. Phys. Soc. London 49, 381 共1937兲.
24
A. Revuelta, L. Sánchez, and A. Liñán, ‘‘The virtual origin as a first-order
correction for the far-field description of laminar jets,’’ Phys. Fluids 14,
1821 共2002兲.
25
N. Riley, ‘‘Asymptotic expansions in radial jets,’’ J. Math. Phys. 41, 132
共1962兲.
21
Downloaded 04 Jan 2005 to 136.167.58.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp