Chapter 8
IDEAL-FLUID FLOW

In the preceding chapters most of the relations have been developed for
one-dimensional flow, i.e., flow in which the average velocity at each cross
section is used and variations across the section are neglected.

Many design problems in fluid flow, however, require more exact
knowledge of velocity and pressure distributions, such as in flow over
curved boundaries along an airplane wing, through the passages of a
pump or compressor, or over the crest of a dam.

An understanding of two- and three-dimensional flow of a nonviscous,
incompressible fluid provides the student with a much broader approach
to many real fluid-flow situations. There are also analogies that permit the
same methods to apply to flow through porous media.

In this chapter the principles of irrotational flow of an ideal fluid are
developed and applied to elementary flow cases. After the flow
requirements are established, Euler's equation is derived and the velocity
potential is defined.
8.1 REQUIREMENTS FOR IDEAL-FLUID FLOW



The Prandtl hypothesis states that for fluids of low viscosity the effects of
viscosity are appreciable only in a narrow region surrounding the fluid
boundaries.
For incompressible flow situations in which the boundary layer remains
thin, ideal-fluid results may be applied to flow of a real fluid to a
satisfactory degree of approximation.
An ideal fluid must satisfy the following requirements:
 1. The continuity equation div q = 0, or

2. Newton's second law of motion at every point at every instant

3. Neither penetration of fluid into, nor gaps between, fluid and
boundary at any solid boundary

If, in addition to requirements 1, 2, and 3, the assumption of
irrotational flow is made, the resulting fluid motion closely resembles
real-fluid motion for fluids of low viscosity, outside boundary layers.

Using the above conditions, the application of Newton's second law to a
fluid particle leads to the Euler equation, which, together with the
assumption of irrotational flow, can be integrated to obtain the
Bernoulli equation. The unknowns in a fluid-flow situation with given
boundaries are velocity and pressure at every point.

Unfortunately, in most cases it is impossible to proceed directly to
equations for velocity and pressure distribution from the boundary
conditions.
8.2 EULER'S EQUATION OF MOTION


Euler's equation of motion along a streamline (one-dimensional) was
developed in Sec. 3.5 by use of the momentum and continuity equations
and Eq. (2.2.5).
In this section it is developed from Eq. (2.2.5) for the xyz-coordinate
system in any orientation, with the assumption that gravity is the only
body force acting. Since Euler's equation is based on a frictionless fluid,
the vector equation (2.2.5)
(2.2.5)

may be reorganized into the proper form. The unit vector j' is directed
vertically upward in the coordinate direction h.

These are the direction cosines of h with respect to the xyz system of
coordinates, and they may be written
Figure 8.1 Arbitrary orientation of xyz
coordinate system

For example, ∂h/∂x is the change in h for unit change in x when y, z,
and t are constant. In equation form,

The operation
applied to the scalar h yields the gradient of h, as in
Eq. (2.2.2).
Eq. (2.2.5) now becomes
(8.2.1)


The component equations of Eq.(8.2.1) are
(8.2.2)

u, v, w are velocity components in the x, y, z directions, respectively, at
any point; du/dt is the x component of acceleration of the fluid particle
at (x, y, z).

Since u is a function of x, y, z, and t and x, y, and z are coordinates of
the moving fluid particle, they become functions of t; hence,

However, dx/dt, dy/dt, and dz/dt are the velocity components of the
particle, so that ax, the x component of the particle acceleration, is

By treating dv/dt and dw/dt in a similar manner, the Euler equations in
three dimensions for a frictionless fluid are
(8.2.3)
(8.2.4)
(8.2.5)

The first three terms on the right-hand sides of the equations are
convective-acceleration terms, depending upon changes of velocity with
space.

The last term is local acceleration, depending upon velocity change
with time at a point.
Natural Coordinates in Two-Dimensional Flow

Euler's equations in two dimensions are obtained from the generalcomponent equations by setting w = 0 and ∂/∂z=0; thus
(8.2.6)
(8.2.7)

By taking particular directions for the x and y axes, they can be reduced to
a form that makes them easier to understand.

The velocity component u is vs, and the component v is vn. As vn is zero at
the point, Eq. (8.2.6) becomes
(8.2.8)

Although vn is zero at the point (s, n), its rates of change with respect to s
and t are not necessarily zero. Equation (8.2.7) becomes
(8.2.9)
Figure 8.2 Notation for natural coordinates

With r the radius of curvature of the streamline at s, from similar triangles
(Fig. 8.2),

Substituting into Eq. (8.2.9) gives
(8.2.10)

For steady flow of an incompressible fluid Eqs. (8.2.6) and (8.2.10) can be
written
(8.2.11)
and
(8.2.12)


Equation (8.2.11) can be integrated with respect to s to produce Eq. (3.6.1),
with the constant of integration varying with n, that is, from one
streamline to another.
Equation (8.2.12) shows how pressure head varies across streamlines. With
vs and r known functions of n, Eq. (8.2.12) can be integrated.
Example 8.1

A container of liquid is rotated with angular velocity ω about a vertical
axis as a solid. Determine the variation of pressure in the liquid.
Solution

n is the radial distance, measured inwardly, dn = -dr, and vs = ωr .
Integrating Eq. (8.2.12) gives
or

To evaluate the constant, if p = p0 when r = 0 and h = 0,

which shows that the pressure is hydrostatic along a vertical line and
increases as the square of the radius.
Integration of Eq. (8.2.11) shows that the pressure is constant for a given h
and vs, that is, along a streamline.

8.3 IRROTATIONAL FLOW; VELOCITY
POTENTIAL

In this section it is shown that the assumption of irrotational flow leads
to the existence of a velocity potential. By use of these relations and the
assumption of a conservative body force, the Euler equations can be
integrated.

The individual particles of a frictionless incompressible fluid initially
at rest cannot be caused to rotate. This can be visualized by
considering a small free body of fluid in the shape of a sphere. Surface
forces act normal to its surface, since the fluid is frictionless, and
therefore act through the center of the sphere.

Similarly, the body force acts at the mass center. Hence, no torque can
be exerted on the sphere, and it remains without rotation. Likewise,
once an ideal fluid has rotation, there is no way of altering it, as no
torque can be exerted on an elementary sphere of the fluid.

An analytical expression for fluid rotation of a particle about an axis
parallel to the z axis is developed. The rotation component may be defined
as the average angular velocity of two infinitesimal linear elements that are
mutually perpendicular to each other and to the axis of rotation.

The two line elements may conveniently be taken as x and y in Fig. 8.3,
although any other two perpendicular elements in the plane through the
point would yield the same result. The particle is at P(x, y), and it has
velocity components u, v in the xy plane. The angular velocities of δx and
δy are sought.

The angular velocity of δx is

and the angular velocity of δy is

if counterclockwise is positive. Hence, by definition, the rotation
component ωz of a fluid particle at (x,y) is
(8.3.1)
Figure 8.3 Rotation in a fluid

Similarly, the two other rotation components, ωx and ωy, about axes
parallel to x and to y are
(8.3.2)

The rotation vector ω is
(8.3.3)
× q, is defined as twice the rotation vector.

The vorticity vector, curl q =
It is given by 2ω.

By assuming that the fluid has no rotation, i.e., it is irrotational, curl q = 0,
or from Eqs. (8.3.1) and (8.3.2)
(8.3.4)

These restrictions on the velocity must hold at every point (except special
singular points or lines).

The first equation is the irrotational condition for two dimensional flow. It
is the condition that the differential expression

is exact, say
(8.3.5)

The minus sign is arbitrary; it is a convention that causes the value of φ to
decrease in the direction of the velocity. By comparing terms in Eq. (8.3.5),

In vector form,
(8.3.6)

is equivalent to
(8.3.7)

The assumption of a velocity potential is equivalent to the assumption of
irrotational flow, as
(8.3.8)

because
differentiation:
. This is shown from Eq. (8.3.7) by cross-

proving

Substitution of Eqs. (8.3.7) into the continuity equation

yields
etc.
(8.3.9)

In vector form this is
(8.3.10)

and is written

Equation (8.3.9) or (8.3.10) is the Laplace equation. Any function that
satisfies the Laplace equation is a possible irrotational fluid-flow case. As
there are an infinite number of solutions to the Laplace equation, each of
which satisfies certain flow boundaries, the main problem is the selection
of the proper function for the particular flow case.

Because appears to the first power in each term, Eq. (8.3.9), is a linear
equation, and the sum of two solutions also is a solution; e.g., if φ1 and φ2
are solutions of Eq. (8.3.9), then φ1 + φ2 is a solution; thus

then

Similarly, if φ1 is a solution, Cφ1 is a solution if C is constant.
8.4 INTEGRATION OF EULER`S EQUATIONS;
BERNOULLI EQUATION

Equation (8.2.3) can be rearranged so that every term contains a partial
derivative with respect to x. From Eq. (8.3.4)

and from Eg. (8.3.7)

Making these substitution into Eq. (8.2.3) and rearranging give

As
the square of the speed,
(8.4.1)

Similarly, for the y and z direction,
(8.4.2)
(8.4.3)


The quantities within the parentheses are the same in Eqs. (8.4.1) to
(8.4.3). Equation (8.4.1) states that the quantity is not a function of x, since
the derivative with respect to x is zero.
Similarly, the other equations show that the quantity is not a function of y
or z. Therefore, it can be a function of t only, say F(t):
(8.4.4)

In steady flow ∂φ/∂t=0 and F(t) becomes a constant E:
(8.4.5)

The available energy is everywhere constant throughout the fluid. This is
Bernoulli's equation for an irrotational fluid.

The pressure term can be separated into two parts, the hydrostatic
pressure ps, and the dynamic pressure pd, so that ps + pd. Inserting in Eq.
(8.4.5) gives

The first two terms can be written

with h measured vertically upward. The expression is a constant, since it
expresses the hydrostatic law of variation of pressure. These two terms
may be included in the constant E. After dropping the subscript on the
dynamic pressure, there remains
(8.4.6)

This simple equation permits the variation in pressure to be determined if
the speed is known or vice versa. Assuming both the speed q0 and the
dynamic pressure p0 to be known at one point,
(8.4.7)
Example 8.2


A submarine moves through water at a speed of 10 m/s. At a point A on the
submarine 1.5 m above the nose, the velocity of the submarine relative to
the water is 15 m/s. Determine the dynamic pressure difference between
this point and the nose, and determine the difference in total pressure
between the two points.
Solution
If the submarine is stationary and the water is moving past it, the velocity
at the nose is zero and the velocity at A is 15 m/s. By selecting the dynamic
pressure at infinity as zero, from Eq. (8.4.6)

For the nose

For point A

Therefore, the difference in dynamic pressure is

The difference in total pressure can be obtained by applying Eq. (8.4.5) to
point A and to the nose n,

Hence

It can also be reasoned that the actual pressure difference varies by 1.5γ
from the dynamic pressure difference since A is 1.5 m above the nose, or
8.5 STREAM FUNCTIONS; BOUNDARY
CONDITIONS
Two-Dimensional Stream Function



If A, P represent two points in one of the flow planes, e.g., the xy
plane (Fig. 8.4), and if the plane has unit thickness, the rate of flow
across any two lines ACP, ABP must be the same if the density is
constant and no fluid is created or destroyed within the region, as a
consequence of continuity.
If A is a fixed point and P a movable point, the flow rate across any
line connecting the two points is a function of the position of P.
If this function is ψ, and if it is taken as a sign convention that it
denotes the flow rate from right to left as the observer views the
line from A looking toward P, then is defined as the stream function.
Figure 8.4 Fluid region showing the
positive flow direction used in the
definition of a stream function
Figure 8.5 Flow between two
points in a fluid region

If ψ1 and ψ2 represent the values of stream function at points Pl and
P2 (Fig. 8.5), respectively, then ψ2 - ψ1 is the flow across PlP2 and is
independent of the location of A.

Taking another point 0 in the place of A changes the values of ψ1
and ψ2 by the same amount, namely, the flow across OA. Then ψ is
indeterminate to the extent of an arbitrary constant.

The velocity components u, v in the x, y directions can be obtained
from the stream function. In Fig. 8.6a, the flow δψ across AP=δy,
from right to left, is -uδy, or
(8.5.1)
(8.5.2)

In plane polar coordinates from Fig. 8.6b.

Comparing Egs. (8.3.3) with Eqs. (8.5.1) and (8.5.2) leads to
(8.5.3)

These are the Cauchy-Riemann equations.

By Eqs. (8.5.3), a stream function can be found for each velocity
potential. If the velocity potential satisfies the Laplace equation, the
stream function also satisfies it. Hence, the stream function may be
considered as velocity potential for another flow case.
Figure 8.6 Selection of path to show relation of
velocity components to stream function
Stokes's Stream Function for Axially Symmetric Flow


In any one of the planes through the axis of symmetry select two
points A, P such that A is fixed and P is variable. Draw a line
connecting AP.
The flow through the surface generated by rotating AP about the
axis of symmetry is a function of the position of P. Let his function
be 2Pψ, and let the x axis of symmetry be the axis of a cartesian
system of reference. Then ψ is a function of x and ω, where

is the distance from P to the x axis. The surfaces ψ = const are
stream surfaces.

The resulting relations between stream function and velocity are
given by

Solving for u and v’ gives
(8.5.4)


The same sign convention is used as in the two-dimensional case.
The relations between stream function and potential function are
(8.5.5)

The stream function is used for flow about bodies of revolution that
are frequently expressed most readily in spherical polar
coordinates. From Fig. 8.7a and b,

from which
(8.5.6)

and
(8.5.7)

These expressions are useful in dealing with flow about spheres,
ellipsoids, and disks and through apertures.
Figure 8.7 Displacement of to show the relation between
velocity components and Stokes' stream function.
Boundary Conditions

At a fixed boundary the velocity component normal to the boundary must
be zero at every point on the boundary (Fig. 8.8):
(8.5.8)

n1 is a unit vector normal to the boundary. In scalar notation this is easily
expressed in terms of the velocity potential
(8.5.9)

at all points on the boundary. For a moving boundary (Fig. 8.9), where the
boundary point has the velocity V, the fluid velocity component normal to
the boundary must equal the velocity of the boundary normal to
boundary; thus
(8.5.10)

or
(8.5.11)
Figure 8.8 Notation for
boundary condition at a fixed
boundary
Figure 8.9 Notation for boundary
condition at a moving boundary
8.6 THE FLOW NET


In two-dimensional flow the flow net is of great benefit; it is taken up
in this section.
The line given by φ(x, y)=const is called an equipotential line. It is a
line along which the value of φ (the velocity potential) does not change.
Since velocity vs in any direction s is given by

The line φ(x, y)=const is a streamline and is everywhere tangent to the
velocity vector. Streamlines and equipotential lines are therefore
orthogonal; i.e., they intersect at right angles, except at singular points.

In Fig. 8.10, if the distance between streamlines is Δn and the distance
between equipotential lines is Δs at some small region in the flow net,
the approximate velocity vs is the given in terms of the spacing of the
equipotential lines [Eq. (8.3.7)],

or in terms of the spacing of streamlines [Eqs. (8.5.1) and (8.5.2)].
Figure 8.10 Elements of a flow net


In steady flow when the boundaries are stationary, the boundaries
themselves become part of the flow net, as they are streamlines. The
problem of finding the flow net to satisfy given fixed boundaries
may be considered purely as a graphical exercise
This is one of the practical methods employed in two-dimensionalflow analysis, although it usually requires many attempts and much
erasing.

Another practical method of obtaining a flow net for a particular
set of fixed boundaries is the electrical analogy. The boundaries in a
model are formed out of strips of nonconducting material mounted
on a flat nonconducting surface, and the end equipotential lines are
formed out of a conducting strip, e.g., brass or copper.

The relaxation method numerically determines the value of
potential function at points throughout the flow, usually located at
the intersections of a square grid. The laplace equation is written as
a difference equation, and it is shown that the value of the potential
function at a grid point is the average of the four values at the
neighboring grid points.
Use of the Flow Net

After a flow net for a given boundary configuration has been
obtained, it may be used for all irrotational flows with
geometrically similar boundaries.

Application of the Bernoulli equation [Eq. (8.4.7)] produces the
dynamic pressure. If the velocity is known, e.g., at A (Fig. 8.10).
With the constant Δc determined for the whole grid in this manner,
measurement of Δs or Δn at any other point permits the velocity to
be computed there,


The concepts underlying the flow net have been developed for
irrotational flow of an ideal fluid.
8.7 TWO-DIMENSIONAL FLOW
Flow around a Corner

The potential function

has as its stream function

in which r and θ are polar coordinates. It is plotted for equalincrement changes in φ and ψ in Fig. 8.11. Conditions at the origin
are not defined, as it is a stagnation point.

The streamlines are rectangular hyperbolas having y=±x as axes
and the coordinate axes as asymptotes. From the polar form of the
stream function it is noted that the two lines θ=0 and θ=π/2 are the
streamline ψ=0.
Figure 8.11 Flow net for flow around a 90o bend
Figure 8.12 Flow net for flow along two inclined surfaces
Source

A line normal to the xy plane, from which fluid is imagined to flow
uniformly in all directions at right angles to it, is a source. It
appears as a point in the customary two-dimensional flow diagram.
The total flow per unit time per unit length of line is called the
strength of the source.

Since by Eq. (8.3.7) the velocity in any direction is given by the
negative derivative of the velocity potential with respect to the
direction,

is the velocity potential, in which r is the distance from the source.
This value of φ satisfies the Laplace equation in two dimensions.

The streamlines are radial lines from the source, i.e.,

From the second equation

Lines of constant φ (equipotential lines) and constant ψ are shown
in Fig. 8.13. A sink is a negative source, a line into which fluid is
flowing.
Figure 8.13 Flow net for
source or vortex
Vortex

In examining the flow case given by selecting the stream function
for the source as a velocity potential,


which also satisfies the Laplace equation, it is seen that the
equipotential lines are radial lines and the streamlines are circles.
The velocity is in a tangential direction only, since ∂φ/∂r=0 . It is

since r ∂φ is the length element in the tangential direction.

In referring to Fig. 8.14, the flow along a closed curve is called the
circulation. The circulation Γ around a closed path C is

The value of the circulation is the strength of the vortex. By
selecting any circular path of radius r to determine the circulation,
hence,
Figure 8.14 Notation for definition of
circulation
Doublet

The two-dimensional doublet is defined as the limiting case as a
source and sink of equal strength approach each other so that the
product of their strength and the distance between them remains a
constant 2πμ. μ is called the strength of the doublet.

In Fig. 8.15 a source is located at (a, 0) and a sink of equal strength
at (-a, 0). The velocity potential for both, at some point P, is

with r1, r2 measured from source and sink, respectively, to the point
P.
Figure 8.15 Notation for derivation of a
two-dimensional doublet

The terms r1, and r2 may be expressed in terms of the polar
coordinates r, θ by the cosine law, as follows:

Rewriting the expression for φ with these relations gives

The series expression

leads to

After simplifying,

If 2am=μ, and if the limit is taken as a approaches zero,

which is the velocity potential for a two-dimensional doublet at the
origin, with axis in the +x direction.

Using the relations

gives for the doublet

After integrating,

The equations in cartesian coordinates are

Rearranging gives

The lines of constant φ are circles through the origin with centers
on the x axis, and the streamlines are circles through the origin
with centers on the y axis, as shown in Fig. 8.16.

The origin is a singular point where the velocity goes to infinity.
Figure 8.16 Equipotantial lines and streamlines for
the two-dimensional doublet
Uniform Flow

Uniform flow in the -x direction, u = -U, is expressed by

In polar coordinates,
Flow around a Circular Cylinder

The addition of the flow due to a doublet and a uniform flow results
in flow around a circular cylinder; thus

As a streamline in steady flow is a possible boundary, the
streamline ψ=0 is given by

which is satisfied by θ=0, π, or by the value of r that makes

If this value is r=a, which is a circular cylinder, then

The potential and stream functions for uniform flow around a
circular cylinder of radius a are, by substitution of the value of μ

for the uniform flow in the -x direction. The equipotential lines and
streamlines for this case are shown in Fig. 8.17

On the surface of the cylinder the velocity is necessarily tangential
and is expressed by ∂ψ/∂r=0 for r=a; thus

For the dynamic pressure zero at infinity, with Eq. (8.4.7)
Figure 8.17 Equipotential lines and streamlines
for flow around a circular cylinder

For points on the cylinder,

A cylindrical pilot-static tube is made by providing three openings
in a cylinder, at 0 and ± 30°, as the difference in pressure between 0
and ±30° is the dynamic pressure ρU2/2.

The drag on the cylinder is shown to be zero by integration of the x
component of the pressure force over the cylinder; thus

Similarly, the lift force on the cylinder is zero.
Flow around a Circular Cylinder with Circulation

The addition of a vortex to the doublet and the uniform flow
results in flow around a circular cylinder with circulation,

The streamline
is the circular cylinder r = a. At
great distances from the origin, the velocity remains u = - U,
showing that flow around a circular cylinder is maintained with
addition of the vortex. Some of the streamlines are shown in Fig.
8.18.
Figure 8.18 Streamlines for flow around a circular
cylinder with circulation

The velocity at the surface of the cylinder, necessarily tangent to the
cylinder, is

Stagnation points occur when q=0; that is,

The pressure at the surface of the cylinder is

The drag again is zero. The lift, however, becomes

The theoretical flow around a circular cylinder with circulation can
be transformed into flow around an airfoil with the same
circulation and the same lift.

The airfoil develops its lift by producing circulation around it due
to its shape. It can be shown that the lift is for any cylinder in twodimensional flow. The angle or inclination of the airfoil relative to
the approach velocity (angle of attack) greatly affects the
circulation. For large angels or attack, the flow does not follow the
wing profile, and the theory breaks down.

It should be mentioned that all two-dimensional ideal-fluid-flow
cases may be conveniently handled by complex-variable theory and
by a system of conformal mapping, which transforms a flow net
from one configuration to another by a suitable complex-variable
mapping function.
Example 8.3

A source with strength 0.2 m3/s·m and a vortex with strength 1
m2/s are located at the origin. Determine the equations for velocity
potential and stream function. What are the velocity components at
x = 1 m, y = 0.5 m?

Solution:
The velocity potential for the source is

and the corresponding stream function is

The velocity potential for the vortex is

and the corresponding stream function is

Adding the respective functions gives

The radial and tangential velocity components are

At (1, 0.5),
, vr = 0.0285 m/s, vθ = 0.143 m/s
Example 8.4

A circular cylinder 2 m in diameter and 20 m long is rotating at 120
rpm in the positive direction (counterclockwise) about its axis. Its
center is at the origin of a cartesian coordinate system. Wind at 10
m/s blows over the cylinder in the positive x direction; t=200C and
p=100 kPa abs. Determine the lift on the cylinder and the location
in the fourth quadrant of the streamline through the stagnation
point.

Solution
The stagnation point has ψ=0 . By selecting increments of R, θ can
be determined from

The lift is given by ρUΓL.