Z. angew. Math. Phys. 49 (1998) 363–3830044-2275/98/030363-21 $ 1.50+0.20/0c© 1998 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

Inverse Euler equations

Jakob J. Keller

Abstract. Following previous work by Keller [2], that is extended to compressible flow, thegeneral time-independent Euler equations for inviscid fluid flow are first written in a perfectlyantisymmetric form, using a pair of stream functions as the dependent variables. In a sec-ond step the equations are written in an inverse form, using the two stream functions and thenatural coordinate as independent variables. As a special case the Bragg-Hawthorne equationfor axisymmetric flow is first extended to compressible flow and also transformed to its inverseform. The main advantage of using these inverse equations is associated with the possibilitiesof using static pressure distributions, Mach number distributions, geometric constraints, etc., orany combination of geometric constraints and specifications of physical quantities to define theboundary conditions. In contrast to conventional inverse methods, that are based on iterativeapproximations to a desired pressure distribution along the surface of a flow device, for example,the use of inverse Euler equations offers the possibility to arrive at the solution for any kindof boundary conditions in a single step. Furthermore, there is no need for complicated gridgeneration procedures, because the domain of definition in inverse space is typically a cube withCartesian coordinates. In the original space, the surfaces on which the natural coordinate isconstant are orthogonal to the streamlines. As a consequence, the computation time can be keptsmall and the accuracy is remarkably high. This semi-orthogonal curvilinear grid is generatedautomatically together with the solution. The density of grid lines is automatically getting largein domains where gradients are large. Possible difficulties with using inverse Euler equationsare mainly related to the topology of the flow field. The transform to inverse coordinates mustcorrespond to a one-to-one mapping. Hence, if the domain of definition is not simply connectedit must be cut suitably to obtain piecewise domains for which one-to-one mappings exist.

Keywords. Inverse design, natural coordinates, stream functions, inverse coordinates.

1. Introduction

This paper presents a transformation of the general time-independent Euler equa-tions for inviscid fluid flow to a fully inverse form, using two stream functionsand the natural coordinate as independent variables. As a special case the Bragg-Hawthorne equation for axisymmetric flow is first extended to compressible flowand also transformed to its inverse form.

The particular merits of inverse equations have been recognized long ago. Atransformation that interchanges the roles of dependent and independent variables

364 J. J. Keller ZAMP

is usually called a hodograph transformation (see Courant and Friedrichs [1], forexample). To solve two-dimensional potential flow problems in the case of free sur-faces, for instance, it is convenient to use the conformal mapping that interchangesthe roles of the complex velocity and the complex space coordinates. In general,transformations that interchange the roles of dependent and independent variableshave been used to simplify equations. A set of reducible second-order equations,for example, can be transformed to a set of linear equations. Unfortunately themerits of this type of transformation have usually been limited due to the factthat the boundary conditions may become quite complicated.

However, the situation is different if the original dependent variables are a pairof stream functions. In this case the boundary conditions become very simpleafter transformation. In a recent paper by J. J. Keller [2] the Euler equations forincompressible time-independent flow have been transformed to an antisymmetricform, using a pair of stream functions as dependent variables. After extendingthese equations to compressible flow we will have a suitable basis to derive thecorresponding inverse set of Euler equations.

In recent years substantial efforts have been made to introduce inverse designtools for airfoils, turbine blade and vane channels, and other flow devices for whichthe optimum combination of aerodynamic forces and losses plays a crucial role.For an adequate overview of the state of the art the reader is referred to work byJameson et al. [3], Giles and Drela [4], Dang and Isgro [5], Dulikravich [6], Pandyaand Baysal [7]. An overview of applications to current turbomachinery design isgiven by Jennions [8]. It should be understood that this list of publications oninverse design methods is far from complete. Furthermore, it should be mentionedthat formulations of the Euler equations using two stream functions have beenpreviously used (see Turner and Giles [9], for example). However, with the excep-tion of genuine inverse methods (Keller et al. [10], [11]) that have been proposedfor certain two-dimensional flows, the “inverse methods” that are presently usedto arrive at an optimum design of three-dimensional flow devices are not reallyinverse. In general “Evolution Strategy”, genetic algorithms or simply Newton’smethod are used to guide an extensive series of flow calculations for graduallyvarying geometry toward some kind of optimum.

The key advantage of using inverse equations instead, is associated with the factthat the geometry of a flow device does not have to be defined in advance. As analternative it is possible to impose the distribution of static pressure, flow speed,temperature, heat flux or any other physical quantity, as a boundary conditionand ask for the geometry (as a part of the solution) that generates the desired dis-tribution of static pressure, etc.. Furthermore, if the original domain of definitionis simply connected, a one-to-one mapping into a cubic domain of definition withCartesian coordinates can be achieved. As a consequence, the generally tedioustask of grid generation for numerical calculations can be avoided.

Vol. 49 (1998) Inverse Euler equations 365

2. Stream functions and natural coordinate

To define boundary conditions and to embed boundary layers in an inviscid bulkflow, for example, it may be useful to introduce the ”natural coordinate” σ thatis related to the arc lengths along stream lines. Following the ideas of Ref. [2]we introduce a pair of stream functions to define a vector potential for the ve-locity field. Without essential loss of generality we restrict our considerations tosolenoidal vector fields for the mass flux. In order to extend the discussion pre-sented in Ref. [2] to compressible flows the mass flux (rather than the velocity)needs to be related to the stream functions. Hence, introducing the two streamfunctions ψ and χ, we can write

ρu = grad (ψ)× grad (χ), (1)

where u is the velocity vector and ρ the density. The natural coordinate can bedefined by

hgrad (σ) =grad (ψ)× grad (χ)|grad (ψ)× grad (χ)| , (2)

where σ is the dimensionless natural coordinate and h = h(σ, ψ, χ) is a nonlinearstretching coefficient, such that hdσ denotes the actual length increment along thestreamlines. As usual the nonlinear stretching needs to be included, in order toreach compatibility between the orthogonality requirement (2) and the uniformdefinition of hdσ as the length increment along the streamlines.

Furthermore, in inverse coordinates we can write

∂x

∂σ· ∂x∂σ

= h2, x = x(σ, ψ, χ), (3)

where x is the coordinate vector in the original space.First we need to express the time-independent Euler equations in terms of the

two stream functions ψ and χ.

3. The governing equations in a rotating frame of reference

An area of major interest to use inverse equations is the design of optimizedturbine and compressor blade channels. For this reason it seems to be appropriateto further extend the governing equations to rotating frames of reference. In aframe of reference rotating steadily at the angular velocity Ω the Euler equationcan be written in the form

(u · grad )u+1ρ

grad p = −2Ω× u− Ω× (Ω× x), (4)

366 J. J. Keller ZAMP

where p refers to the static pressure. The terms on the right-hand side of (4)contribute to the fictitious volume force. The first term accounts for the Coriolisforce and the second for the centrifugal force. Making use of the identities

(u · grad )u = grad(

12u · u

)− u× ω (5)

and

−Ω× (Ω× x) = grad(

12

[Ω× x] · [Ω× x]), (6)

where ω refers to the vorticity,

ω = rot (u) = rot(

1ρ

grad (ψ)× grad (χ))

=1ρ

rot(grad (ψ)× grad (χ)

)+ grad

(1ρ

)×[grad (ψ)× grad (χ)

],

(7)

we can rewrite (4) as

grad(

12u · u− 1

2[Ω× x] · [Ω× x]

)− u× [ω + 2Ω] +

1ρ

grad p = 0. (8)

The first law of thermodynamics can be expressed as

Tds = cV dT −p

ρ

dρ

ρ, (9)

where T is the temperature, s denotes the entropy per unit mass and cV refers tothe specific heat per unit mass at constant volume. The equation of state is

p

ρ= (cp − cV )T, (10)

and cp refers to the specific heat per unit mass at constant pressure. Combining(8), (9) and (10) yields

grad(cpT +

12u · u− 1

2[Ω× x] · [Ω× x]

)− u× [ω + 2Ω]− Tgrads = 0. (11)

Introducing the total enthalpy

H = cpT +12u · u− 1

2[Ω× x] · [Ω× x] (12)

equation (11) can now be expressed in the form of Crocco’s theorem in the rotatingframe:

gradH − u× [ω + 2Ω]− Tgrads = 0. (13)

Vol. 49 (1998) Inverse Euler equations 367

By virtue of (1) the condition of mass conservation is identically satisfied. More-over, noting that the validity of the Euler equations is restricted to adiabatic flows,it is obvious that the entropy, s is constant on streamlines. Hence, the entropy isan integral of motion (in any frame of reference),

s = s(ψ, χ) ⇔ u · grads = 0. (14)

Furthermore, it is obvious that

(u× [ω + 2Ω]) · u = 0. (15)

By virtue of (14) and (15) the scalar product of the velocity vector with equation(13) yields

u · gradH = 0 ⇔ H = H(ψ, χ) (16)

This is Bernoulli’s theorem in the rotating frame of reference. It shows that thetotal enthalpy in the rotating frame is an integral of motion. Making use of (1),(14) and (16) equation (13) can be rewritten as(

∂H

∂ψ− T ∂s

∂ψ

)gradψ +

(∂H

∂χ− T ∂s

∂χ

)gradχ

− 1ρ

[gradψ × gradχ]× [ω + 2Ω] = 0(17)

Furthermore, using the identity

− [gradψ × gradχ]× [ω + 2Ω] =([ω + 2Ω] · gradχ)gradψ − ([ω + 2Ω] · gradψ)gradχ,

(18)

the vector products of (17) with gradψ and gradχ, respectively, lead to the fol-lowing generalized Euler equations in terms of two stream functions:

∂H

∂ψ− T ∂s

∂ψ+

1ρ

[ω + 2Ω] · gradχ = 0 (19)

and∂H

∂χ− T ∂s

∂χ− 1ρ

[ω + 2Ω] · gradψ = 0. (20)

For the subsequent discussion it is useful to introduce the mixed product that isdefined by

abc ≡ [a× b] · c. (21)

An aim of the present paper is to derive the inverse forms of Euler’s equations,using the two stream functions ψ and χ and the natural (streamline) coordinateσ as independent variables.

368 J. J. Keller ZAMP

4. Mapping to inverse coordinates

In this section we consider the mapping

x↔ (σ, ψ, χ). (22)

The derivatives in the two coordinate systems are related by

grad (ψ)× grad (χ) =1J

∂x

∂σ,

grad (χ)× grad (σ) =1J

∂x

∂ψ, (23)

grad (σ) × grad (ψ) =1J

∂x

∂χ,

where the mixed product

J =∂x

∂σ

∂x

∂ψ

∂x

∂χ

(24)

may be interpreted as Jacobi’s determinant. Moreover it can be shown that

grad (σ) =1J

[∂x

∂ψ

]×[∂x

∂χ

],

grad (ψ) =1J

[∂x

∂χ

]×[∂x

∂σ

], (25)

grad (χ) =1J

[∂x

∂σ

]×[∂x

∂ψ

].

For a function f(x) = g(σ, ψ, χ) the conversion of derivatives can be expressed as

grad (f) =1J

([∂x

∂ψ

]×[∂x

∂χ

])∂g

∂σ

+([

∂x

∂χ

]×[∂x

∂σ

])∂g

∂ψ+([

∂x

∂σ

]×[∂x

∂ψ

])∂g

∂χ

,

(26)

and for a vector a(x) = b(σ, ψ, χ) we can write

div (a) =1J

([∂x

∂ψ

]×[∂x

∂χ

])· ∂b∂σ

+([

∂x

∂χ

]×[∂x

∂σ

])· ∂b∂ψ

+([

∂x

∂σ

]×[∂x

∂ψ

])· ∂b∂χ

,

(27)

and

rot (a) =1J

([∂x

∂ψ

]×[∂x

∂χ

])× ∂b

∂σ

+([

∂x

∂χ

]×[∂x

∂σ

])× ∂b

∂ψ+([

∂x

∂σ

]×[∂x

∂ψ

])× ∂b

∂χ

.

(28)

Vol. 49 (1998) Inverse Euler equations 369

Using (25), (26) and (28) it may be shown that

grad (χ) · rot (grad (ψ)× grad (χ))

=[

1J

∂x

∂ψ

]·(∂

∂σ

[1J

∂x

∂σ

])−[

1J

∂x

∂σ

]·(∂

∂ψ

[1J

∂x

∂σ

]).

(29)

Similarly (or by analogy) we obtain

grad (ψ) · rot (grad (ψ)× grad (χ))

= −[

1J

∂x

∂χ

]·(∂

∂σ

[1J

∂x

∂σ

])+[

1J

∂x

∂σ

]·(∂

∂χ

[1J

∂x

∂σ

]).

(30)

Furthermore, using (25) it may be shown that

Ω · grad (χ) =1J

∂x

∂σ

∂x

∂ψΩ

=

∂x∂σ

∂x∂ψΩ

∂x∂σ

∂x∂ψ

∂x∂χ

(31)

and

Ω · grad (ψ) =1J

∂x

∂σΩ∂x

∂χ

=

∂x∂σΩ ∂x

∂χ

∂x∂σ

∂x∂ψ

∂x∂χ

. (32)

Making use of (7), the identity

a× (b× c) = (a · c)b− (a · b)c (33)

and (29) to (32) the equations of motion, (19) and (20), can be written as

∂H

∂ψ− ∂

∂ψ

(12

[1ρJ

∂x

∂σ

]·[

1ρJ

∂x

∂σ

])− T ∂s

∂ψ

+[

1ρJ

∂x

∂ψ

]·(∂

∂σ

[1ρJ

∂x

∂σ

])+

2ρJ

∂x

∂σ

∂x

∂ψΩ

= 0(34)

and∂H

∂χ− ∂

∂χ

(12

[1ρJ

∂x

∂σ

]·[

1ρJ

∂x

∂σ

])− T ∂s

∂χ

+[

1ρJ

∂x

∂χ

]·(∂

∂σ

[1ρJ

∂x

∂σ

])− 2ρJ

∂x

∂σΩ∂x

∂χ

= 0.

(35)

Note that the mapping described in this section is thus far valid for any choice ofthe variable σ. No use has been made yet of the fact that σ should ultimately beinterpreted as the natural coordinate as defined by (2) and (3).

370 J. J. Keller ZAMP

The subsequent part of the analysis shall now account for the special signifi-cance of σ as a natural coordinate. From (2), (3), (23) and (25) we obtain

∂x

∂σ=h2

J

[∂x

∂ψ

]×[∂x

∂χ

]. (36)

Hence, (36) implies that the covariant and contravariant forms of ∂x/∂σ differ bythe factor h2. Furthermore, (36) implies

∂x

∂σ· ∂x∂σ

= h2,∂x

∂σ· ∂x∂ψ

= 0,∂x

∂σ· ∂x∂χ

= 0. (37)

The first relation (37) is the same as (3). Using (37) the equations (34) and (35)can be simplified to

∂H

∂ψ− 1ρJ

∂

∂ψ

(h2

ρJ

)− T ∂s

∂ψ+

2ρJ

∂x

∂σ

∂x

∂ψΩ

= 0 (38)

and∂H

∂χ− 1ρJ

∂

∂χ

(h2

ρJ

)− T ∂s

∂χ− 2ρJ

∂x

∂σΩ∂x

∂χ

= 0. (39)

It is quite remarkable that for irrotational isentropic flow,

dH = 0, ds = 0, Ω = 0, (40)

equations (38) and (39) can be integrated once.

5. Thermodynamic relations

Making use of (12) and the definitions

H ≡ cpT0, u ≡ √u · u, W ≡ Ω× x, W ≡√W ·W,

γ ≡ cp/cV , c2 ≡ (cp − cV )γT, M ≡ u/c,(41)

we can writecpT0 = cpT +

12u2 − 1

2W 2 (42)

orT

T0=

1 + W2

2cpT0

1 + γ−12 M2

. (43)

T0 is the stagnation temperature in the absolute system, c is the sound speed andM the Mach number in the rotating frame. Hence,

T0 = T0(ψ, χ). (44)

Vol. 49 (1998) Inverse Euler equations 371

Combining (14) with the first law of thermodynamics and the equation of statewe can show that the isentropic relations

ρ

ρ0=[T

T0

] 1γ−1

,p

p0=[T

T0

] γγ−1

,c

c0=[T

T0

] 12

, (45)

hold along streamlines. In these expressions ρ0, p0 and c0 should be interpreted asthe stagnation density, the stagnation pressure and the stagnation sound speed,respectively, in the absolute system and, by virtue of (14),

ρ0 = ρ0(ψ, χ), p0 = p0(ψ, χ), c0 = c0(ψ, χ). (46)

It is important to note that with the help of (14) and (44) to (46) all thermodynamicquantities can be related to integrals of motion! For the mass flux we obtain (see(3) and the first equation (23))

m2 ≡ ρ2u2 =[grad (ψ)× grad (χ)

]·[grad (ψ)× grad (χ)

]=h2

J2 = ρ20c

20

[1 +

W 2

2cpT0

] γ+1γ−1 M2[

1 + γ−12 M2

] γ+1γ−1

.(47)

We may now note that the equations of motion (38), (39), together with the condi-tion (36), that defines the natural coordinate σ, and the thermodynamic relationsconstitute a complete set of equations in inverse coordinates.

6. Cylindrical polar coordinates as dependent variables

For many applications, including flows through turbines, it is convenient to expressthe coordinate vector x in terms of cylindrical polar coordinates. Introducingcylindrical polar coordinates (x, r, θ) it may be shown that Jacobi’s determinantassumes the form

J =

∣∣∣∣∣∣∣∂x∂σ

∂x∂ψ

∂x∂χ

∂r∂σ

∂r∂ψ

∂r∂χ

r ∂θ∂σ r ∂θ∂ψ r ∂θ∂χ

∣∣∣∣∣∣∣=∂x

∂σ

∂r

∂ψr∂θ

∂χ+∂x

∂ψ

∂r

∂χr∂θ

∂σ+∂x

∂χ

∂r

∂σr∂θ

∂ψ

− ∂x

∂σ

∂r

∂χr∂θ

∂ψ− ∂x

∂ψ

∂r

∂σr∂θ

∂χ− ∂x

∂χ

∂r

∂ψr∂θ

∂σ

(48)

Assuming that the angular velocity vector Ω is parallel to the axis and points inits positive direction, we find

∂x

∂σ

∂x

∂ψΩ

= Ωr∂r

∂σ

∂θ

∂ψ− Ωr

∂r

∂ψ

∂θ

∂σ(49)

372 J. J. Keller ZAMP

and ∂x

∂σΩ∂x

∂χ

= Ωr

∂r

∂χ

∂θ

∂σ− Ωr

∂r

∂σ

∂θ

∂χ. (50)

The relation (37), that defines the natural coordinate σ, may be expressed as

(∂x

∂σ

)2+(∂r

∂σ

)2+(r∂θ

∂σ

)2= h2,

∂x

∂σ

∂x

∂ψ+∂r

∂σ

∂r

∂ψ+(r∂θ

∂σ

)(r∂θ

∂ψ

)= 0, (51)

∂x

∂σ

∂x

∂χ+∂r

∂σ

∂r

∂χ+(r∂θ

∂σ

)(r∂θ

∂χ

)= 0.

Inserting (48) to (50) in (38) and (39) leads to a complete set of equations in cylin-drical polar coordinates, if the equations (51), that define the natural coordinate,and the thermodynamic relations are added.

7. Cartesian coordinates as dependent variables

In most cases, including inverse design of wings, for example, the coordinate vectoris preferably expressed in terms of Cartesian coordinates and the frame of referenceis not rotating. Hence, we can set Ω = 0. Introducing Cartesian coordinates(x, y, z) Jacobi’s determinant assumes the form

J =∂x

∂σ

∂y

∂ψ

∂z

∂χ+∂x

∂ψ

∂y

∂χ

∂z

∂σ+∂x

∂χ

∂y

∂σ

∂z

∂ψ− ∂x∂σ

∂y

∂χ

∂z

∂ψ− ∂x∂ψ

∂y

∂σ

∂z

∂χ− ∂x∂χ

∂y

∂ψ

∂z

∂σ(52)

The relation (37), that defines the natural coordinate σ, may be expressed as

(∂x

∂σ

)2+(∂y

∂σ

)2+(∂z

∂σ

)2= h2,

∂x

∂σ

∂x

∂ψ+∂y

∂σ

∂y

∂ψ+∂z

∂σ

∂z

∂ψ= 0, (53)

∂x

∂σ

∂x

∂χ+∂y

∂σ

∂y

∂χ+∂z

∂σ

∂z

∂χ= 0.

Vol. 49 (1998) Inverse Euler equations 373

8. Isentropic flows

As pointed out earlier, the equations (38) and (39) can be substantially simplifiedif we restrict the consideration to isentropic flows. In this case the conditions (40)lead to

∂

∂ψ

(h2

ρJ

)= 0 ⇒ h2

ρJ= f1(σ, χ)

∂

∂χ

(h2

ρJ

)= 0 ⇒ h2

ρJ= f2(σ, ψ)

⇒ h2

ρJ= f(σ), (54)

where f1 and f2 are functions of integration. Moreover, as we have not yet definedthe scaling of the natural coordinate σ the function f(σ) can be “absorbed” byre-scaling the natural coordinate,

f(σ)dσ → dσ. (55)

For convenience we may also assume that the normalization of the density isincluded in this stretching of the natural coordinate and from now on consider ρas the density that is normalized with the stagnation density. Hence we can write

h2 = ρJ (56)

and

ρ = ρ(M) =[1 +

γ − 12

M2] −1γ−1

. (57)

On the other hand, by suitable scaling of the stream functions, the stagnationsound speed can also be absorbed and (47) may be normalized accordingly,

h2

J2 = m2(M) =M2[

1 + γ−12 M2

] γ+1γ−1

. (58)

Combining (36) and (56) yields

1ρ

∂x

σ=[∂x

∂ψ

]×[∂x

∂χ

]. (59)

Eliminating J and h from (56), (58) and the first equation (37) leads to

∂x

∂σ· ∂x∂σ

=( ρm

)2. (60)

We may now regard (59) as the equation for x = x(σ, ψ, χ), where the density ρ isrelated to |∂x/∂σ| by (57), (58) and (60). In other words, (57) and (58) may be

374 J. J. Keller ZAMP

considered as a parametric representation (in terms of the Mach number) of thefunction m = m(ρ) that can be inserted in (60) to obtain the desired expressionfor the density.

The structure of the equations becomes still more transparent by consideringthe special case of plane two-dimensional flows. This simplification is achieved,for example, by introducing Cartesian coordinates, x = (x, y, z) and defining thestream function χ by

χ = z. (61)

In this case (59) can be reduced to

∂x

∂σ= ρ

∂y

∂ψ,

∂y

∂σ= −ρ ∂x

∂ψ(62)

and (57), (58) and (60) can be put into the form(∂x

∂σ

)2+(∂y

∂σ

)2=( ρm

)2, ρ = ρ(M), m = m(M). (63)

The Jacobian reduces toJ =

∂x

∂σ

∂y

ψ− ∂x

∂ψ

∂y

∂σ. (64)

Combining (56), (58), (62) and (64) it may be shown that(∂x

∂ψ

)2+(∂y

∂ψ

)2=

1m2 . (65)

Considering now an incremental stream tube it is easy to show that the equations(62), (63) and (65) are in agreement with the equations of motion. Eliminatingρdσ from (62) and (63) we obtain the correct relation between massflow vector andvelocity vector. On the other hand (65) relates the massflow to the cross-sectionalarea of the incremental stream tube. The massflow m reaches its maximum atM = 1. Hence, the term on the right-hand side of (65) reaches its minimum atM = 1. Considering an incremental Laval nozzle with slowly varying width dyalong a line y = constant, for example, the convergent-divergent character of thecross-section is immediately apparent from (65). Nevertheless, it is surprising thatthe system of equations (62) is generally elliptic. It appears that the hyperboliccharacter of supersonic flow is hidden in the combination of (62) and (63) with∂m/∂M < 0.

Finally, for the special case of incompressible flows (62) reduces to Laplace’sequation,

∂x

∂σ=∂y

∂ψ,

∂y

∂σ= − ∂x

∂ψ⇒ ∂2x

∂σ2 +∂2x

∂ψ2 = 0,∂2y

∂σ2 +∂2y

∂ψ2 = 0, (66)

which is in agreement with the properties of a special type of conformal mappingthat is usually called hodograph transformation.

Vol. 49 (1998) Inverse Euler equations 375

9. General limitations with respect to supersonic flow

An important limitation that applies to any coordinate inversion and is relevantfor supersonic flow appears if the Jacobian vanishes,

J = 0. (67)

Flows that satisfy the condition (67) are not included in the solution space ofthe inverse equations. The reader is referred to the corresponding discussion byCourant and Friedrichs [1]. Flows that satisfy (67) are called simple waves. Anexpansion fan in an isentropic flow, for example, is not a solution of the equations(57) to (60). To obtain a complete set of solutions we need to add the solutions of(67) to the solution space of any set of inverse equations. In Cartesian coordinates,for example, the general solution of (67) can be expressed as

G(x, y, z) = 0. (68)

The validity of (68) is easily confirmed by introducing (52) in (67) and inserting(68).

10. Axisymmetric flows

For the special case of axisymmetric flow in the absolute frame of reference theequations (19) and (20) reduce to a generalized form of the Bragg-Hawthorneequation. Without giving the details of deriving the equation we simply state theresults. For axisymmetric flows we introduce a single stream function ψ and thespecific angular momentum C as an additional integral of motion:

ρu =1r

∂ψ

r, ρv = −1

r

∂ψ

∂x, C = C(ψ) = rw, (69)

where x and r are the axial and radial coordinates, and u, v and w denote theaxial, radial and azimuthal velocity components, respectively. In this case theEuler equation is found to be

1ρr

∂

∂r

[1ρr

∂ψ

∂r

]+

1ρr2

∂

∂x

[1ρ

∂ψ

∂x

]=dH

dψ− T ds

dψ− C

r2dC

dψ, (70)

where the total enthalpy

H = H(ψ) = cpT +12

[u2 + v2 + w2] (71)

376 J. J. Keller ZAMP

and the entropy s = s(ψ) are again integrals of motion. Equation (47) reduces to

ρ2[u2 + v2 + w2] =[

1r

∂ψ

∂x

]2+[

1r

∂ψ

∂r

]2+C2

r2ρ2

0[1 + γ−1

2 M2] 2γ−1

=ρ2

0c20M

2[1 + γ−1

2 M2] γ+1γ−1

.

(72)

The remaining equations are the same as in section 5. We may note that, regardingthe density ρ and the entropy s as constants, replacing the total enthalpy (71) bythe total head,

H = H(ψ) =p

ρ+

12

[u2 + v2 + w2], (73)

and eliminating the density from the definition of the stream function,

u =1r

∂ψ

∂r, v = −1

r

∂ψ

∂x, (74)

(70) reduces to the well-known Bragg-Hawthorne equation for incompressible axi-symmetric flow:

∂2ψ

∂r2 −1r

∂ψ

∂r+∂2ψ

∂x2 = r2 dH

dψ− C dC

dψ. (75)

At this point we introduce the mapping

(x, r) ↔ (σ, ψ) (76)

to inverse coordinates and, for simplification, we restrict the consideration to in-compressible flow. In this case the streamline coordinate σ is defined by

h∂σ

∂x=

∂ψ∂r√(

∂ψ∂x

)2+(∂ψ∂r

)2, h

∂σ

∂r=

−∂ψ∂x√(∂ψ∂x

)2+(∂ψ∂r

)2, (77)

where h = h(σ, ψ) is the stretching function that provides the basis for compatibil-ity between orthogonality of the curves σ(x, r) = constant and ψ(x, r) = constantand the uniform meaning of hdσ,

hdσ =√

(dx)2 + (dr)2 ⇒(∂x

∂σ

)2+(∂r

∂σ

)2= h2, (78)

as the length increment along streamlines. The condition for the curvilinear coor-dinates to be orthogonal is again

∂x

∂σ

∂x

∂ψ+∂r

∂σ

∂r

∂ψ= 0. (79)

Vol. 49 (1998) Inverse Euler equations 377

Introducing Jacobi’s determinant,

J ≡ ∂x

∂σ

∂r

∂ψ− ∂r

∂σ

∂x

∂ψ, (80)

the partial derivatives of σ(x, r) and ψ(x, r) can be expressed as

∂σ

∂x=

1J

∂r

∂ψ,

∂σ

∂r= − 1

J

∂x

∂ψ,

∂ψ

∂x= − 1

J

∂r

∂σ,

∂ψ

∂r=

1J

∂x

∂σ. (81)

For a function f(x, r) = g(σ, ψ) the conversion of derivatives can be expressed as

∂f

∂x=

1J

[∂r

∂ψ

∂g

∂σ− ∂r

∂σ

∂g

∂ψ

],

∂f

∂r= − 1

J

[∂x

∂ψ

∂g

∂σ− ∂x

∂σ

∂g

∂ψ

]. (82)

Inserting (81) in (82) we obtain the conversion of the second derivatives:

∂2ψ

∂x2 =1J

[∂r

∂σ

∂

∂ψ

(1J

∂r

∂σ

)− ∂r

∂ψ

∂

∂σ

(1J

∂r

∂σ

)],

∂2ψ

∂r2 =1J

[∂x

∂σ

∂

∂ψ

(1J

∂x

∂σ

)− ∂x

∂ψ

∂

∂σ

(1J

∂x

∂σ

)].

(83)

Adding the second derivatives according to (83), and making use of (78), (79) and(80), integration by parts yields

∂2ψ

∂x2 +∂2ψ

∂r2 =1J

∂

∂ψ

(h2

J

). (84)

Furthermore, from (78), (79) and (80) it follows that

∂x

∂σ=h2

J

∂r

∂ψ⇒ 1

r

∂ψ

∂r=

1rJ

∂x

∂σ=

h2

rJ2∂r

∂ψ. (85)

Combining (75) with (84) and (85) we finally obtain the inverse Bragg-Hawthorneequation

r

J

∂

∂ψ

(h2

rJ

)= r2 dH

dψ− CdC

dψ. (86)

A complete system of equations is obtained by adding (78) and (79). The equation(86) is again remarkably simple. To check the previous results a somewhat moreelaborate analysis has been chosen in this case to derive the inverse equation. Itis tedious, but rather straightforward to eliminate the Jacobian from (86) in favor

378 J. J. Keller ZAMP

of partial derivatives of the radius. Making use of (78), (79) and (80) the equation(86) can be written in the form(

∂r

∂ψ

)2∂2r

∂σ2 +∂r

∂σ

∂r

∂ψ

∂2r

∂σ∂ψ+

[h2 −

(∂r

∂σ

)2][

∂2r

∂ψ2 +1r

(∂r

∂ψ

)2]

− 1h

∂r

∂σ

∂r

∂ψ

[∂r

∂σ

∂h

∂ψ+∂r

∂ψ

∂h

∂σ

]+ h2

(∂r

∂ψ

)3r2 dH

dψ− CdC

dψ

= 0

(87)

and the derivative of the coordinate stretching factor with respect to the streamfunction can be expressed as

∂h

∂ψ=

∂r∂ψ

h2 −(∂r∂σ

)2 [∂h∂σ ∂r∂σ − h ∂2r

∂σ2

]. (88)

The idea is now to solve (87) by means of a Gauss-Seidel iteration scheme anddetermine the coordinate stretching factor h using a stepwise integration of (88).From (74), (78), (81) and (85) we obtain

u2 + v2 =1r2

[(∂ψ

∂x

)2+(∂ψ

∂r

)2]

=1

r2J2

[(∂x

∂σ

)2+(∂r

∂σ

)2]

=h2

r2J2 =h2 −

(∂r∂σ

)2h2r2

(∂r∂ψ

)2

(89)

The flow speed can now be expressed as

√u2 + v2 + w2 =

√u2 + v2 +

(C

r

)2=

√√√√√ h2 −(∂r∂σ

)2h2r2

(∂r∂ψ

)2 +(C

r

)2. (90)

11. Difficulties encountered with applications of inverse equa-tions

For the transformation of the equations to their inverse form it is essential to choosethe domain of definition such that a one-to-one mapping is generated. For simplyconnected domains of definition, such as turbine blade channels, for example, thisdifficulty would generally not appear. However, in the case of flow fields thatare not simply connected the difficulty arises that the entire flow field must besuitably cut to obtain domains for which a one-to-one mapping exists. A further

Vol. 49 (1998) Inverse Euler equations 379

00 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1.2

1.4

inner wall outer wallfl

ow

spee

d

arc length from inlet along respective wall

Figure 1.Distributions of flow speed along the two walls of an annular diffuser. The solid lines markcurves of maximum adverse pressure gradient. Smooth transitions between these curves havebeen generated using fourth-order polynomials, in order to define suitable distributions of flowspeed along the diffuser walls.

point that should be addressed concerns the chosen domain of definition for thestream functions. In this context the reader is referred to Keller [12] for moredetails. If a wake appears within the domain of definition there may be a problemwith streamlines, originating on the downstream side, for which the integrals ofmotion have not been defined. For this reason it is a good idea to include a checkin a numerical solution procedure that warns the user in cases where integrals ofmotion have been taken beyond their intended domain of definition.

12. Two simple examples

To demonstrate the capabilities of inverse equations we consider two simple axi-symmetric problems. The design of a compressor diffuser of a gas turbine is oftenstrongly restricted by the space requirements of turbine and combustor. As aconsequence there is usually the necessity of choosing a curved diffuser. Althoughthere is broad empirical knowledge on the performance of straight annular diffusers(see Kline, Abbott and Fox [13], for example), it may often be difficult to apply thisinformation to arbitrary curved diffusers. However, knowing the relation betweenthe largest acceptable adverse pressure gradient and the ratio of (local) length

380 J. J. Keller ZAMP

0.80 0.1 0.2 0.3 0.4 0.8 0.6 0.7 0.8 0.9 1

0.9

1

1.1

1.2

1.3

1.4

r

x

Figure 2.Map of streamlines and curves of constant σ that corresponds to the boundary conditions definedby the distributions of flow speed shown in figure 1.

and inlet width of a diffuser we can specify restrictions for the static pressuredistributions along the inner and outer walls of a curved annular diffuser. Asindicated in figure 1, a possible strategy to design a curved diffuser may be definedas follows. Near the diffuser inlet the flow speeds along both walls first followa curve that corresponds to the largest acceptable pressure gradient. After acertain arc length a smooth transition to a different curve of maximum pressureincrease is chosen for the outer diffuser wall. For a certain distance the flow speeddistributions along the two walls follow two different curves of largest pressureincrease. This domain will generate the curved portion of the diffuser. Towardthe exit of the diffuser a smooth transition of the flow speed along the innerwall is introduced, in order to obtain a velocity profile at the diffuser exit thatis as uniform as possible. The second transition brings the flow speed along theinner wall back to a common curve of maximum pressure increase. Knowing theflow speed distributions along both diffuser walls enables us to express ∂r/∂σ interms of ∂r/∂ψ, making use of equation (90). At the inlet to the diffuser we set∂r/∂σ = 0, and at the exit of the diffuser we choose a linear transition for ∂r/∂σbetween the corresponding values at the endpoints of the two walls. Choosing anykind of distributions for the total head and the circulation at the diffuser inletdoes not cause additional difficulties, because H = H(ψ) and C = C(ψ) appearas integrals of motion in the governing equations (87) and (88). It is now easy tofind a numerical solution with the help of the Gauss-Seidel iteration scheme. For

Vol. 49 (1998) Inverse Euler equations 381

0.80 0.2 0.4 0.6 0.8 1

1

1.2

1.4

1.6

1.8

2

r

x

Figure 3.Map of streamlines and curves of constant σ of a turbine inlet flow.

more details the reader is referred to Keller et al. [10]. For the simple examplepresented it has been assumed that the flow is incompressible, the total head isuniform and the flow is free of swirl. Figure 2 illustrates a streamline map thatcorresponds to the boundary conditions defined by the flow speed distributionsshown in figure 1. Furthermore, figure 2 shows lines of constant σ. It should bepointed out that, according to the definition of the natural coordinate, lines ofconstant σ are always orthogonal to streamlines (i.e., lines of constant ψ and χ).Thus we may observe that one of the remarkable spin-offs of using inverse equationsis the automatic generation of an orthogonal contour-fitted coordinate system inthe original space. Although the actual computation is done in a cubic domainof definition with Cartesian coordinates in inverse space, a curvilinear orthogonalgrid is simultaneously generated together with the solution in the original space.

382 J. J. Keller ZAMP

Furthermore, the density of grid lines gets automatically high in domains of largegradients. For these reasons the accuracy of the method is remarkably high (for afixed number of grid points) and the convergence is very fast.

The second example (see figure 3) is a turbine inlet flow. In this case thegeometry of an annular duct is imposed as a boundary condition, combined with alinear variation of the flow angles at the inlet and exit, respectively. According toABB’s convention the flow is from right to left. This is one of the examples thathas been used to check the equations. To include and discuss three-dimensionalexamples of inverse design would exceed the scope of this paper. Such results willbe presented elsewhere.

13. Concluding remarks

With the transformation of the Euler equations to their inverse form, using a pairof stream functions and the natural coordinate as independent variables, a theo-retical basis has been introduced for a broad class of extremely powerful inversecomputation methods. Instead of searching for a geometry of a flow device thatwill lead to certain physical properties, the physical properties can be defined inadvance and the solution of the inverse equations directly leads to the correspond-ing geometry. It has become easier to arrive at an optimized design. However, thedifficulty of defining an optimum remains.

From a practical point of view it may be convenient to proceed as follows. Asa preparation for the optimum design of a certain flow device we introduce a firstguess for its geometry, that may satisfy certain basic constraints. Secondly we usethe inverse method for a “backward computation”. In other words, we specify thegeometry according to the first guess and determine the static pressure distributionalong the surface. In a second step we modify the static pressure distribution alongthe surface according to the requirements of the optimization and introduce themodified static pressure distribution as the boundary condition for the “forwardcomputation”. It should be noted that the inverse equations provide an excellentbasis for using both physical quantities or the geometry of a flow device to definethe boundary conditions. In fact it is even easy to use combinations of geometricaland physical boundary conditions.

Hence, it seems that there is justified hope that computation methods thatare based on inverse equations will become extremely efficient tools to arrive atoptimized flow devices.

It might appear that the existence of integrals of motion is essential to derivea set of antisymmetric equations of the form (19), (29). However, it is interestingthat the extension to non-adiabatic flows of both antisymmetric equations and thetransformation to inverse coordinates is relatively simple, including the case of thegeneral time-independent Navier-Stokes equations. A discussion of this issue ispostponed to another paper.

Vol. 49 (1998) Inverse Euler equations 383

References

[1] R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Interscience, New York1967.

[2] J. J. Keller, A pair of stream functions for three-dimensional vortex flows, Z. angew. Math.Phys. 47 (1996), 821.

[3] A. Jameson, N. A. Pierce, and L. Martinelli, Optimum aerodynamic design using the Navier-Stokes equations, AIAA paper 97-0101, 35th Aerospace Sciences Meeting, Reno, Nevada1997.

[4] M. B. Giles and M. Drela, Two-dimensional transonic aerodynamic design method, AIAAJournal 25 (1987), 1199.

[5] T. Dang and V. Isgro, Inverse method for turbomachine blades using existing time-marchingtechniques, ASME paper 94-GT-20, International Gas Turbine and Aeroengine Congress,The Hague, Netherlands, 1994.

[6] G. S. Dulikravich, Shape inverse design and optimization for three-dimensional aerodynam-ics, AIAA paper 95-0695, 33rd Aerospace Sciences Meeting , Reno, Nevada 1995.

[7] M. J. Pandya and O. Baysal, Gradient-based aerodynamic shape optimization using ADImethod for large-scale problems, AIAA paper 96-0091, 34th Aerospace Sciences Meeting ,Reno, Nevada 1996.

[8] I. K. Jennions, The role of CFD in the design process, AGARD Paper , 7 Rue Ancelle,92200 Neuilly sur Seine, France, Lecture Series 195, May to June 1994.

[9] M. G. Turner and M. B. Giles, Design and analysis of internal flow fields using a two streamfunction formulation, ASME Winter Annual Meeting 1990, FED Vol. 103, 203.

[10] J. J. Keller, W. Egli, and J. Exley, Force- and loss-free transitions between flow states, Z.angew. Math. Phys. 36 (1985), 854.

[11] J. J. Keller, W. Egli, and R. Althaus, Vortex breakdown as a fundamental element of vortexdynamics, Z. angew. Math. Phys. 39 (1988), 404.

[12] J. J. Keller, On the interpretation of vortex breakdown, Phys Fluids 7 (7) (1995), 1696.[13] S. J. Kline, D. E. Abbott and R. W. Fox, Optimum design of straight-walled diffusers, J.

Basic Eng., Trans. ASME 81, Series D (Sept. 1959).

Jakob J. KellerABB Power Generation5401 Baden, Switzerland(Fax: +4156-632 7139)

(Received: April 15, 1997; revised: January 20, 1998)