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On Complex-Lamellar Motion of a Prim Gas
Journal of Mathematical Analysis and Applications 266, 55᎐69 Ž2002. doi:10.1006rjmaa.2001.7685, available online at http:rrwww.idealibrary.com on
On Complex-Lamellar Motion of a Prim Gas C. Rogers and W. K. Schief School of Mathematics, The Uni¨ ersity of New South Wales, Sydney, NSW 2052, Australia
and W. H. Hui Department of Mathematics and Center for Scientific Computation, Hong Kong Uni¨ ersity of Science and Technology, Hong Kong, China Submitted by William F. Ames Received June 11, 2001
An intrinsic formulation of Prim’s canonical equations of gas dynamics is employed to establish a generalization of Crocco’s theorem to complex-lamellar motion. This new result is then employed to establish the existence of a Heisenberg spin-type equation in such motion. The optimal coordinate systems in the generalized Lagrangian method which are known to exist only in complex-lamellar motion are then shown by intrinsic geometric methods to correspond to a marching 䊚 2002 Elsevier Science direction along geodesics on generalized Bernoulli surfaces.
1. INTRODUCTION The term ‘‘complex-lamellar’’ appears to go back at least to Kelvin. It is here used in a hydrodynamics context to designate motions in which the velocity and vorticity vectors are everywhere orthogonal w1x. In this case, there exists a family of what are termed Beltrami surfaces which are normal to the streamlines. In geometric terms, the complex-lamellar condition reduces, in regions excluding stagnation points, to ⍀ s s t ⭈ curl t s 0, where ⍀ s is the abnormality of the t-field and t is the unit tangent to a generic streamline. It is remarked that the complex-lamellar condition also arises in both the study of orthotomic systems of rays in optics w2x and in electromagnetic wave propagation w3x. 55 0022-247Xr02 $35.00 䊚 2002 Elsevier Science All rights reserved.
56
ROGERS, SCHIEF, AND HUI
In hydrodynamics, complex-lamellar motions have been the subject of a number of studies Žsuch as in w4᎐7x., and potentialrcomplex-lamellar decompositions have also received attention w8, 9x. In recent work, the complex-lamellar condition has been shown by Hui and He w10x to be a necessary and sufficient condition for the existence of an optimal coordinate system within a generalized Lagrangian framework in spatial gas dynamics w11᎐15x. In this context, an optimal coordinate system is a three-dimensional orthogonal coordinate system with stream surfaces as two sets of coordinate surfaces. The existence of an optimal coordinate system is of fundamental importance in the numerical computation of steady gas flow. The computation then marches along the streamlines and is thus robust, accurate, and efficient w10, 15x. On the other hand, in w16x, it has been established recently that there exists a remarkable connection between complex-lamellar hydrodynamic motions and a Heisenberg spin equation which, in a certain reduction, is associated with the nonlinear Schrodinger equation. This connection was ¨ subsequently discussed in w17x. Here, an intrinsic formulation with origins in hydrodynamics and recently exploited in soliton theory w18, 19x is adopted to establish a generalization of Crocco’s theorem to complex-lamellar motion. This result is then used to establish a novel connection between such complex-lamellar motions in gas dynamics and a Heisenberg spin equation.
2. THE CANONICAL EQUATIONS Here we consider the steady, anisentropic gas dynamics system div q s 0,
Ž 2.1.
q ⭈ grad q q grad p s 0,
Ž 2.2.
q ⭈ grad s s 0
Ž 2.3.
s Ž p, s . ,
⭸p ⭸
) 0,
Ž 2.4.
s
where q denotes the velocity vector, is the gas density, p is the gas pressure, and s is the specific entropy. Scalar multiplication of Ž2.2. by q yields, taking into account Ž2.4., the Bernoulli integral q 2 q 2 h Ž p, s . s a2 s 2 h Ž p 0 , s . ,
Ž 2.5.
COMPLEX-LAMELLAR MOTION OF A PRIM GAS
57
wherein h denotes the specific enthalpy, h Ž p, s . s
dp ˜
p
H0
Ž ˜ p, s .
.
Ž 2.6.
Here, a is the limiting or ultimate velocity magnitude and p 0 designates the stagnation pressure. In the above, q ⭈ ⵜa s 0,
Ž 2.7.
whence, in view of Ž2.4. and Ž2.5., it is seen that q ⭈ ⵜp 0 s 0.
Ž 2.8.
The quantity hŽ p 0 , s . is commonly termed the stagnation enthalpy or total head. Munk and Prim w20, 21x showed that for a gas having a product-type equation of state,
s P Ž p. SŽ s. ,
Ž 2.9.
the gas dynamic system Ž2.1. ᎐ Ž2.4. is invariant under the substitution principle qU s
1 m
q,
U s m2 ,
pU s p,
sU s Sy1 m2 S Ž s . , Ž 2.10.
where q ⭈ ⵜm s 0.
Ž 2.11.
The underlying group structure of this transformation has recently been elucidated by Ovsiannikov w22x. The invariance principle Ž2.10. led Munk and Prim w21, 23x to introduce the reduced velocity vector w s qra
Ž 2.12.
into the gas dynamics system. Insertion of Ž2.12. into the continuity equation Ž2.1. gives div P Ž p . w s 0,
Ž 2.13.
while substitution into the equation of motion Ž2.2. yields w ⭈ grad w q
1
a2
grad p s 0.
Ž 2.14.
58
ROGERS, SCHIEF, AND HUI
The Bernoulli integral Ž2.5. shows that w 2 a2 q 2 h Ž p, s . s a2 ,
Ž 2.15.
where h Ž p, s . s H Ž p . rS Ž s . ,
H Ž p. s
p
H0
dp ˜ PŽ ˜ p.
.
Ž 2.16.
Elimination of a2 between Ž2.14. and Ž2.15. yields w ⭈ grad w q
1 2h
Ž 1 y w 2 . grad p s 0,
Ž 2.17.
whence w ⭈ grad w q 12 Ž 1 y w 2 . grad ln H Ž p . s 0.
Ž 2.18.
The canonical system consisting of Ž2.13. and Ž2.17. of Prim w21x constitutes four equations in four unknowns, namely the pressure p and the three components of the reduced velocity vector w. Once a solution of this system has been obtained, the stagnation pressure is computed via the relation H Ž p0 . s H Ž p . r Ž 1 y w 2 . .
Ž 2.19.
If a2 is regarded as assigned on each streamline, then the entropy distribution is determined via the relation S Ž s . s 2 H Ž p 0 . ra2 ,
Ž 2.20.
while the density distribution is obtained through the prevailing Prim gas law Ž2.9.. The actual velocity q is recovered through relation Ž2.12..
3. THE COMPATIBILITY CONDITIONS The canonical equation of motion Ž2.18. may be rewritten, on use of Ž2.19., as w = curl w 1yw
2
s
1 2
grad ln H Ž p 0 . ,
Ž 3.1.
Thus, for gases with a product equation of state Ž2.9., what Prim w21x termed generalized Bernoulli surfaces exist which contain the reduced
COMPLEX-LAMELLAR MOTION OF A PRIM GAS
59
velocity and vorticity lines. These generalized Bernoulli surfaces are surfaces of constant stagnation pressure. Moreover, the equation of motion Ž3.1. is seen to have the necessary compatibility condition, curl
w = curl w 1 y w2
s 0.
Ž 3.2.
In the sequel, geometric consequences of Ž3.2. are investigated via an intrinsic formulation developed in w24x in a kinematic study of hydrodynamics. Therein, the orthonormal basis Žt, n, b. is introduced along the tangent, principal normal, and binormal directions to the vector lines of a non-vanishing three-dimensional vector field. In the present context, this will be the reduced velocity field w s wt.
Ž 3.3.
If ␦r␦ s, ␦r␦ n, and ␦r␦ b denote directional derivatives in the tangential, principal normal, and binormal directions, respectively, then ŽMarris and Passman w24x.
␦
t 0 n s y ␦s b 0
ž/ ž ž/ ž/
0 y
0 0
0 t yn s n s ␦n b yŽ ⍀ b q .
␦
0 t ⍀ n s nq ␦b b y b s
␦
t n , b
/ž / n s 0
⍀b q ydiv b
div b
0
t n , b
0ž /
yŽ ⍀ n q . 0
bs q div n
y Ž q div n .
0
Ž 3.4. t n , b
0ž /
where n s and b s are geometric quantities originally introduced by Bjørgum w25x via
n s s n ⭈
␦t ␦n
,
bs s b ⭈
␦t ␦b
,
Ž 3.5.
while ⍀ s s t ⭈ curl t, ⍀ n s n ⭈ curl n, ⍀ b s b ⭈ curl b are termed the abnormalities of the t, n, an b fields, respectively.
60
ROGERS, SCHIEF, AND HUI
It is readily shown that div t s n s q b s ,
␦n
div n s y q b ⭈
␦b
,
div b s yb ⭈
␦n ␦n
, Ž 3.6.
curl t s ⍀ s t q b, curl n s y Ž div b . t q ⍀ n n q n s b,
Ž 3.7.
curl b s Ž q div n . t y b s n q ⍀ b b. The identity curl grad s 0, on use of the relations Ž3.7., now yields
␦ 2 ␦n ␦b ␦ 2 ␦b␦s ␦ 2 ␦s ␦n
␦ 2
y
␦b␦n
y y
␦ 2 ␦s ␦b ␦ 2 ␦n ␦s
sy sy sy
␦ ␦s
⍀s q
␦
⍀n q
␦n ␦ ␦s
y
␦ ␦n ␦ ␦b
␦ ␦n
␦
div b y
Ž q div n . ,
␦b
bs ,
n s y
Ž 3.8. ␦ ␦b
⍀b.
These relations are crucial in the sequel and the intrinsic decomposition of the compatibility condition Ž3.2.. In that connection, use of the relation Ž3.7.1 originally obtained by Masotti w26x shows that the reduced vorticity admits the intrinsic decomposition curl w s w⍀ s t q
␦w ␦b
n q w y
ž
␦w ␦n
/
b,
Ž 3.9.
b.
Ž 3.10.
whence w = curl w s w
ž
␦w ␦n
y w n q w
/
␦w ␦b
Accordingly, excluding generalized Beltrami flows, the normal N to the generalized Bernoulli surfaces is parallel to the principal normal n to the streamlines iff
␦w ␦b
s 0,
w / 0.
Ž 3.11.
Thus, the streamlines are geodesics on the generalized Bernoulli surfaces if the reduced velocity magnitude is constant along binormal lines.
COMPLEX-LAMELLAR MOTION OF A PRIM GAS
61
On expansion of the compatibility condition Ž3.2. one obtains curl
w = curl w 1yw
s w curl w ⭈ ⵜ x
2
ž
w 1yw
y Ž curl w . div
ž
2
/
1
y
w 1 y w2
/
1 y w2
Ž w ⭈ ⵜ . curl w
,
whence, on setting ⌳ [ ⌳ s t q ⌳ n n q ⌳ b b s curl w, it is seen that, on use of Ž3.4., ⌳n
␦
w
ž
␦n 1 y w
2
/
q
w 1yw
2
q ⌳b
␦
w
ž
␦b 1 y w
2
/
y div
ž
⌳ sw 1 y w2
t s 0.
/
Ž 3.12. w 1yw
2
w ⌳ b ⍀ n q ⌳ n bs x q
2
w ⌳ n ⍀ b y ⌳ b n s x s
w 1yw
␦
ž ž
⌳ nw
␦ s 1 y w2 ␦
⌳ bw
␦ s 1 y w2
/ /
s 0,
Ž 3.13.
s 0.
Ž 3.14.
4. COMPLEX-LAMELLAR MOTION: A GENERALIZED CROCCO THEOREM It has been established by Hui and He w10x that the necessary and sufficient condition for the existence of optimal coordinates within the generalized Lagrangian formulation of the steady spatial motion of a polytropic gas is that the motion must be complex-lamellar, that is, q ⭈ curl q s 0,
Ž 4.1.
⍀ s s t ⭈ curl t s 0.
Ž 4.2.
or, if q / 0,
Here, geometric consequences of the condition Ž4.2. are determined in the more general setting of the spatial anisentropic motion of a Prim gas. Thus, on assumption of the complex-lamellar condition Ž4.2., the relation Ž3.9. shows that ⌳ s s 0, and the compatibility condition Ž3.12. reduces to
␦w
␦
ž
w
␦ b ␦ n 1 y w2
/
q
w 1 y w2
q w y
ž
␦w ␦n
/ž
1 q w2 1 y w2
/
s 0,
62
ROGERS, SCHIEF, AND HUI
whence
␦w ␦b
w s 0.
Ž 4.3.
Accordingly, if w / 0 and geometries with s 0 are excluded, it follows that the complex-lamellar condition implies
␦w ␦b
s 0.
Ž 4.4.
The streamlines are then necessarily geodesics on the generalized Bernoulli surfaces. The condition Ž4.4. in turn implies that the principal normal component ⌳ n of the reduced vorticity vector vanishes so that ⌳ s ⌳b and the residual compatibility conditions Ž3.13., Ž3.14. reduce to ⍀ n s 0,
⌳/0
Ž 4.5.
and
n s s
␦ ␦s
ln
1 y w2 ⌳w
.
Ž 4.6.
The condition Ž4.5. guarantees the existence of a system of surfaces containing the s-lines Žstreamlines . and b-lines Žthe binormal lines.. These are the generalized Bernoulli surfaces, and thereon the streamlines are geodesics, and their orthogonal trajectories, the b-lines, are parallels. If the generalized Bernoulli surfaces ⌺ are parameterized in terms of these geodesics and parallels, then the surface metric becomes I ⌺ s ds 2 q gdb 2 ,
Ž 4.7.
where
␦ ␦s
Ž ln
g 1r2 . s b s .
Ž 4.8.
The canonical continuity equation Ž2.13. together with the relation Ž4.8. delivers
␦ ␦s
ln P Ž p . wg 1r2
q n s s 0,
Ž 4.9.
COMPLEX-LAMELLAR MOTION OF A PRIM GAS
63
whence
␦ ␦s
⌳rP Ž p . g 1r2 Ž 1 y w 2 . s 0.
Ž 4.10.
Accordingly, we obtain the following generalization to complex-lamellar motion of the classical Crocco theorem: THEOREM 1. In complex-lamellar motion so that ⍀ s s 0, if / 0 the streamlines and ¨ orticity lines are, respecti¨ ely, geodesics and parallels on the generalized Bernoulli surfaces ⌺. Moreo¨ er, if these cur¨ es are taken as coordinate lines on the surfaces ⌺ then the metric adopts the form I ⌺ s ds 2 q gdb 2 and the ¨ orticity magnitude ⌳ s < ⌳b < is such that ⌳ s ␣ P Ž p . H Ž p . g 1r2 ,
Ž 4.11.
where ␦␣r␦ s s 0. The relation Ž4.11. represents a generalization of the classical Crocco theorem recorded in 1937 w27x. This was originally obtained for plane isoenergetic flows of a perfect gas and later generalized by Prim in 1948 to plane flows having non-uniform stagnation entropy w28x. An extension of Crocco’s pressure theorem to axially symmetric flow was set down in w21x. It is noted that the complex-lamellar condition includes both the plane and axially symmetric geometries. The three compatibility conditions embodied in Ž3.2. provide necessary conditions on the motion. Let us now return to the original canonical continuity equation Ž2.13. and equation of motion Ž2.18.. Their intrinsic decomposition yields
␦
ln Ž P Ž p . w . q div t s 0,
␦s w
␦w ␦s
q
w 2 q
1 2
1 2
Ž1 y w2 .
Ž1 y w2 . ␦p ␦b
␦
ln H Ž p . s 0,
␦s ␦
␦n
ln H Ž p . s 0,
s 0.
Ž 4.12. Ž 4.13. Ž 4.14. Ž 4.15.
64
ROGERS, SCHIEF, AND HUI
wIt is noted that, if s 0, then Ž4.14. yields
␦p ␦n
s 0,
and the relation Ž3.8.1 leads to the complex-lamellar condition ⍀s s 0 if div t / 0.x The canonical continuity equation Ž4.12. yields, on use of the relation Ž3.8. 2 together with Ž4.4., Ž4.5., and Ž4.14.,
␦ ␦b
div t s 0.
Ž 4.16.
The complex-lamellar condition ⍀ s s 0 represents the necessary and sufficient condition for the existence of a family of so-called Beltrami surfaces w4x containing curves tangential to the principal normal n and principal binormal b directions, namely so-called n-lines and b-lines. It is equivalent to the requirement t s ,
Ž 4.17.
where the family has constituent members s constant and represents the distance function. Certain geometric properties of the Beltrami surfaces are set down below. The identity div curl t s 0, together with the Masotti᎐Emde᎐Bjørgum relation Ž3.7.1 , shows that div Ž ⍀ s t q b . s 0, whence for complex-lamellar motions div b s 0.
Ž 4.18.
The mean curvature ᑤ s 2 M of the Beltrami surfaces s const. is given by ᑤ s ydivsconst. N s ydiv N s ydiv t.
Ž 4.19.
Similarly, the Gaußian curvature is shown to be K s n s b s y 2 .
Ž 4.20.
COMPLEX-LAMELLAR MOTION OF A PRIM GAS
65
In general,
q 2 s w 2 a2 s P Ž p . w 2 a2 S s 2 P Ž p . w 2 H Ž p0 . s
2w2 1 y w2
P Ž p. H Ž p. .
Ž 4.21.
However, in complex-lamellar motion ␦ wr␦ b s 0 Ž / 0., and this condition together with ␦ pr␦ b s 0 shows that w s w Ž p . on the Beltrami surfaces s const. Accordingly, the relation Ž4.21. implies that
q2 s
2 w2 Ž p.
on s const.
P Ž p. H Ž p.
1 y w2 Ž p.
Ž 4.22.
This both makes explicit and extends to Prim gases the result of Hui and He w10x. In addition, the condition Ž4.16. together with the relation Ž4.19. shows that the mean curvature ᑤ of the Beltrami surfaces varies only with the pressure thereon. In a similar manner, q 2 and w on the generalized Bernoulli surfaces depend only on the pressure. It should be noted that both q 2 and p are invariants under Prim’s substitution principle. In conclusion, it is natural to enquire under what conditions
q2 s ⌽ Ž p.
Ž 4.23.
throughout a spatial flow. In this case, the intrinsic equation of motion Ž4.13., namely,
q
␦q ␦s
q
␦p ␦s
s 0,
together with Ž4.23. immediately produces
␦p ␦s
s
␦q ␦s
s
␦ ␦s
s 0,
whence div t s 0.
Ž 4.24.
66
ROGERS, SCHIEF, AND HUI
Let us now consider the behavior of the Mach number in the motion. Equation Ž2.5. shows that M 2 q 2 H Ž p . P X Ž p . s a2 SP X Ž p . ,
Ž 4.25.
whence, on use of Ž2.20., M 2 s 2 PX Ž p . H Ž p0 . y H Ž p . s
2 PX Ž p . H Ž p . w 2 1 y w2
.
Ž 4.26.
Hence, on a Beltrami surface s const., M s 2
2 PX Ž p . H Ž p . w 2 Ž p . 1 y w2 Ž p.
,
Ž 4.27.
so that, indeed, M s M Ž p, . ,
Ž 4.28.
as established by Hui and He w10x.
5. A HEISENBERG SPIN EQUATION It has been seen that in complex-lamellar motion div b s 0,
Ž 5.1.
while, with the use of the relation Ž3.8.1 , div Ž ⌳b . s s
␦⌳ ␦b ␦ ␦b
q ⌳ div b
ž
w y
␦w ␦n
q w y
/ ž
␦w ␦n
/
div b
s w div b. Accordingly, div Ž ⌳b . s 0,
Ž 5.2.
COMPLEX-LAMELLAR MOTION OF A PRIM GAS
67
whence, assuming non-vanishing div b, combination with Ž4.1. yields
␦ ␦b
Ž ⌳r . s 0,
so that ⌳ s ,
Ž 5.3.
where ␦r␦ b s 0. The generalized Crocco relation Ž4.11., together with the expression Ž5.3. for the reduced vorticity ⌳, now shows that
␣ g 1r2 s  ,
Ž 5.4.
where
␦␣r␦ s s 0,
␦r␦ b s 0.
Ž 5.5.
Now, if r is the generic position vector to an individual generalized Bernoulli surface, then
⭸r ⭸b
s g 1r2
⭸r ⭸b
s

ž / ␣
b s
 ␣
ž
t=
⭸t ⭸s
/
,
Ž 5.6.
whence, on removal of ␣ by scaling on b, it is seen that Ž5.6. delivers the Heisenberg spin-type equation,
⭸t ⭸b
s
⭸ ⭸s
 t=
ž
⭸t ⭸s
/
Ž ⭸r⭸ b s 0 . ,
Ž 5.7.
where t is the unit tangent to the streamlines on the generalized Bernoulli surfaces. These streamlines constitute geodesics thereon. It is remarked that, accordingly, the marching direction in the optimal coordinate system of Hui and He w10x is along geodesics on the generalized Bernoulli surfaces. It is interesting to note that in steady Newton᎐Busemann gas flow wherein the Mach number M ª 1 and ␥ s 1, the streamlines are geodesics on the body surface w29x, but this is not so for general gas flow with finite Mach number.
ACKNOWLEDGMENT Professor Hui acknowledges with gratitude support from the Research Council of Hong Kong.
68
ROGERS, SCHIEF, AND HUI
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26. A. Masotti, Decomposizione intrinseca del vortice a sue applicazioni, Inst. Lombardo Sci. Lett. Rendiconti Ž 20 . 60 Ž1927., 869᎐874. 27. L. Crocco, Eine neue Stromfunction fur ¨ die Erforschung der Bewegung der Gase mit Rotation, Zamm 17 Ž1937., 1᎐7. 28. R. C. Prim, Extension of Crocco’s theorems to flows having a non-uniform stagnation enthalpy, Phys. Re¨ . 73 Ž1948., 186. 29. W. D. Hayes and R. F. Probstein, ‘‘Hypersonic Flow Theory,’’ Academic Press, San Diego, 1966.
On Complex-Lamellar Motion of a Prim Gas C. Rogers and W. K. Schief School of Mathematics, The Uni¨ ersity of New South Wales, Sydney, NSW 2052, Australia
and W. H. Hui Department of Mathematics and Center for Scientific Computation, Hong Kong Uni¨ ersity of Science and Technology, Hong Kong, China Submitted by William F. Ames Received June 11, 2001
An intrinsic formulation of Prim’s canonical equations of gas dynamics is employed to establish a generalization of Crocco’s theorem to complex-lamellar motion. This new result is then employed to establish the existence of a Heisenberg spin-type equation in such motion. The optimal coordinate systems in the generalized Lagrangian method which are known to exist only in complex-lamellar motion are then shown by intrinsic geometric methods to correspond to a marching 䊚 2002 Elsevier Science direction along geodesics on generalized Bernoulli surfaces.
1. INTRODUCTION The term ‘‘complex-lamellar’’ appears to go back at least to Kelvin. It is here used in a hydrodynamics context to designate motions in which the velocity and vorticity vectors are everywhere orthogonal w1x. In this case, there exists a family of what are termed Beltrami surfaces which are normal to the streamlines. In geometric terms, the complex-lamellar condition reduces, in regions excluding stagnation points, to ⍀ s s t ⭈ curl t s 0, where ⍀ s is the abnormality of the t-field and t is the unit tangent to a generic streamline. It is remarked that the complex-lamellar condition also arises in both the study of orthotomic systems of rays in optics w2x and in electromagnetic wave propagation w3x. 55 0022-247Xr02 $35.00 䊚 2002 Elsevier Science All rights reserved.
56
ROGERS, SCHIEF, AND HUI
In hydrodynamics, complex-lamellar motions have been the subject of a number of studies Žsuch as in w4᎐7x., and potentialrcomplex-lamellar decompositions have also received attention w8, 9x. In recent work, the complex-lamellar condition has been shown by Hui and He w10x to be a necessary and sufficient condition for the existence of an optimal coordinate system within a generalized Lagrangian framework in spatial gas dynamics w11᎐15x. In this context, an optimal coordinate system is a three-dimensional orthogonal coordinate system with stream surfaces as two sets of coordinate surfaces. The existence of an optimal coordinate system is of fundamental importance in the numerical computation of steady gas flow. The computation then marches along the streamlines and is thus robust, accurate, and efficient w10, 15x. On the other hand, in w16x, it has been established recently that there exists a remarkable connection between complex-lamellar hydrodynamic motions and a Heisenberg spin equation which, in a certain reduction, is associated with the nonlinear Schrodinger equation. This connection was ¨ subsequently discussed in w17x. Here, an intrinsic formulation with origins in hydrodynamics and recently exploited in soliton theory w18, 19x is adopted to establish a generalization of Crocco’s theorem to complex-lamellar motion. This result is then used to establish a novel connection between such complex-lamellar motions in gas dynamics and a Heisenberg spin equation.
2. THE CANONICAL EQUATIONS Here we consider the steady, anisentropic gas dynamics system div q s 0,
Ž 2.1.
q ⭈ grad q q grad p s 0,
Ž 2.2.
q ⭈ grad s s 0
Ž 2.3.
s Ž p, s . ,
⭸p ⭸
) 0,
Ž 2.4.
s
where q denotes the velocity vector, is the gas density, p is the gas pressure, and s is the specific entropy. Scalar multiplication of Ž2.2. by q yields, taking into account Ž2.4., the Bernoulli integral q 2 q 2 h Ž p, s . s a2 s 2 h Ž p 0 , s . ,
Ž 2.5.
COMPLEX-LAMELLAR MOTION OF A PRIM GAS
57
wherein h denotes the specific enthalpy, h Ž p, s . s
dp ˜
p
H0
Ž ˜ p, s .
.
Ž 2.6.
Here, a is the limiting or ultimate velocity magnitude and p 0 designates the stagnation pressure. In the above, q ⭈ ⵜa s 0,
Ž 2.7.
whence, in view of Ž2.4. and Ž2.5., it is seen that q ⭈ ⵜp 0 s 0.
Ž 2.8.
The quantity hŽ p 0 , s . is commonly termed the stagnation enthalpy or total head. Munk and Prim w20, 21x showed that for a gas having a product-type equation of state,
s P Ž p. SŽ s. ,
Ž 2.9.
the gas dynamic system Ž2.1. ᎐ Ž2.4. is invariant under the substitution principle qU s
1 m
q,
U s m2 ,
pU s p,
sU s Sy1 m2 S Ž s . , Ž 2.10.
where q ⭈ ⵜm s 0.
Ž 2.11.
The underlying group structure of this transformation has recently been elucidated by Ovsiannikov w22x. The invariance principle Ž2.10. led Munk and Prim w21, 23x to introduce the reduced velocity vector w s qra
Ž 2.12.
into the gas dynamics system. Insertion of Ž2.12. into the continuity equation Ž2.1. gives div P Ž p . w s 0,
Ž 2.13.
while substitution into the equation of motion Ž2.2. yields w ⭈ grad w q
1
a2
grad p s 0.
Ž 2.14.
58
ROGERS, SCHIEF, AND HUI
The Bernoulli integral Ž2.5. shows that w 2 a2 q 2 h Ž p, s . s a2 ,
Ž 2.15.
where h Ž p, s . s H Ž p . rS Ž s . ,
H Ž p. s
p
H0
dp ˜ PŽ ˜ p.
.
Ž 2.16.
Elimination of a2 between Ž2.14. and Ž2.15. yields w ⭈ grad w q
1 2h
Ž 1 y w 2 . grad p s 0,
Ž 2.17.
whence w ⭈ grad w q 12 Ž 1 y w 2 . grad ln H Ž p . s 0.
Ž 2.18.
The canonical system consisting of Ž2.13. and Ž2.17. of Prim w21x constitutes four equations in four unknowns, namely the pressure p and the three components of the reduced velocity vector w. Once a solution of this system has been obtained, the stagnation pressure is computed via the relation H Ž p0 . s H Ž p . r Ž 1 y w 2 . .
Ž 2.19.
If a2 is regarded as assigned on each streamline, then the entropy distribution is determined via the relation S Ž s . s 2 H Ž p 0 . ra2 ,
Ž 2.20.
while the density distribution is obtained through the prevailing Prim gas law Ž2.9.. The actual velocity q is recovered through relation Ž2.12..
3. THE COMPATIBILITY CONDITIONS The canonical equation of motion Ž2.18. may be rewritten, on use of Ž2.19., as w = curl w 1yw
2
s
1 2
grad ln H Ž p 0 . ,
Ž 3.1.
Thus, for gases with a product equation of state Ž2.9., what Prim w21x termed generalized Bernoulli surfaces exist which contain the reduced
COMPLEX-LAMELLAR MOTION OF A PRIM GAS
59
velocity and vorticity lines. These generalized Bernoulli surfaces are surfaces of constant stagnation pressure. Moreover, the equation of motion Ž3.1. is seen to have the necessary compatibility condition, curl
w = curl w 1 y w2
s 0.
Ž 3.2.
In the sequel, geometric consequences of Ž3.2. are investigated via an intrinsic formulation developed in w24x in a kinematic study of hydrodynamics. Therein, the orthonormal basis Žt, n, b. is introduced along the tangent, principal normal, and binormal directions to the vector lines of a non-vanishing three-dimensional vector field. In the present context, this will be the reduced velocity field w s wt.
Ž 3.3.
If ␦r␦ s, ␦r␦ n, and ␦r␦ b denote directional derivatives in the tangential, principal normal, and binormal directions, respectively, then ŽMarris and Passman w24x.
␦
t 0 n s y ␦s b 0
ž/ ž ž/ ž/
0 y
0 0
0 t yn s n s ␦n b yŽ ⍀ b q .
␦
0 t ⍀ n s nq ␦b b y b s
␦
t n , b
/ž / n s 0
⍀b q ydiv b
div b
0
t n , b
0ž /
yŽ ⍀ n q . 0
bs q div n
y Ž q div n .
0
Ž 3.4. t n , b
0ž /
where n s and b s are geometric quantities originally introduced by Bjørgum w25x via
n s s n ⭈
␦t ␦n
,
bs s b ⭈
␦t ␦b
,
Ž 3.5.
while ⍀ s s t ⭈ curl t, ⍀ n s n ⭈ curl n, ⍀ b s b ⭈ curl b are termed the abnormalities of the t, n, an b fields, respectively.
60
ROGERS, SCHIEF, AND HUI
It is readily shown that div t s n s q b s ,
␦n
div n s y q b ⭈
␦b
,
div b s yb ⭈
␦n ␦n
, Ž 3.6.
curl t s ⍀ s t q b, curl n s y Ž div b . t q ⍀ n n q n s b,
Ž 3.7.
curl b s Ž q div n . t y b s n q ⍀ b b. The identity curl grad s 0, on use of the relations Ž3.7., now yields
␦ 2 ␦n ␦b ␦ 2 ␦b␦s ␦ 2 ␦s ␦n
␦ 2
y
␦b␦n
y y
␦ 2 ␦s ␦b ␦ 2 ␦n ␦s
sy sy sy
␦ ␦s
⍀s q
␦
⍀n q
␦n ␦ ␦s
y
␦ ␦n ␦ ␦b
␦ ␦n
␦
div b y
Ž q div n . ,
␦b
bs ,
n s y
Ž 3.8. ␦ ␦b
⍀b.
These relations are crucial in the sequel and the intrinsic decomposition of the compatibility condition Ž3.2.. In that connection, use of the relation Ž3.7.1 originally obtained by Masotti w26x shows that the reduced vorticity admits the intrinsic decomposition curl w s w⍀ s t q
␦w ␦b
n q w y
ž
␦w ␦n
/
b,
Ž 3.9.
b.
Ž 3.10.
whence w = curl w s w
ž
␦w ␦n
y w n q w
/
␦w ␦b
Accordingly, excluding generalized Beltrami flows, the normal N to the generalized Bernoulli surfaces is parallel to the principal normal n to the streamlines iff
␦w ␦b
s 0,
w / 0.
Ž 3.11.
Thus, the streamlines are geodesics on the generalized Bernoulli surfaces if the reduced velocity magnitude is constant along binormal lines.
COMPLEX-LAMELLAR MOTION OF A PRIM GAS
61
On expansion of the compatibility condition Ž3.2. one obtains curl
w = curl w 1yw
s w curl w ⭈ ⵜ x
2
ž
w 1yw
y Ž curl w . div
ž
2
/
1
y
w 1 y w2
/
1 y w2
Ž w ⭈ ⵜ . curl w
,
whence, on setting ⌳ [ ⌳ s t q ⌳ n n q ⌳ b b s curl w, it is seen that, on use of Ž3.4., ⌳n
␦
w
ž
␦n 1 y w
2
/
q
w 1yw
2
q ⌳b
␦
w
ž
␦b 1 y w
2
/
y div
ž
⌳ sw 1 y w2
t s 0.
/
Ž 3.12. w 1yw
2
w ⌳ b ⍀ n q ⌳ n bs x q
2
w ⌳ n ⍀ b y ⌳ b n s x s
w 1yw
␦
ž ž
⌳ nw
␦ s 1 y w2 ␦
⌳ bw
␦ s 1 y w2
/ /
s 0,
Ž 3.13.
s 0.
Ž 3.14.
4. COMPLEX-LAMELLAR MOTION: A GENERALIZED CROCCO THEOREM It has been established by Hui and He w10x that the necessary and sufficient condition for the existence of optimal coordinates within the generalized Lagrangian formulation of the steady spatial motion of a polytropic gas is that the motion must be complex-lamellar, that is, q ⭈ curl q s 0,
Ž 4.1.
⍀ s s t ⭈ curl t s 0.
Ž 4.2.
or, if q / 0,
Here, geometric consequences of the condition Ž4.2. are determined in the more general setting of the spatial anisentropic motion of a Prim gas. Thus, on assumption of the complex-lamellar condition Ž4.2., the relation Ž3.9. shows that ⌳ s s 0, and the compatibility condition Ž3.12. reduces to
␦w
␦
ž
w
␦ b ␦ n 1 y w2
/
q
w 1 y w2
q w y
ž
␦w ␦n
/ž
1 q w2 1 y w2
/
s 0,
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ROGERS, SCHIEF, AND HUI
whence
␦w ␦b
w s 0.
Ž 4.3.
Accordingly, if w / 0 and geometries with s 0 are excluded, it follows that the complex-lamellar condition implies
␦w ␦b
s 0.
Ž 4.4.
The streamlines are then necessarily geodesics on the generalized Bernoulli surfaces. The condition Ž4.4. in turn implies that the principal normal component ⌳ n of the reduced vorticity vector vanishes so that ⌳ s ⌳b and the residual compatibility conditions Ž3.13., Ž3.14. reduce to ⍀ n s 0,
⌳/0
Ž 4.5.
and
n s s
␦ ␦s
ln
1 y w2 ⌳w
.
Ž 4.6.
The condition Ž4.5. guarantees the existence of a system of surfaces containing the s-lines Žstreamlines . and b-lines Žthe binormal lines.. These are the generalized Bernoulli surfaces, and thereon the streamlines are geodesics, and their orthogonal trajectories, the b-lines, are parallels. If the generalized Bernoulli surfaces ⌺ are parameterized in terms of these geodesics and parallels, then the surface metric becomes I ⌺ s ds 2 q gdb 2 ,
Ž 4.7.
where
␦ ␦s
Ž ln
g 1r2 . s b s .
Ž 4.8.
The canonical continuity equation Ž2.13. together with the relation Ž4.8. delivers
␦ ␦s
ln P Ž p . wg 1r2
q n s s 0,
Ž 4.9.
COMPLEX-LAMELLAR MOTION OF A PRIM GAS
63
whence
␦ ␦s
⌳rP Ž p . g 1r2 Ž 1 y w 2 . s 0.
Ž 4.10.
Accordingly, we obtain the following generalization to complex-lamellar motion of the classical Crocco theorem: THEOREM 1. In complex-lamellar motion so that ⍀ s s 0, if / 0 the streamlines and ¨ orticity lines are, respecti¨ ely, geodesics and parallels on the generalized Bernoulli surfaces ⌺. Moreo¨ er, if these cur¨ es are taken as coordinate lines on the surfaces ⌺ then the metric adopts the form I ⌺ s ds 2 q gdb 2 and the ¨ orticity magnitude ⌳ s < ⌳b < is such that ⌳ s ␣ P Ž p . H Ž p . g 1r2 ,
Ž 4.11.
where ␦␣r␦ s s 0. The relation Ž4.11. represents a generalization of the classical Crocco theorem recorded in 1937 w27x. This was originally obtained for plane isoenergetic flows of a perfect gas and later generalized by Prim in 1948 to plane flows having non-uniform stagnation entropy w28x. An extension of Crocco’s pressure theorem to axially symmetric flow was set down in w21x. It is noted that the complex-lamellar condition includes both the plane and axially symmetric geometries. The three compatibility conditions embodied in Ž3.2. provide necessary conditions on the motion. Let us now return to the original canonical continuity equation Ž2.13. and equation of motion Ž2.18.. Their intrinsic decomposition yields
␦
ln Ž P Ž p . w . q div t s 0,
␦s w
␦w ␦s
q
w 2 q
1 2
1 2
Ž1 y w2 .
Ž1 y w2 . ␦p ␦b
␦
ln H Ž p . s 0,
␦s ␦
␦n
ln H Ž p . s 0,
s 0.
Ž 4.12. Ž 4.13. Ž 4.14. Ž 4.15.
64
ROGERS, SCHIEF, AND HUI
wIt is noted that, if s 0, then Ž4.14. yields
␦p ␦n
s 0,
and the relation Ž3.8.1 leads to the complex-lamellar condition ⍀s s 0 if div t / 0.x The canonical continuity equation Ž4.12. yields, on use of the relation Ž3.8. 2 together with Ž4.4., Ž4.5., and Ž4.14.,
␦ ␦b
div t s 0.
Ž 4.16.
The complex-lamellar condition ⍀ s s 0 represents the necessary and sufficient condition for the existence of a family of so-called Beltrami surfaces w4x containing curves tangential to the principal normal n and principal binormal b directions, namely so-called n-lines and b-lines. It is equivalent to the requirement t s ,
Ž 4.17.
where the family has constituent members s constant and represents the distance function. Certain geometric properties of the Beltrami surfaces are set down below. The identity div curl t s 0, together with the Masotti᎐Emde᎐Bjørgum relation Ž3.7.1 , shows that div Ž ⍀ s t q b . s 0, whence for complex-lamellar motions div b s 0.
Ž 4.18.
The mean curvature ᑤ s 2 M of the Beltrami surfaces s const. is given by ᑤ s ydivsconst. N s ydiv N s ydiv t.
Ž 4.19.
Similarly, the Gaußian curvature is shown to be K s n s b s y 2 .
Ž 4.20.
COMPLEX-LAMELLAR MOTION OF A PRIM GAS
65
In general,
q 2 s w 2 a2 s P Ž p . w 2 a2 S s 2 P Ž p . w 2 H Ž p0 . s
2w2 1 y w2
P Ž p. H Ž p. .
Ž 4.21.
However, in complex-lamellar motion ␦ wr␦ b s 0 Ž / 0., and this condition together with ␦ pr␦ b s 0 shows that w s w Ž p . on the Beltrami surfaces s const. Accordingly, the relation Ž4.21. implies that
q2 s
2 w2 Ž p.
on s const.
P Ž p. H Ž p.
1 y w2 Ž p.
Ž 4.22.
This both makes explicit and extends to Prim gases the result of Hui and He w10x. In addition, the condition Ž4.16. together with the relation Ž4.19. shows that the mean curvature ᑤ of the Beltrami surfaces varies only with the pressure thereon. In a similar manner, q 2 and w on the generalized Bernoulli surfaces depend only on the pressure. It should be noted that both q 2 and p are invariants under Prim’s substitution principle. In conclusion, it is natural to enquire under what conditions
q2 s ⌽ Ž p.
Ž 4.23.
throughout a spatial flow. In this case, the intrinsic equation of motion Ž4.13., namely,
q
␦q ␦s
q
␦p ␦s
s 0,
together with Ž4.23. immediately produces
␦p ␦s
s
␦q ␦s
s
␦ ␦s
s 0,
whence div t s 0.
Ž 4.24.
66
ROGERS, SCHIEF, AND HUI
Let us now consider the behavior of the Mach number in the motion. Equation Ž2.5. shows that M 2 q 2 H Ž p . P X Ž p . s a2 SP X Ž p . ,
Ž 4.25.
whence, on use of Ž2.20., M 2 s 2 PX Ž p . H Ž p0 . y H Ž p . s
2 PX Ž p . H Ž p . w 2 1 y w2
.
Ž 4.26.
Hence, on a Beltrami surface s const., M s 2
2 PX Ž p . H Ž p . w 2 Ž p . 1 y w2 Ž p.
,
Ž 4.27.
so that, indeed, M s M Ž p, . ,
Ž 4.28.
as established by Hui and He w10x.
5. A HEISENBERG SPIN EQUATION It has been seen that in complex-lamellar motion div b s 0,
Ž 5.1.
while, with the use of the relation Ž3.8.1 , div Ž ⌳b . s s
␦⌳ ␦b ␦ ␦b
q ⌳ div b
ž
w y
␦w ␦n
q w y
/ ž
␦w ␦n
/
div b
s w div b. Accordingly, div Ž ⌳b . s 0,
Ž 5.2.
COMPLEX-LAMELLAR MOTION OF A PRIM GAS
67
whence, assuming non-vanishing div b, combination with Ž4.1. yields
␦ ␦b
Ž ⌳r . s 0,
so that ⌳ s ,
Ž 5.3.
where ␦r␦ b s 0. The generalized Crocco relation Ž4.11., together with the expression Ž5.3. for the reduced vorticity ⌳, now shows that
␣ g 1r2 s  ,
Ž 5.4.
where
␦␣r␦ s s 0,
␦r␦ b s 0.
Ž 5.5.
Now, if r is the generic position vector to an individual generalized Bernoulli surface, then
⭸r ⭸b
s g 1r2
⭸r ⭸b
s

ž / ␣
b s
 ␣
ž
t=
⭸t ⭸s
/
,
Ž 5.6.
whence, on removal of ␣ by scaling on b, it is seen that Ž5.6. delivers the Heisenberg spin-type equation,
⭸t ⭸b
s
⭸ ⭸s
 t=
ž
⭸t ⭸s
/
Ž ⭸r⭸ b s 0 . ,
Ž 5.7.
where t is the unit tangent to the streamlines on the generalized Bernoulli surfaces. These streamlines constitute geodesics thereon. It is remarked that, accordingly, the marching direction in the optimal coordinate system of Hui and He w10x is along geodesics on the generalized Bernoulli surfaces. It is interesting to note that in steady Newton᎐Busemann gas flow wherein the Mach number M ª 1 and ␥ s 1, the streamlines are geodesics on the body surface w29x, but this is not so for general gas flow with finite Mach number.
ACKNOWLEDGMENT Professor Hui acknowledges with gratitude support from the Research Council of Hong Kong.
68
ROGERS, SCHIEF, AND HUI
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