Lester Spielvogel.

Incompressible cnoidal waves in circular channels online

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IMM 376
May 1969



Courant Institute of
Mathematical Sciences



Incompressible Cnoidal Waves
in Circular Channels

Lester Spielvogel



Prepared under Contract Nonr-285(55) with
the Office of Naval Research NR 062-160

Distribution of this document is unlimited.



New York University



NEW YO«K UNivrnwir
CCX*ANT INSTITUT£ • LHJKMiV



NR 062-160 IMM 376

May 1969



New York University
Courant Institute of Mathematical Sciences



INCOMPRESSIBLE CNOIDAL WAVES IN CIRCULAR CHANNELS

Lester Spielvogel



This report represents results obtained at the Courant Institute
of Mathematical Sciences, New York University, with the Office
of Naval Research, Contract Nonr-285(55) .

Reproduction In whole or In part is permitted for any purpose of
the United States Government.

Distribution of this document Is unlimited.



' NEW YORK UNIVBB8ITY
COURAST INSTITUTE- LIBRARY



TABLE OF CONTENTS

Chapter page

I INTRODUCTION - CONCLUSION 1

II EQUATIONS OF MOTION: ROTATIONAL AND IRROTATIONAL . . 5

III IRROTATIONAL STATIONARY CNOIDAL WAVES 11

IV IRROTATIONAL PROGRESSING CNOIDAL WAVES

(CONSTANT VELOCITY) 33

V LIMITING CASE: STRAIGHT CHANNEL 44

VI ROTATIONAL CNOIDAL WAVES 48

BIBLIOGRAPHY 55



111



ABSTRACT

This report is concerned with the incompressible,
Irrotational flow of fluid in circular channels or circular
troughs. Fluid circulates through these channels and is
contained in them by the action of a gravitational body
force. Under certain velocity conditions, permanent waves
of large magnitude and constant angular velocity may be
solutions to the differential equations of motion. These
waves are closely related to the cnoidal and solitary waves
of the infinite straight channel. Expressions for their
shape, the critical velocity condition and their relation
to waves in straight channels are derived and discussed.

The irrotational case is treated thoroughly and the
equations for the rotational case are derived and one
solution is developed in detail.



A CK NOWLEDGEMENT

I would like to thank Professor Arthur S. Peters
of the Courant Institute of Mathematical Sciences for
his helpful and stimulating advice in the preparation
of this report.



vil



Chapter I
INTRODUCTION - CONCLUSIONS

Since l844 when Scott Russel made his first observation of
a solitary wave on a stream, much theoretical work has been done
on the problem of solitary waves in a liquid of constant density.
Almost all of this work has been in two dimensional flow, or in
Infinite straight channels. There has also been some work done
on the associated cnoidal waves for these surfaces and also on
the surfaces of infinite cylinders and toroidal channels with
radial gravitation. In this report we shall work with a different
geometry (the circular channel or bowl) which is described later in
this chapter.

In general, the solutions will have several things in
common with previous results. The first of these is the charac-
teristic critical speed . For certain velocity profiles the equa-
tions will exhibit a bifurcation i.e., there will be two families
of solutions. The first family, which will be called the steady-
state flow, will exhibit no wave in the direction of the flow.
This family of solutions is also common to flows at non-critical
speeds and is an exact solution of the differential equations of
fluid flow. The second family of solutions will be waves of a
permanent type, that is, the basic shape of the wave is fixed
although it may move over the surface of the water. The second
thing in common with the previous work in this subject will be
the shape of the wave; it will in general be the characteristic



cnoldal profile (Jacob! Elliptic Function) In the direction of
the flow. A third thing In common with previous work is that
the specific wave shape depends on the solution of an elliptic
Neumann problem in the equilibrium or steady state cross section
of the fluid.

The steadily progressing waves discussed here will be waves
whose shape remains fixed as viewed from a coordinate system which
is rotating with respect to the fixed coordinate system at a con-
stant angular velocity. A special case of this will be stationary
waves, i.e., where this angular velocity is zero. Let me note
here that we will depart from the main body of previous works by
describing all these waves from a fixed coordinate system. We
will find this to our advantage within the confines of the geometry
of this problem.

We will be interested in waves in circular channels, see
Diagram 2-1. Consider a fixed curve (s-G(p) = 0) in a cylindri-
cal coordinate system where (p) is the radial coordinate, (a) is
the angular coordinate and (s) is the axial coordinate. This fixed
curve is rotated about the axis of the coordinate system and thus
it generates a trough or circular channel. Fluid will be contained
in this trough by the action of an axial constant gravitational
force.

It is not a priori clear that cnoidal/solitary waves exist
in circular channels since their existence in straight infinite
channels is strongly coupled with the fact that waves of very large
wave length exist in such channels. The infinite length of the



straight channel. In which It has been proved that cnoldal waves
can exist, stands In contrast to the finite length of the liquid
filled circular channel considered here. The existence of these
waves In finite channels Is suggested by observations. In par-
ticular, the solitary waves observed for finite times under
natural circumstances and also under laboratory conditions in
finite channels strongly suggest that these permanent waves do
exist. It is for these reasons that we apply slight variations of
the techniques for infinite straight channels to the problem of
circular troughs. What we do, basically, is associate a stretching
parameter to some of the dependent and independent variables so
as to emphasize the angular direction and angular velocity. We
then expand each of the new dependent variables in a power series
in this parameter. The basic assumption is that these power
series converge to solutions. We substitute these series into the
equations of fluid motion and then extract the equations coeffi-
cient to different powers of the stretching parameter. That is,
the equation L(x, e,(|){x, e ) ) = 0, where (L) is a differential opera-
tor (or boundary condition), and (x) represents time and space
variables, and (e) is the stretching parameter, and (• 0). To
solve for Case 3 we merely transform (+Y) directly into (-Y), and
then equation (3-63) falls into Case 2. Therefore^ we need only
discuss the solutions of Case 2.



Case 2 : m, <

To get the simplest form of the solution, we let



(3-71)



Z = Y + m^/m.



This transforms (3-63) to



(3-72)

Let

(3-73)



m^Z = m-,ZZ .



A = -3mQ/m-j^ >• ,



25



(3-7M



c = -m^/yEgm^ *■



Then (3-52) reduces to



(3-75)



^1 =^2



1 -



2^
7 c

— ?

P J



7Z

P



We will now show that Z{9) exhibits the characteristic
cnoidal shape, and we will determine which parameters are arbitrary.
We begin to solve for (Z) by integrating (3-72) or (3-76), multi-
plying by (Z) and then integrating again.



(3-76)



?v Z = -3ZZ



(3-77)



A(Z)^ = -Z^ + A^Z + Aq = P(Z)



Let us look at the roots of [P(Z)). (A ) is the product of
the roots. Looking at (P(0)), we see that {(sgn A ) = ( sgn A)).
This gives us our first relation



(3-78)



1,2,3
1 r (Roots of P) > .



The absence of the quadratic term in (p(Z)) Implies the
second relation



(3-79)



1,2,3

> (Roots of P) = .



These two relations imply that two roots are negative, one is
positive. Let



(3-80)



Ql >■ qg *" •



26



The roots of [P(Z)} can now be written as follows:
(3-81) +q^, -qg, ^2" ^1 •

Equation (3-77) can be written as (3-82).

(3-82) -h{zf = ?{Z) = (q3^-Z)(q2 + Z)(q^- qg+Z)

2 2 ^

= q-Lq2(qi - ^2^ ^ ^^i ' ^^1^2 '^^2 ^^ " ^

Without loss of generality, let us order the negative roots. Let

q-L - ^2 ^ ^2



or



q^ ^ 2q2



With the roots thus specified, {P(Z)} appears as shown in Diagram
3-2.

We make the substitution (3-83) in (3-82) to get the
differential equation into the more familiar form (3-84).

2 2

(3-83) Z = q, cos O - qg sin O

2

= -q2+ (q]_ + q2) ^°^ ^
o = o(0)

f^ R)l^ ^dn^2 4A , ^1 "^ ^2 . 2 ^
(>S^) (dF) 2q^ - qg = 1 - 2q^ - q^ ^^^^ "

We simplify the notation with the following substitution of
constants.



27



k =



2^1 - ^2



? =



2^1 - ^2



Let us note that (k ) and (i) are related as In (3-86)



(v_ -



l^^+l = 3A



Equation (3-84) now reads as


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Online LibraryLester SpielvogelIncompressible cnoidal waves in circular channels → online text (page 1 of 3)