AIAA 2009-4291
39th AIAA Fluid Dynamics Conference
22 - 25 June 2009, San Antonio, Texas
Single Scalar Function Satisfying the Conservation of Mass
for Three-Dimensional Flows
Yongho Lee 1
Embry-Riddle Aeronautical University, Daytona Beach, FL 32114
In the present work, a novel stream function satisfying the three-dimensional continuity
equation is defined in the form of a single scalar function and proved to represent the
streamlines of certain three-dimensional flows. Consequently, the analysis and the results
produce a limited counterevidence to the statement about the nonexistence of such function
for three-dimensional flows. This new stream function is considered for both incompressible
and compressible flows, and the examples of irrotational and rotational flows are discussed
in connection with the characteristics of the stream function. In the three-dimensional
irrotational incompressible flow considered herein, it is verified that the new stream function
satisfies the three-dimensional Laplace equation and is orthogonal to the potential function,
as in the case of the Lagrange’s two-dimensional stream function and the corresponding
velocity potential. Unsteady incompressible flows are also discussed in this paper.
Nomenclature
r
t
u
v
v
vr
vz
vθ
w
x
y
z
=
=
=
=
=
=
=
=
=
=
=
=
radial position
time
velocity component in x-direction
velocity component in y-direction
velocity vector
radial component of velocity vector
z-component of velocity in polar or cylindrical coordinate system
circumferential component of velocity vector
velocity component in z-direction
x-coordinate
y-coordinate
z-coordinate
θ
ρ
φ
ψ
=
=
=
=
=
angle in circumferential direction
density
velocity potential
stream function
gradient operator
∇
I
I. Introduction
t is well known that the stream function was first introduced by Louis de Lagrange in the late 18th century for
two-dimensional flows, in which the stream function replaces two unknown variables by identically satisfying the
conservation of mass and, as a result, eliminates the need to solve conservation of mass and momentum
simultaneously. Expressed in terms of the stream function, the governing equations are reduced to one scalar
equation if, for example, the flow is two-dimensional and incompressible, where the stream function (ψ) is defined
as following:
1
Assistant Professor, Dep. of Mechanical, Civil and Engineering Sciences, 600 S. Clyde Morris Blvd., Daytona
Beach, FL 32114, Member AIAA.
1
American Institute of Aeronautics and Astronautics
Copyright © 2009 by Yongho Lee. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
u≡
∂ψ
∂y
v≡−
∂ψ
,
∂x
with u and v denoting the velocity components in the two spatial coordinates, x and y, respectively. The motivation
of this definition of stream function, which is not unique, is to make ψ identically satisfy the conservation of mass,
∂u ∂v
+
= 0.
∂x ∂y
It has long been noticed that two independent stream functions can describe a three-dimensional flow field (see,
for example, Ref. 1). Reference 2 states explicitly, and Ref. 3 rather implicitly and indirectly, that the (single) stream
function is not defined in three-dimensional flows, where the conservation of mass, compared with the above twodimensional equation, involves one more term – the derivative of the third velocity component with respect to the
third spatial variable. Many textbooks confine the discussion of stream function to two-dimensional flows without
associating it with three-dimensional flows. Reference 4 concluded through an extensive review of published
literature that the single stream function satisfying continuity equation for three-dimensional flows is not available,
hence they attempted to define three-dimensional stream function vector with three scalar components. However,
the statements about the nonexistence of a single scalar stream function describing a three-dimensional flow are
partly disproved by a new definition of stream function, considered and exemplified in the present work, which
concentrates only on the existence of such function and its characteristics.
II. Analysis and Results
The fact that the single scalar potential function associated with the velocity components exists for a threedimensional irrotational flow signifies the existence of the single scalar stream function, at least, in some threedimensional irrotational flows. If a flow is three-dimensional and incompressible, the nondimensionalized
conservation of mass is
∇•v=
∂u ∂v ∂w
+
+
=0,
∂x ∂y ∂z
(1)
where ∇ is the gradient operator, v is the velocity vector, and u, v, and w are the velocity components in x, y, and
z-direction, respectively, in the Cartesian coordinate system. Let us define a new scalar stream function, ψ, for threedimensional flow as
u≡
∂ 2ψ
∂y∂z
v≡
∂ 2ψ
∂x∂z
w ≡ −2
∂ 2ψ
,
∂x∂y
(2)
which is readily proven to satisfy the three-dimensional continuity equation, Eq. (1). This definition of ψ, which is
not unique as the counterpart of two-dimensional flows, is valid for both steady-incompressible and unsteadyincompressible flows, inasmuch as both types of flows are described by the same conservation of mass, Eq. (1). The
continuity equation, if written in terms of ψ defined in Eq. (2), is equivalent to
∂ 3ψ
∂ 3ψ
∂ 3ψ
+
−2
=0.
∂x∂y∂z ∂x∂y∂z
∂x∂y∂z
(3)
Note that any smooth scalar function ψ (or more restrictively, any scalar function that is differentiable once with
respect to each spatial variable) satisfies this relation, since the left-hand side of the equation is identically equal to
zero for any such function, ψ. In the case of the well-known Lagrange’s stream function defined somewhat
differently for two-dimensional flows, the continuity equation is also satisfied by any smooth scalar function ψ
defined appropriately for such flows.
By analogy with Lagrange’s stream function in two-dimensional flows, one can deduce that a three-dimensional
steady compressible flow satisfying the conservation of mass,
2
American Institute of Aeronautics and Astronautics
∇ • ( ρ v) =
∂ ( ρ u ) ∂ ( ρ v ) ∂ ( ρ w)
+
+
= 0,
∂x
∂y
∂z
with ρ representing the density, can also have a single scalar stream function defined as
ρu ≡
∂ 2ψ
∂y∂z
ρv ≡
∂ 2ψ
∂x∂z
ρ w ≡ −2
∂ 2ψ
.
∂x∂y
In the cylindrical coordinate system, the conservation of mass for three-dimensional incompressible flows is, for
both steady and unsteady flows,
∇•v=
1 ∂ (r v r ) 1 ∂ ( vθ ) ∂ (v z )
+
+
= 0,
∂z
r ∂r
r ∂θ
where vr, vθ, and vz are respectively the velocity components in the radial direction, r, the circumferential direction,
θ, and the third direction, z, which is orthogonal to both r and θ-directions. From this equation, we can define the
single stream function as
vr ≡
1 ∂ 2ψ
r ∂θ ∂z
vθ ≡
∂ 2ψ
∂r∂z
vz ≡ −
2 ∂ 2ψ
.
r ∂r∂θ
If the flow is compressible but independent of time, the three-dimensional continuity equation in cylindrical
coordinate system is
∇ • ( ρ v) =
1 ∂ (rρ v r ) 1 ∂ ( ρ vθ ) ∂ ( ρ v z )
+
+
=0,
r
r ∂θ
∂r
∂z
and this equation allows us to introduce the single scalar stream function defined as
ρ vr ≡
1 ∂ 2ψ
r ∂θ ∂z
ρ vθ ≡
∂ 2ψ
∂r∂z
ρ vz ≡ −
2 ∂ 2ψ
.
r ∂r∂θ
As the first example of the new stream function, let us consider a simple three-dimensional flow field described
by the three nondimensional velocity components,
u=x
v= y
w = −2 z .
(4)
This steady flow is an incompressible irrotational (or potential) flow since ∇ • v = 0 and ∇ × v = 0 . Integration of
the three second-order partial differential equations in Eq. (2) with the flow in Eq. (4) leads to the following scalar
function:
ψ = xyz + f 1 ( x) + f 2 ( y ) + f 3 ( z ) ,
where f 1 ( x) , f 2 ( y ) , and f 3 ( z ) are arbitrary functions that are differentiable once with respect to x, y, and z,
respectively. It can be easily verified by using Eq. (2) that this single scalar stream function is the correct
representation of the three-dimensional irrotational incompressible flow in Eq. (4). The three arbitrary functions can
be determined by enforcing the general condition for the surfaces of ψ = constant to be tangent to the velocity
vectors in the flow field:
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American Institute of Aeronautics and Astronautics
v • ∇ψ = u
∂ψ
∂ψ
∂ψ ∂ 2ψ ∂ψ ∂ 2ψ ∂ψ
∂ 2ψ ∂ψ
+v
+w
=
+
−2
= 0,
∂x
∂y
∂z ∂y∂z ∂x ∂x∂z ∂y
∂x∂y ∂z
since the direction of ∇ ψ is normal to the surface represented by ψ = constant. When applied to the flow in Eq. (4),
this tangency condition – notice that it is not equivalent to Eq. (3) satisfied by any smooth function – is simplified to
xf 1′( x) + yf 2′ ( y ) − 2 zf 3′ ( z ) = 0 ,
an obvious solution of which states that all three functions, f 1 ( x) , f 2 ( y ) , and f 3 ( z ) , are arbitrary constants. Thus,
the stream function of this flow can be rewritten in the following simple form consistent with Eq. (4):
ψ = xyz .
We can interpret the surfaces described by ψ = constant as the stream surfaces that are the collections of the stream
lines tangent to the velocity vectors in the flow field. The flow in Eq. (4) associated with the simple stream function
obtained above is somewhat similar to, but more complex than, the two-dimensional inviscid flow near a stagnation
point on a flat plate perpendicular to the free stream. If the momentum equation is applied to Eq. (4), the pressure
variation is also found to be similar to that of the two-dimensional inviscid flow near the stagnation point.
It is well known that, if a flow is irrotational ( ∇ × v = 0 ), a single scalar velocity potential function, φ, exists
such that v = ∇ φ , i.e.,
u=
∂φ
∂x
v=
∂φ
∂y
w=
∂φ
,
∂z
(5)
and, since ∇ • v = ∇ • ∇ φ = 0 for incompressible irrotational flows, the velocity potential of such flow is the
solution of the Laplace equation written as
∂ 2φ
∂x 2
+
∂ 2φ
∂y 2
+
∂ 2φ
∂z 2
=0.
This property indicates that the irrotational incompressible flow in Eq. (4) is an inviscid (or frictionless) flow,
because the viscous term in the incompressible Navier-Stokes equation with constant viscosity is identically equal to
zero. By integrating the three equations in Eq. (5) with the velocity components in Eq. (4), the velocity potential is
determined to be
φ=
1 2 1 2
x + y − z2 .
2
2
If and only if the stream surfaces represented by ψ = constant are orthogonal to the equipotential surfaces for which
φ = constant, we have
∇φ • ∇ ψ =
∂φ ∂ψ ∂φ ∂ψ ∂φ ∂ψ
+
+
= v • ∇ψ = 0 .
∂x ∂x ∂y ∂y ∂z ∂z
This condition is satisfied automatically at all points in the flow field if the flow is, as in the flow of Eq. (4),
irrotational-incompressible and the surfaces of ψ = constant are tangent to the velocity vectors. In addition, it can be
readily proved that the stream function obtained above, for the simple irrotational flow in Eq. (4), also satisfies the
three-dimensional Laplace equation for ψ:
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American Institute of Aeronautics and Astronautics
∂ 2ψ
∂x
2
+
∂ 2ψ
∂y
2
+
∂ 2ψ
∂z 2
= 0.
Figure 1(a) shows two stream surfaces for ψ = 0.1 and ψ = 1, plotted together with the velocity vectors of the
example flow of Eq. (4), in the domain corresponding to 0 < x ≤ 2, 0 < y ≤ 2, and 0 < z ≤ 2. The velocity vectors
are observed to be tangent to the stream surfaces. Apparently demonstrated in Fig. 1(b) is the orthogonality between
the stream surfaces (only ψ = 1 shown for simplicity) and the equipotential surfaces (φ = 1 and φ = −1). If the
velocity potential of this idealized flow is expressed in cylindrical coordinates, it is a function of only two spatial
variables, r and z. However, the stream function and the velocity components depend on θ as well as r and z, whence
the flow in equation (4) is truly a three-dimensional flow. To some extent, the flow pattern in Fig. 1(a) is akin to part
of the inviscid corner flow with a stagnation point at the origin, and partly similar to the inviscid flow approaching
an object with the shape of the cross-head screwdriver aligned with the flow direction. The flow in the region where
z < 0 is the mirror image of the flow in z > 0.
2
φ = −1
1.5
2
1.5
1
z
2
1
0.5
1.5
z
0.5
0
2
0.5
0
1
y
0.5
ψ=1
φ=1
1
1
x
1.5
1
0.5
x
1.5
2
0
1.5
0.5
y
20
(a)
(b)
Figure 1. Plots of the irrotational flow in Eq. (4): (a) two stream surfaces and velocity vectors, (b) a
stream surface compared with two equipotential surfaces.
As the second example of the new stream function, let us consider a rotational flow that is steady, threedimensional, incompressible, and described by the nondimensional velocity components,
u = 2x 2 + 6
v = 4 x ( y + 3)
w = −8 xz ,
(6)
which obviously yield ∇ • v = 0 and ∇ × v ≠ 0 . In this rotational flow, the velocity potential does not exist, however
the single scalar stream function can still be obtained as following, by repeating the procedure described earlier for
the flow in Eq. (4):
ψ = (2 x 2 + 6) ( y + 3) z .
This single stream function is consistent with the three velocity components in Eq. (6), satisfy the conservation of
mass in Eq. (1), and v • ∇ ψ = 0 , indicating that the surfaces of ψ = constant are tangent to the velocity vectors
throughout the flow field. Figure 2 exhibits three stream surfaces of the flow in Eq. (6) for ψ = 10, ψ = 25, and ψ =
5
American Institute of Aeronautics and Astronautics
40, and the velocity vectors in the octant formed by 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, and 0 ≤ z ≤ 2. Both flows in Fig. 1
and 2 have the infinite velocity in the regions far away from the origin, as the well-known two-dimensional (either
inviscid or viscous) solution of the stagnation-point flow.
2
y
1
0
2
1.5
z
1
0.5
0
0
0.5
1
x
1.5
2
Figure 2. Stream surfaces and velocity vectors of the steady rotational incompressible flow in Eq. (6).
The two simple flows discussed above are the special cases of more general, three-dimensional, steady,
incompressible flows of which velocity components can be written in the form of
u = g1 ( x) g 2 ( y ) g 3 ( z )
v = g 4 ( x) g 5 ( y ) g 6 ( z )
w = g 7 ( x) g 8 ( y) g 9 ( z ) ,
(7)
where g 1 through g 9 are any differentiable functions that make the velocity components satisfy the conservation of
mass in Eq. (1), with the spatial dependence specified in Eq. (7); each of the functions, g 1 through g 9 , can contain
a separate additive constant term in it. For the flows of the type in Eq. (7), the stream function can be determined by
following the aforementioned procedure in the two examples. If the velocity components cannot be written in the
separable form as in Eq. (7), the integration of Eq. (2) and the application of the tangency condition may be more
complicated, or it may be even impossible to determine the single scalar stream function of which isosurfaces are
tangent to the velocity vectors. However, even in the case where the surfaces of ψ = constant do not represent the
stream surfaces tangent to the velocity vectors, the single scalar function ψ still satisfies the conservation of mass of
three-dimensional flows, and reduces the number of unknowns in the continuity equation. The same procedures and
comments apply to unsteady incompressible flows, if the velocity components are as following:
u = g 1 ( x, t ) g 2 ( y , t ) g 3 ( z , t )
v = g 4 ( x, t ) g 5 ( y , t ) g 6 ( z , t )
w = g 7 ( x, t ) g 8 ( y , t ) g 9 ( z , t ) .
For instance, let us consider a simple unsteady, incompressible, rotational, three-dimensional flow. It can be proved
that a nondimensional scalar function,
ψ =−
( x + t ) 2 ( y + 2 sin 2 t ) z
2(t 2 + 1)
,
corresponds to the stream function of the following velocity field, and satisfies the continuity equation and the
orthogonality condition between v and the surfaces of ψ = constant:
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American Institute of Aeronautics and Astronautics
u=−
(x + t) 2
v=−
2
2(t + 1)
( x + t ) ( y + 2 sin 2 t )
2
t +1
w=
2( x + t ) z
t 2 +1
.
(8)
Figure 3(a) and 3(b) display the velocity vectors of this flow with two representative stream surfaces at t = 1 and 6.
In Fig. 3(a), the stream surfaces are presented for ψ = 1 and 1/3, and in Fig. 3(b), ψ = 1/7 and 1/30. The figure
clearly shows the time-dependence of the velocity vectors and the stream surfaces that are tangent to each other.
0
0
-0.5
z
-1
2
-1.5
1.5
-2
0
1
y
-0.5
z
-1
2
-1.5
1.5
-2
0
0.5
1
0.5
1
x
0.5
1.5
1
x
2
0.5
1.5
2
0
(a)
y
0
(b)
Figure 3. Stream surfaces and velocity vectors of the unsteady rotational incompressible flow in Eq.
(8): (a) t = 1, (b) t = 6.
III. Conclusion
It was attempted to define a new stream function – a single scalar function – for three-dimensional flows;
incompressible (steady or unsteady) and steady compressible flows. Both Cartesian and cylindrical coordinate
systems were considered. This single stream function obtained by the novel definition is verified to possess the
characteristics similar to those of Lagrange’s two-dimensional stream function – it is tangent to the velocity vectors,
and in irrotational incompressible flows it is orthogonal to the potential function. The orthogonality between the
two-dimensional stream function and potential function has been well known, and the existence of the threedimensional potential function in irrotational flows was established long time ago, however the existence of the
single scalar stream function orthogonal to the potential function of a three-dimensional flow is a new finding,
according to the knowledge of the present author.
Albeit the single scalar function attested for the streamlines of general three-dimensional flows has not been
found, the stream function defined in the present work has desirable characteristics and applies to, at least, certain
types of three-dimensional flows identified in the present work. Even when the newly defined function does not
represent the stream surfaces tangent to the velocity vectors, the single scalar function still satisfies the conservation
of mass, reducing the number of unknowns in the governing equations of three-dimensional flows.
At present, general comments cannot be made about the use of the new scalar function in solving threedimensional fluid flow problems, because, for example, it involves higher-order derivatives compared with the twodimensional version. In the case of a two-dimensional incompressible flow, the velocity components are the firstderivatives of stream function, and the momentum equation becomes a fourth-order partial differential equation of
stream function. On the contrary, in three-dimensional flows discussed herein, the velocity components are the
second-derivatives of the scalar function, which increases the order of derivatives in momentum equation even
further than that of the two-dimensional flow.
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American Institute of Aeronautics and Astronautics
References
1
Karamcheti, K. (1966), Principles of Ideal-Fluid Aerodynamics, John Wiley & Sons, Inc., 1966, Chap. 4.
2
White, F. M., Fluid Mechanics, 2nd ed., McGraw-Hill, New York, 1988, Chap. 4.
3
Anderson, J. D., Fundamentals of Aerodynamics, McGraw-Hill, New York, 1984, Chap. 3.
4
Elshabka, A. M. and Chung, T. J., “New definition of three-dimensional stream function vector, verification of theory, and
associated physics,” 1st AIAA Theoretical Fluid Mechanics Meeting, AIAA Paper 96-2130, 1996.
8
American Institute of Aeronautics and Astronautics