morino_1986 Helmholtz decomposition revisited; Vorticity generation and trailing edge condition

Computational Mechanics (1986) 1, 65-90

Computational Mechanics © Springer-Verlag 1986

Helmholtz decomposition revisited: Vorticity generation and trailing edge condition

Part I: Incompressible flows

L. Morino Department of Aerospace and Mechanical Engineering, Boston University, Boston, Mass. 02215, USA

Abstract. The use of the Helmholtz decomposition for exterior incompressible viscous flows is examined, with special emphasis on the issue of the boundary conditions for the vorticity. The problem is addressed by using the decomposition for the infinite- space; that is, by using a representation for the velocity that is valid for both the fluid region and the region inside the boundary surface. The motion of the boundary is described as the limiting case of a sequence of impulsive accelerations. It is shown that at each instant of velocity discontinuity, vorticity is generated by the boundary condition on the normal component of the velocity, for both inviscid and viscous flows. In viscous flows, the vorticity is then diffused into the surroundings : this yields that the no-slip conditions are thus automatically satisfied (since the presence of a vortex layer on the surface is required to obtain a velocity slip at the boundary). This result is then used to show that in order for the solution to the Euler equations to be the limit of the solution to the Navier-Stokes equations, a trailing-edge condition (that the vortices be shed as soon as they are formed) must be satisfied. The use of the results for a computational scheme is also discussed. Finally, Lighthill's transpiration velocity is interpreted in terms of Helmholtz decomposition, and extended to unsteady compressible flows.

I Introduction

This paper deals with the use of the Helmholtz decomposition theorem in fluid dynamics, with special emphasis on the boundary conditions. The motivation for the work is originally computational: the issue of the boundary conditions must be addressed carefully before attempting a numerical solution of the problem. However, in the process of analyzing the boundary conditions, several issues, which are of more general interest than the computational solution of the problem, were uncovered and clarified. The objective of this paper is to present these conceptual results, both because of the general theoretical interest, and because of their implications in developing computational schemes. In particular, issues that are clarified include that of generation of vorticity. Also, much attention is given to the limiting process of how the solution of the Euler equations may be obtained as the limit of the solution of the Navier Stokes equations; this reveals the need for a trailing-edge condition for inviscid flows that is physically equivalent to, but conceptually different from, the classical Kutta- Joukowski condition. The concept of transpiration velocity is also addressed.

The problem considered in this paper is that of a body which moves in a fluid. The analysis is limited to incompressible flows (compressible flows are considered in Part II of this work, currently in preparation; compressibility is included here only in Appendix C, in the discussion of transpiration velocity). At time t < 0, both body and fluid are assumed to be at rest unless otherwise stated. The frame of reference will always be connected with the undisturbed flow. The surface of the body is assumed to be impermeable (although prescribed flow-through may be easily incorporated in the formulation). The fluid region is assumed to be simply-connected, unless otherwise stated.

The problem is addressed by using the Helmholtz decomposition theorem, which states that, under suitable asymptotic behavior at infinity, any vector field v(x) may be decomposed into two parts. The first one is irrotational (but, in general, not solenoidal) and may be expressed as the gradient of a scalar function (called scalar potential). The second one is solenoidal (but, in general, rotational) and may be expressed as the curl of a vector function (called vector potential). As shown

66 Computational Mechanics 1 (1986)

later, this decomposition is unique only for the infinite space (i. e., the whole space, R 3, without any boundary). If the vector field is defined in a region different from the infinite space, then the decomposition is not unique: an irrotational solenoidal field (which is identically equal to zero in the infinite space, if v = 0 at infinity) may be expressed either as the gradient of a scalar potential or as the curl of a vector potential. Hence boundary conditions on the potentials are needed in order to obtain a unique decomposition.

A very thorough analysis of the specific issue of the boundary conditions for the scalar/vector potential decomposition of a general vector field was given by Weyl (1940), and Bykhovskiy and Smirnov (1960) who show how, under suitable boundary conditions, a vector field may be uniquely decomposed into several orthogonal vector spaces (see Appendix A). Related approaches are used, in the field of computational fluid dynamics, by several authors, notably Hirasaki and Hellum (1968, 1970), Wu (e. g., 1976, 1982, 1984), Richardson and Cornish (1977), and Quartapelle and Valz-Gris (1981). Their methods (known as potential-vorticity method or scalar/vector-potential method) differ mainly because of boundary conditions (Appendix B).

A different approach is used in this paper. The issue of the boundary conditions on the potentials is addressed by studying the velocity field in the infinite space, including the region inside the flow boundary, where the velocity is assumed to be prescribed (for simplicity this region will be referred to as the solid region). In other words, the system under consideration is thought of as a continuum which occupies the infinite space and is part fluid and part solid. Having transformed the fluid problem into an infinite-space continuum problem, the Helmholtz decomposition is unique and the analysis simplifies considerably. It will be shown that for both viscous and inviscid flows, using this representation for the solution, the only boundary conditions required on the body are those on the normal component of velocity (for the sake of conciseness, this condition will be referred to as the normal boundary condition). It will also be shown that, under special circumstances (e. g., sudden changes in velocity of the boundary), the no-slip conditions (on the tangential components of the velocity) cannot be satisfied, whereas under ordinary circumstances, they are automatically satisfied by the representation used.

More specifically, one obtains that for a given vorticity distribution at time t = ti, an impulsive change in the boundary conditions at t = ti yields a Zero-thickness vortex-layer on the surface of the body. This vortex-layer is the same for both inviscid and viscous flows: it will be shown that this is the essence of the mechanism of vorticity generation which is therefore independent of the presence of viscosity. The difference between viscous and inviscid flows is due to the fact that in viscous flows, the vortex layer is immediately diffused, so that, at time t = t + , the no-slip condition is obtained. For inviscid flows, on the other hand, the vortex layer remains on the surface. This argument is then extended to the case of a continuous motion, by describing such motion as the limit of a piece-wise continuous one.

It is apparent that this approach is particularly suited for a computational scheme: this aspect of the work is addressed by Del Marco (1986). This paper is limited to the mathematical foundation and the physical interpretation of the approach, with particular emphasis on vorticity generation and on the conditions under which the solution of the Euler equations is the limit of the solution of the Navier-Stokes equations. (The computational scheme is actually closely related to that of Wu (e. g., 1976, 1982, 1984); the relationship of the proposed approach to that of Wu is examined in detail in Appendix B.)

The mechanism of vorticity generation for a body under impulsive start has been addressed by Lighthill (1963) and Batchelor (1967). Their formulation (based on physical rather than mathematical considerations) clearly describes the phenomenon as diffusion of the zero-thickness vortex layer due to the slip at the boundary. The generation of vorticity for the steady-state case is dealt with by Lighthill (1963, p. 84), who shows that "cross-stream vorticity is created at the rate UU', and the streamwise vorticity at the rate uZ~s '', where U is the magnitude of the external velocity, the prime indicates directional derivative along the streamlines, and t¢ s is the curvature of those streamlines in a plane tangential to the surface. The issue of generation of vorticity is also addressed by Morton (1984), who tries to reconcile the two mechanisms described by Lighthill (1963) and Batchelor (1967) by examining specific cases and dividing them into two groups: fast motion (with instantaneous generation of vorticity) and slow motion (with continuous generation of vorticity). The results

L. Morino: Helmholtz decomposition revisited: Vorticity generation and trailing edge condition. Part 1 67

obtained here tend to indicate that the mechanism of vorticity generation can always be explained (even for the case of uniform motion) in terms of a phenomenon similar to that described by Lighthill (1963) and Batchelor (1967) for the limited case of impulsive start.

The results outlined above are valid for an arbitrary shape. However, special emphasis will be given here, especially in Sect. 3 and 6, to the limiting case of inviscid flow around a body with a sharp trailing edge: for simplicity, this case will be referred to as the 'wing problem', even though it actually occurs on a broader class of problems (such as windmills, propellers, and helicopter rotors). In this case, an additional issue arises because, it will be shown, for zero viscosity vorticity is actually shed at the trailing edge and goes into the fluid field: a mechanism for the formation of vorticity in the fluid region for inviscid flows. The conditions under which the solution to the Euler equations is the limit of that to the Navier-Stokes equations (and in particular the trailing-edge condition for the wing problem) are included in the discussion.

Since the solution for the fluid region is imbedded into an infinite-space solution, this approach will be referred to as the infinite-space approach. This paper is divided into seven sections. Preliminary considerations on Helmholtz decomposition, fundamental to this paper, are reviewed in Sect. 2. In Sect. 3, the problem of a wing in potential flow, essential for the understanding of the paper, is reviewed: it will be shown how vorticity is indeed generated on the surface of the wing and shed into the flow field at the trailing edge, even within the framework of potential flow theory. Sections 4 and 5 deal with rotational flows (for inviscid and viscous fluids, respectively) around general bodies. The emphasis is again on the wing problem in Sect. 6, where it is shown that at least for attached flows, the solution to the Euler equations is the limit of the solution of the Navier-Stokes equations, provided that a condition on vortex shedding of the trailing edge is satisfied. Comments and recommendations for future work are given in Sect. 7. Three Appendices are also included. Appendix A deals with the relationship of the current work to the classical decomposition of Weyl (1940) and Bykhovskiy and Smirnov (1960). In Appendix B, a comparison with other works in fluid dynamics such as Quartapelle and Valz-Griz (1981) and Wu (e. g., 1976, 1982, and 1984) is provided. Finally, the Helmholtz decomposition is used to provide a theoretical basis for a new formulation for the transpiration velocity (introduced by Lighthill, 1958): the advantage of the new formulation is that it shows that the expression for the transpiration velocity, introduced by Lighthill for incompressible flows, is indeed valid for unsteady compressible flows as well (in contrast to the common expression used for steady compressible flows; see, e.g., Lemmerman and Sonnad (I 979)).

It should be emphasized that at some stages in this paper, intuitive physical reasoning is used at the expense of mathematical rigor. At times this is done for the sake of clarity (in particular, the use of functional analysis is avoided, as much as possible, because the mathematical complexity may obscure the physical significance of the results; the more mathematically complex issues, encountered for instance, in the comparison with related formulations, have been included in Appendices A and B). At other times, this author was not able to obtain a rigorous formulation and the results are shown to be valid under certain simplifying conditions or using physical reasoning. In either case, a plausible approach to be used to obtain a rigorous proof of the results and/or the need for additional analysis are indicated. A rigorous approach to some of the issues discussed here is presented by Del Marco (1986).

2 Helmholtz decomposition

For the sake of clarity the essential elements of the Helmholtz scalar/vector potential decomposition are briefly presented here. Given an arbitrary vector field (which is here assumed to be differentiable) it is possible to decompose the vector field as

v=V~b+VxA

where 4) is called scalar potential whereas A is called vector potential, and is such that

V . A = 0

(2.1)

(2.2)


According to Lamb (1932), this result is to be attributed to Helmholtz and is usually referred to in the literature as the Helmholtz decomposition theorem. It is apparent that V~b is irrotational, whereas V x A is solenoidal. In addition, taking the divergence of Eq. (2.1) one obtains that, necessarily,

V2~ = 0 (2.3)

where

0 = V. v (2.4)

is the expansion of the velocity field. Also taking the curl of Eq. (2.1), noting that, for any vector a,

v x (v x a)= V(V. a)-V2a

and using Eq. (2.2), one obtains that, necessarily

V2 A = -co

where co is the vorticity

~ = V X V

(2.5)

(2.6)

(2.7)

The proof of the Helmholtz decomposition theorem is easily established as follows. Find a particular solution Ao to V × (V × A) = 0. IfV. A0 4 = 0, consider A - A0 + V~/with ~/such that V2~/= - V . A0 (note that V × V~/~0, therefore the value ofq does not affect the value of v). Thus A satisfies Eqs. (2.2) and (2.6). Next note that using Eqs. (2.2, 2.5, 2.6) and (2.7),

V x (v - V x A)=oJ - o ) = 0 (2.8)

that is, v - V x A is irrotational and therefore there exists a potential 4) such that

v - V x A =V4) (2.9)

(where 4) satisfies Eq. 2.3). Equation 2.9 is equivalent to the desired Eq. (2.1) and therefore the proof of the theorem is complete. The only condition used is that v is differentiable. Note that the decomposition is not unique since the solution of Eq. (2.6) is not unique without appropriate boundary conditions. In particular, any irrotational solenoidal field may be represented either as V4) or as V × A. However the decomposition may be made unique by the addition of suitable boundary conditions, as shown in Appendix A.

On the other hand, the decomposition is actually unique for the infinite-space case. More precisely, assume that the vector field is defined in the whole three-dimensional space, and is differentiable and that 0 and co vanish at infinity of order r -3. In this case, adding the inessential conditions that 4) and A go to zero at infinity, the solutions to Eqs. (2.3) and (2.6) are actually unique and given by:

0(y) 4)(x)--I I! 4 lx_y I dV(y) (2.10)

and

o(y) A(x)=j'~.. 4rc[x--~ dV(y) (2.1')

Combining Eqs. (2.J, 2.3, 2.5, 2.10) and (2.11) one obtains the elegant expression

Vy x v(y) V~. v(y) dV(y)+Vx x 555 4rc l~yT dV(y) (2.12) v(x)=-v, a lx-yl

Note that the only assumptions used are that v is differentiable and 0 and o) vanish at infinity of order r- 3.

It may be worth noting that the second integral in Eq. (2.12) yields, in the limit, the well known law of Blot and Savart, for the velocity "induced" by a vortex filament. Therefore this integral may be thought of as a Biot and Savart law for distributed vorticity. It is understood that (in contrast to

L. Morino: Helmholtz decomposition revisited: Vorticity generation and trailing edge condition. Part I 69

classical electromagnetism) "induced" is meant here more as a mathematical term than as a physical one.

Only the case of an incompressible fluid (i. e., 0 = 0) is considered here; the compressible flow problem is addressed in Part 2 of this work. Particularly important is the case in which the vorticity is concentrated into an infinitesimally thin layer (on a surface o-), possibly bound by a vortex filament. In this case Eq. (2.12) yields, in the limit,

v(x)=VxX~ T ( Y ) da(y)+VxX~ F(y) 4 lx-y[ c 4 lx-yl dy (2.13) o

where Cis the boundary of a: F is not necessarily constant, but the combined system ofvorticity (on a and C) is solenoidal. It should be noted that in this case, v may also be expressed in terms of a scalar potential, because of the well known equivalence between doublet layers and vortex layers (Batchelor 1967, Campbell 1973, Soohoo et al. 1978)

3#o D(y)N 4 lx-yl with

~ = n xV~D

where V~ is the surface gradient.

1 1 ) 4~[x-y[ d~(y)+~ c D(y) 4~ lx_y [ dy (2.14)

(2.15)

Equation (2.15) seems to contradict the uniqueness of the decomposition for the infinite-space case. The apparent contradiction is due to the fact that Eq. (2.10) gives a solution of Eq. (2.3) which is unique under the implicit assumption that q~ is continuous (and twice differentiable) in the infinite space: on the other hand the potential representation used in Eq. (2.15) yields a potential function which is discontinuous on a. The discussion may be carried further by noting that Eq. (2.12) is actually valid, in the limit, under the much less restrictive assumption that v belongs to the L2 space (i. e., that v is square-summable in Lebesgue sense, Bykhovskiy and Smirnov, 1960). In this case, Eq. (2.13) is obtained as the limit of Eq. (2.12). The doublet representation, on the other hand, is not obtained as a limit of Eq. (2.12).

Finally it should be emphasized that, even though the decomposition is not unique, the velocity in a volume Vis uniquely determined by the vorticity to in V, the expansion 0 in V, and the condition on the normal velocity on the boundary of V (Lamb, 1932, § 147). For, given a particular solution of Eq. (2.6), the difference (Eq. 2.9) is irrotational and therefore it may be expressed as V~, with q~ satisfying the Neumann exterior problem for the Poisson equation, Eq. (2.3), which has a unique solution.

3 Potential flows

The formulation presented in this paper was obtained by this author in the process of trying to understand how his work on potential flows could be extended to viscous flows. For the sake of clarity, the formulation is presented by following the same logical process, starting from the potential- flow formulation and building it up to viscous flows (this approach is particularly useful in discussing the relationship between the Navier-Stokes and the Euler equations). Therefore, the potential-flow formulation introduced by this author is reviewed in this section, and reinterpreted in terms of Helmholtz decomposition. In this section, emphasis is given to the problem of a wing with a sharp trailing edge.

The problem of inviscid incompressible flows around a solid region (bounded by a surface o-) is governed by the continuity equation

V . v = 0 (3.1) and the Euler equations Dv 1 D~-- QVp (3.2)

where D/Dt is the substantial or material derivative 8/St + v. V. The surface a is assumed to be impermeable; this implies that


v . n = v b . n on cr b (3.3)

(where Vb is the velocity of the point x of the boundary surface ab). The boundary conditions at infinity are v = 0 and p = p ~ . The initial conditions are

v (x, 0) = Vo (X) (3.4)

In this section, it is assumed that Vo(X)=0. This implies that the flow is initially irrotational. Taking the curl of Eq. (3.2), with Q constant, one obtains the vorticity transport equation

Do9 Dt o9 Vv (3.5)

This implies that the flow, which has been assumed to be initially irrotational, remains irrotational at all times, with the possible exception of those points that come in contact with the surface of the body. More specifically, consider the case of a wing with a sharp trailing edge. In this case the flow is irrotational at all points except for those emanating from the points of the trailing edge (Morino et al. 1985). These points form a streak surface that is called wake.

If the flow is irrotational, then

v = V ¢ (3.6)

(if, as assumed, the flow region is simply connected then q5 is single valued). Using Eq. (3.6), the continuity equation reduces to

V 2¢=0 (3.7)

with the boundary conditions (Eq. 3.3)

~ n = V b . n on ob (3.8)

and q~=0 at infinity. The Euler equations may be integrated to yield Bernoulli's theorem

8¢ +.1 Ir12 q p _ p ~ (3.9) ~t Q 0

Finally, on the wake, using the conservation of mass and momen tum across a surface of discontinuity, i.e., the conditions Ap = 0 and A (v. n) = 0, one obtains Morino et al. 1985, or Morino and Bharadvaj 1985 where a more detailed and more rigorous derivation is presented)

DV¢ Dt (A4~) = 0 (3.10)

where Dw/Dt indicates the substantial derivative following a wake point having velocity v w = (V 1 -1-V2)/2 (where 1 and 2 indicate the two sides of the wake). The initial condition for .this equation is A¢ = 0 at t = 0 (no other initial conditions are required for the potential formulation). Equation (3.10) may be integrated to yield

Aq~ = constant (3.11)

following a wake point. The constant is equal to the value of Aq~t e at the time when the wake point left the trailing edge (or the value that Aq~ had at time t = 0, i.e., A~b = 0 in our case). In addition one needs a trailing edge condition: this condition is discussed later in this section after introducing an integral formulation for the problem (the generalization of this condition to rotational inviscid flows is one of the main objectives of this paper).

The above problem may be addressed by using an integral equation introduced by Morino (1973, 1974) for unsteady compressible potential flows. Given its bearing on this work, the integral equation is briefly reviewed here for the case of incompressible flows and then reformulated in terms of vector potential.

Using Green's function method, with a surface surrounding both body and wake, letting the closed surface around a wake collapse into two infinitesimally close surfaces and using the condition


A (v. n)= 0 on the wake, one obtains

~b 1 da (y )+ 8 ( 1 ) da(y)

+If A4) N d<y) (3.12) ~ w

where eb is the closed surface of the body (e.g., a wing), aw is the open surface of the wake, A4) = 4)1 - 4)2 is the discontinuity of the potential across this surface and ~/~n = n (y). Vr. Note that the unit normal n on a is pointed outward, whereas that on o-w it points from side 2 to side 1. Also,

E(x)= 1 4n (3.13)

where f2 is the solid angle described by the surface of the body as seen by an observer in x. In particular, E = 1 outside o-b, E = 0 inside o'b, and E = 1/2 on a smooth point of the surface of the body (i.e., a point where the normal is continuous).

If x is on the surface of the body, Eq. (3.12) is an integral equation relating the potential 4) on ab to the normal derivative (which is known from the boundary condition, Eq. 3.8) and to the potential discontinuity across the wake (which is known from the preceding time history, see Eq. 3.11). The integral equation may be used to evaluate 4) on the body. Then Eq. (3.12) may be used to evaluate 4) and hence the velocity anywhere in the field, in particular for the points of the wake. From this, the position of the wake at time t + dt may be calculated and the process may be repeated. (This formulation was actually implemented by Morino and Kuo (1974) for the prescribed-wake analysis of unsteady compressible flows around wings and by Morino et al. (1985) for the free-wake analysis of unsteady incompressible flows around helicopter rotors.)

An important consideration should be made at this point: an additional condition is required at the trailing edge. Because of the equivalence between doublet layers and vortex layers introduced in Sect. 2 (Eq. 2.14), the representation given by Eq. (3.12) indicates the presence of two line-vortices at the trailing edge: the first is generated by the doublet distribution over the wing and has intensity equal to (4), - 4)~) at the trailing edge (with u and l indicating the upper and lower surfaces of the wing, which are continued into sides 1 and 2 of the wake), whereas the second is generated by the doublet distribution over the wake and has intensity A4)te. Assuming that the flow is smooth at the trailing edge, the two line-vortices must cancel each other out, which implies that at a point xt, of the trailing edge

A4) (Xte) = 4)u (Xte) -- 4)l(Xte) (3.14)

This condition is one of the key issues addressed here and is generalized to rotational inviscid flows later in this paper. In order to avoid confusion with the conditions of Kutta (1902) and Joukowski (1907) (who introduced a physically equivalent condition with a different mathematical objective: eliminating the nonuniqueness of two-dimensional steady incompressible potential flows), the assumption that no vortices exist at the trailing edge will be referred to as the 'trailing-edge condition for the Euler equations'.

In order to appreciate the importance of the trailing-edge condition, it is interesting to discuss what happens at time t = 0 +. As mentioned above, both the fluid and the solid regions are assumed to be at rest for t < 0, hence, at time t = 0 +, the wake has not had time to form, and therefore at time t = 0 +, the potential is continuous and satisfies the exterior Neumann problem for the Laplace equation. Since the fluid region is assumed to be simply connected, the solution exists and is unique (Batchelor, 1967), and is obtained by solving Eq. (3.12) (with O4)/~n given by Eq. (3.8) and with the last integral identically equal to zero). Such a solution, for the case of a wing with a sharp trailing edge, typically exhibits the presence of a line-vortex at the trailing edge (a rigorous proof appears desirable here). The solution thereafter depends upon the condition that one imposes at the trailing edge. If we impose that the vortex remains at the trailing edge, we obtain one solution; if we assume that the vortex leaves the trailing edge, we obtain a different one. The trailing-edge condition used in this paper is that the vortex is shed immediately after it is formed. One of the objectives of this paper is to show


that this is the correct trailing-edge condition to impose if we want our solution to be the limiting case of the solution of the Navier-Stokes equations (Sect. 6).

It should be pointed out that from the mathematical point of view, the integral Eq. (3.12) may be thought of as a compatibility condition between ¢ and ~a/an for a given A~p distribution. The main objective of this paper is to show that this integral equation (or equivalent ones) is the key to the generation of vorticity, not only for potential flows but for viscous flows as well.

In order to show this for the simple case of potential flow, it is convenient, following Lamb (1932), to introduce a fictitious internal flow, vt(x) = Vxcp~(x). This is obtained by writing an equation similar to Eq. (3.12) for the volume inside a. Adding the two expressions one obtains (Lamb 1932)

,b an / 4n[x-yl - ( ¢ - ¢ ' ) ~ 4~lx-y [ da(y)

+ ~wfl A¢ ~nn -Yl da(y) (3.15)

where (Eq. 3.13)

~0 (x) = E(x) ¢ (x) + (1 - e(x)) ~,(x)

= qS(x) outside ab

= ~b~(X) inside ab

=½ (~b(x) + ~bi(x)) on smooth point of ob (3.16)

Consider for simplicity the case in which the solid region moves in arbitrary rigid-body translation (the more general case is considered in Sect. 4, see Eq. 4.8). Hence, vb(x, t )= vs(t)= VCs, where qSs = vs" x. Setting

¢ , = ¢ ~ (3.17)

Eq. (3.8) yields

an an (3.18)

and Eq. (3.15) reduces to

~ 0 ( x ) = ~ b ( ¢ - ¢ ~ ) ~ rclx_y I &r(y)+ A~b~ rclx_y I dtr(y) (3.19)

If x is on trb Eq. (3.19) is an integral equation (equivalent to Eq. 3.12) which may also be used to evaluate ¢ on O-b. (This equation was actually used by Maskew (1982) as a basis for a computational scheme.) The advantage of Eq. (3.19) over Eq. (3.t2) in the present context is that it gives a representation of the velocity field which is valid for both the fluid and the solid region.

More importantly, using the equivalence between doublet layer and vortex layer, Eq. (2.14), and using Eq. (3.14), one can see that the velocity field may be expressed as

r(y) d~(y)+vx × SI r(y) v ( x ) = V ~ x ~ 4r t lx_y [ ~,~ 4~lx-rl da(y) (3.20)

where, according to Eq. (2.15),

~,(y)=nxV,(qS-~b~) on o-b (3.21)

~ , (y )=nxV, (A¢) on ~

It may be noted that Eq. (3.20) yields, on the surface of the body

vb(x).n(x)=n(x). V~x ~ 47tgl!Y)l" Y[ da(y)+n(x). V~x j'~ 4n~!Y)l-- Y[ da(y) (3.22)


which is a third integral equation (equivalent to Eqs. 3.13 and 3.19) that may be used to evaluate 7 on %, given the velocity of the body and the vorticity distribution on the wake. (This equation is actually the basis for the well known vortex-lattice method.)

Note that the velocity given by Eq. (3.19) and hence by Eq. (3.20) is correct for both the fluid and the solid region. Therefore Eq. (3.22) yields the correct velocity discontinuity between the fluid and the solid.

An alternate way to interpret 7 on ab (in Eqs. 3.20 and 3.22) is as a zero-thickness layer of vorticity that resides completely within the fluid region (and not between the fluid and the solid) : a zero-thickness boundary layer, so to speak. This point of view will facilitate considerably the physical interpretation of the comparison between the solutions to the Euler equations and the Navier-Stokes equations and will be used throughout this paper. The boundary between the fluid and the solid regions is then interpreted to be immediately inside the vortex layer. Accordingly, the velocity of the fluid on ab is equal to that of the body: the no-slip conditions are thus automatically satisfied. The velocity jump occurs within the fluid region and is caused by the zero-thickness layer of vorticity. With the above interpretation, Eq. (3.20) predicts the correct distribution of the vorticity, for inviscid flows, not only on the wake but also on the surface of the body. (It may be worth noting that if Eq. (3.14) was not satisfied, then, according to Eq. (2.14), Eq. (3.19) would be equivalent to Eq. (3.20) plus an additional trailing-edge vortex filament of intensity qS,-qS~-A~b.)

It is interesting to interpret Eqs. (3.19) and (3.20) in terms of the Helmholtz decomposition. In Eq. (3.19) the velocity is expressed exclusively in terms of the scalar potential; the vorticity in the wake is expressed as a doublet layer. On the other hand, in Eq. (3.20) the velocity is expressed exclusively in terms of the vector potential. This second approach is a particular case of the infinite-space approach (outlined in Sect. 1) : Eq. (3.20) gives the correct representation for the velocity field not only in the fluid region but in the solid one as well. Having embedded the solution in the infinite space, the decomposition is now unique (Sect. 2).

Note that there seems to be a contradiction between the fact that the decomposition for the infinite space is unique (Sect. 2) and the fact that v may be represented (in the infinite space) in terms of either scalar potential (Eq. 3.19) or a vector potential (Eq. 3.20) : this is due to the fact that in proving the uniqueness, we assumed ~b to be continuous in the infinite space, a condition violated by Eq. (3.19) (see the paragraph following Eq. 2.15 for a more detailed discussion).

In summary, it appears that the potential-flow formulation is capable of "generating vorticity." The real question is "how good is this generation"; in other words, how close is the vorticity distribution obtained from a potential flow analysis to the one obtained from a viscous flow analysis ? In order to address this issue, the potential-flow formulation will be extended to rotational flows. It is apparent that this extension can be more easily obtained starting from Eq. (3.20) rather than Eq. (3.19), since in the former, the representation is already given in terms of the vorticity. This extension is accomplished first for rotational inviscid flows (Sect. 4) and then for viscous flows (Sect. 5). The comparison between the solutions (and vorticity generation) for inviscid and viscous flows is addressed in Sect. 6.

4 Rotational inviscid flows

In this section it will not be assumed, in general, that the flow is initially irrotational (in particular, that the fluid is at rest at time t < 0). If the vorticity is not equal to zero at time t -- 0, the flow field, though inviscid, is not potential. Extending the infinite-space approach introduced for potential flows, an expression is sought for the velocity which is valid in the infinite space, i. e., both fluid and solid region. The velocity field in the solid region is assumed to be known (it may be worth noting that any velocity field in the solid region that assumes a prescribed value on the boundary surface is equivalent for the analysis presented in this paper). For the sake of simplicity, initially the velocity field in the solid region is assumed to be solenoidal

V.v~=O (4.1)

(the general case is examined later in this section, see Eq. 4.7).


According to the results of Section 2, the velocity is simply given by

It is understood that in Eq. (4.2) the vorticity is distributed in the solid region, in the fluid region, as well as on the surface of the body (as a zero-thickness vortex layer responsible for the slip). At any instant, the vorticity distribution in the solid region and that in the fluid region are known, either from V x v~ (with v~ assumed to be prescribed) or from the integration of the vorticity transport equation, Eq. (3.5) (note that in general, akin to potential flows, the vorticity in an inviscid flow field may include a vortex layer on the wake, where a tangential discontinuity in the velocity exists). The vorticity distribution on the surface of the boundary, however, is not known, and, as shown below (Eq. 4.6), is determined by the normal boundary conditions. In order to proceed, it is convenient to separate the three vorticity fields (in the fluid region, in the solid region, and on the boundary, respectively).

However, if we do this, the three individual vorticity fields are not necessarily solenoidal in R 3 ; this "induces" individual velocity fields that are not solenoidal : for instance, if ms" n # 0, the velocity "induced" by the vorticity in the solid region is not solenoidal (Lamb 1932). In order to avoid dealing with nonsolenoidal velocity fields, following Lamb, let us add an infinitesimally thin region, Ls, to the solid region and let us "continue" the vorticity field into this region so that the vorticity itself is solenoidal in V~ + L~. Similarly, the vorticity in V: may be made solenoidal by "continuing" it into the thin region L: (this process may be made rigorous if performed in the limit sense, as discussed, for instance, by Bykhovskiy and Smirnov 1960). This yields

v(x) = w:(x) + w~(x) + wb(x) (4.3) where

w:(x)=Vx x ~ 4~rlc°~(Y2yldV(y) (4.4a) Vf Jr Lf

w,(x)=Gx j'j'~ 4~l°(Y2yl.. dV(y) (4.4b) Vs + Ls

and

(4.4c) o" b

Note that, because of the use made of the vorticity in the thin layers Ls and L:, the vorticity on the boundary is also solenoidal, and therefore satisfies the equation

V~. ~ = 0 (4.5)

where V¢. is the surface divergence operator. Taking the normal component of Eqs. (4.3) and using Eqs. (3.3) and (4.4c), one obtains

Vb'n=wf'n+Ws'n+n'Vxx~ 41r:(Y)y do-(y) (4.6, O" b

which is an integral equation of the same nature as Eq. (3.22) for potential flows, and may be used to obtain ~ on O'b.

If the solid region has 0 = V. vs # 0, one should simply add the contribution of the expansion of 0 to Eq. 4.3 (Eqs. 2.1 and 2.10) to obtain

ISI4°4Y)y dV(y) (4.7) V(X)-~- W f (X)--[- Ws(X)--~- Wb (X)-~- V x Vs

Note that no source surface distribution should be included on the surface a: this would yield a discontinuity of the normal component of the velocity, indicating penetration or separation.


[It is now easy to go back to the problem of Section 3 and present the formulation for potential flows for a wing in arbitrary motion. Potential flows are obtained when o~ = 0 for t < 0. This implies o~ = 0 at all times in the fluid region (except for the surface of the wake). Therefore, in this case, the velocity is given by (taking the limit of Eq. (4.5a) as the vorticity in the wake tends to a. vortex layer)

~(Y) ~(Y) v(x)=V~ x ~b 4rtlx--Yt da(y)+V~ x ~" 4n tx_y ] do'(y)

O" w

0 ( ) +V~ IIf 4rcl,Y_~yl~, dV(y)+Vxx IIf 4~lx_y [ dV(y) (4.80 Vs V~ + Ls

which is an obvious extension of Eq. (3.20). It is now apparent that even for rotational flows, if the boundary surface has a sharp trailing edge, wy must include the contribution of a vortex sheet, which is obtaine8 as the limit of a continuous distribution of vorticity (in other words, Eq. (4.4a) should be interpreted in terms of the theory of generalized functions). This point is examined further in Sec- tion 6, with emphasis on the trailing-edge condition.]

In summary, the conclusions are equivalent to those for potential flows. Equation (4.3) gives the velocity field both in the solid region and in the fluid region. Therefore, the sum of the vorticity on and those in Ly and L~ is responsible for the actual velocity discontinuity (as mentioned above, this vortex layer is here interpreted as actual vorticity in the fluid region, i.e., a zero-thickness boundary layer).

Finally, in.order to deepen our understanding of the representation given by Eq. (4.3) (and to facilitate the comparison with the works of Weyl 1940 and Bykhovskiy and Smirnov 1960, see Appendix A), in the remainder of this section, the analysis is carried a bit further. Because of the linearity of the integral equation, Eq. (4.3) may be rewritten as

w = w} + w; + w/, (4.9)

where

wk(x) = wax) + vx x f5 b 4 lx -Yl da(y)

w:(x) = w~(x) + Vx × IJ'~b 4 r t ~ ~ da(y)

b(y)

with ~¢, ~ and ~b such that

w)-. n=O

wg. n = O

W ; " n = v b • !I

respectively. Comparing Eq. (4.6) to Eqs. (4.9-4.11.c) we see that

(4.10a)

(4.lOb)

(4.10.c)

(4.11.a)

(4.11.b) (4Al.c)

(4A2) Note that (see Eq. 4.4.a and 4.10.a), w} is potential inside ab and satisfies Eq. 4.11.a on o-b, and

therefore

w) = 0 inside ab (4.13)

Note that a solenoidal velocity field in a volume Vis uniquely determined by the vorticity in Vand by the normal boundary condition (see end of Section 2). Hence w; + w~, coincides with vs in Vs, because they have the same vorticity in V~ and the same normal component on O'b. Therefore, Eq. (4.9) gives the correct representation for the velocity of the body, since w} = 0 inside ab (Eq. 4.13).

Uging a similar reasoning, one can see that

w~-O outside ab (4.14)


and that w) + w; coincides with vy in Vy, since they have the same vorticity in Vy and the same normal component on ab. Hence, Eq. (4.9) gives the correct representation in the fluid region, since w; ~ 0 outside ab.

It may be worth noting that, according to Eq. (4.13), the contribution of w) is identically equal to zero outside ab and therefore it could be eliminated completely from the formulation if our interests were limited to solving the problem of the fluid flow around ab. However, the resulting vortex layer would have no relationship with the physical vortex layer: therefore, if we are interested in the vorticity generation (for instance, to compare the solutions of the Euler and the Navier-Stokes equations, see Section 6), this term must be retained in the formulation. It may be worth noting that this term is identically equal to zero for a body in translation (and, therefore, typically negligible in aircraft aerodynamics). However, this term is essential to understand vorticity generation for rotating bodies, such as helicopter rotors. This issue is addressed further in Appendix A, where the relationship of the present formulation with the classical decomposition of Bykhovskiy and Smirnov (1960) is also presented. It is apparent, that, in any event, Eqs. (4.10.a) and (4.14) may be used, from a computational point of view, to substitute the evaluation of the volume integrals over the solid region, with the evaluation of the equivalent surface integrals.

5 Viscous flows

The issue that arises next is whether the above analysis may be extended to viscous flows. In this case, except for special cases discussed below, the tangential discontinuity is equal to zero (no-slip condition) and the boundary condition (combining the normal and the tangential components) is

v=vb on a (5.1)

The no-slip condition implies that

~oy. n = a~s" n (5.2)

and hence the two vorticity fields in the two infinitesimal layers, Lf and Lb, may be chosen to be equal and opposite. As a consequence one obtains that ~, must be identically equal to zero, otherwise a tangential discontinuity occurs and the no-slip condition is violated. Therefore, we find ourselves in a situation where we have more boundary conditions to satisfy than in the inviscid case (one on the normal, as well as two on the tangential components of the velocity), but apparently less degrees of freedom, since now ~ = 0. The only possible answer is that o~ in the field adjusts itself to satisfy both the normal and the tangential boundary conditions. The mechanism whereby this process occurs is not immediately obvious and is clarified in this section: this is the key issue of this work.

Incompressible viscous flows are governed by the continuity equation, Eq. (3.1), and by the Cauchy equations of motion

Dv Q Dt = - V p + d i v V (5.3)

where V is the viscous stress tensor. In particular, if

V=2#D (5.4)

where D is the deformation tensor, one recovers the Navier-Stokes equations for incompressible flows

Dv Dt =-VP~-#V2v (5.5)

In order to derive the results presented here, it is not necessary to limit the formulation to the Navier- Stokes equations. However, for the sake of clarity, this case is discussed first and then generalized to general viscous flows. By taking the curl of Eq. (5.5), one obtains the vorticity evolution equation

L. Morino: Helmholtz decomposition revisited: Vorticity generation and trailing edge condition. Part i 77

O¢.o - e~- Vv + vV2o) (5.6)

Dt

with v = #/~. [It should be emphasized that the Navier-Stokes equations and hence Eq. (5.6) are valid whether

or not the flow is turbulent; therefore the analysis of this section is valid for turbulent flows as well, at least from a theoretical point of view. Of course, considerable complications arise if one wants to use the theoretical results to develop a computational scheme. This point is examined further in Section 7. In addition, the issue of turbulence becomes important, even from a theoretical point of view, if one wants to study the relationship between the solution to the Euler equations and the limit of the solution to the Navier-Stokes equations, an issue examined in Section 6.]

In order to facilitate the description of the phenomenon, it is convenient to start from the exception (rather than the rule) and examine an anomalous case in which the no-slip boundary condition is violated even for viscous flows. Such a case arises any time the surface of the boundary is subject to an abrupt change in velocity. This of course violates the laws of physics, because it implies an infinite acceleration. Nonetheless, the study of this limiting case clarifies the essence of the phenomenon of vorticity generation and is the core of the present work.

To simplify things even further, consider the problem of an impulsive start, studied by LighthilI (1963) and Batchelor (1967). Assume that at time t < 0, both the fluid and the solid are at rest. At time t = 0 +, let the solid be subject to a pure translation with finite velocity v0. The vorticity at time t -- 0 + is zero in the solid region (subject to translation) as well as in the fluid region (initial conditions for the differential equation of vorticity transport). This is true whether the fluid is viscous or inviscid. Therefore the vorticity field is completely known except for a surface distribution ofvorticity (which is solenoidal, and therefore may be expressed in terms of a single function, D, see Eq. 2.15). It is apparent that the only unknown is the function D, and therefore only one scalar boundary condition (not three, i.e., v = Vo) may be satisfied. Since the surface is impermeable, the condition on the normal component of the velocity must be satisfied and hence no other conditions may be satisfied. This implies that in general, for the case under consideration (sudden start from rest configurations), we have to give up the no-slip condition, at least at time t = 0 +.

It is unders tood that for viscous flows the zero-thickness layer of vorticity is immediately diffused into the surrounding (at time t = dt, the thickness of the vortex layer is ~/~dt, see Batchelor 1967). (As pointed out by Batchelor 1967, this phenomenon is analogous to that encountered in heat conduction when a body of constant temperature To is suddenly immersed in a fluid of temperature T1). This implies that, immediately after t = 0 +, the boundary conditions on the tangential components of the velocity are automatically satisfied: for, the presence of slip on the boundary would require the presence of a zero-thickness vortex layer, which does not exist for t immediately after t = 0 +. The above analysis is essentially identical to those by Lighthill (1963) and Batchelor (1967), who, however do not carry this analysis any further.

The question that arises next is whether there exist other exceptional instants t + where a similar phenomenon occurs. The answer is yes: anytime the velocity is discontinuous in time, a zero-thickness vortex layer may be generated (and consequently the no-slip conditions are not satisfied). In order to see this, note that Eq. (4.3), derived for inviscid flows, is applicable to this case as well. The contribution of the vorticity in the fluid is still given by Eq. (4.4.a), where o~ now is still given by the "initial conditions" at time t + (arrived at through the integration of the vorticity transport, Eq. 5.6). The contribution of the vorticity in the solid region is still given by Eq. (4.4.b), where o~ is known from the prescribed mot ion of the solid region. Therefore, the only unknown in Eq. (4.3) is the solenoidal vorticity distribution ~,. Again, since the surface is impermeable, the normal boundary condition is responsible for the vorticity generation, and in general the no-slip condition cannot be satisfied.

The result obtained thus far pertains to the generation ofvorticity at anomalous instants times, t +, where the velocity of the boundary is subject to sudden changes. However, this result is the key to the formulation presented in this paper, and will be used to understand the process ofvorticity generation in general (and to draw important conclusions about the relationship between viscous and inviscid flows, Section 6).

In order to study the process of vorticity generation at a regular instant t (as distinct from the anomalous instants ti, where the velocity of the points of the boundary has a time discontinuity), one


may utilize the newly acquired understanding of vorticity generation at anomalous instants, by using the following limiting process: replace the continuous motion of the solid region with a "jerky" motion, i.e., assume that the continuous time-history of the velocity field is replaced with a piece-wise continuous time-history, consisting of a sequence of impulsive accelerations. Within this modified motion, we see that there exist a sequence of anomalous instants, infinitesimally close to each other, with generation of infinitesimal vortex layers, which immediately diffuse during the infinitesimal interval, dri, between two successive anomalous instants. Summarizing, if the actual motion is replaced by a piece-wise continuous one (with instants of discontinuities ri, infinitesimally close and velocity-discontinuities infinitesimally small), we can interpret the process of vorticity generation as caused by the boundary condition on the normal derivative (with slip on the boundary) at the instants "ci, which are followed by infinitesimally small intervals when diffusion (of the vorticity thus generated) occurs.

It seems physically intuitive that as the intervals dv~ go to zero, the discontinuous motion tends to the continuous motion: the vorticity is then continuously generated (by the normal boundary condition) and immediately diffused (thereby automatically satisfying the no-slip condition). Some points need further development, for instance the issue of vorticity generation between r~ and vi + a : it is acknowledged that vorticity is also generated continuously between two successive instances, ~ and ~+1; however it seems plausible that this amount of vorticity generated continuously is small compared to that generated at the instant ~. This issue is clearly important for the conceptual understanding of the continuous generation of the vorticity in the case of steady state flow around an object. This case may be addressed by requiring the vorticity generated between ~ and ~ + 1 to be equal to zero, by letting the solid region move with the fluid, i.e., applying, in the intervals dry, the equations of fluid to the solid region as well. In any event, a rigorous mathematical proof of the limiting process is obviously desirable. Nonetheless, in the following, the above results (obtained using the above physically intuitive approach) are assumed to be valid, and some consequences that may be inferred from the mechanism of vorticity generation outlined here are discussed.

It should be noted that the only concept used to differentiate between the Navier-Stokes equations and the Euler equations is that of diffusion of vorticity, which is apparent from Eq. (5.6). However, the diffusion of vorticity is not limited to the Navier-Stokes equations (Eq. 5.5), but is valid for any viscous flow (Eq. 5.3). Therefore, the results presented here appear to be indeed valid for any viscous flow.

6 Potential solution vs. viscous solution

The analysis of vorticity generation presented above is used next to clarify the relationship between the Euler equations and the Navier-Stokes equations. The emphasis in this section is on the case of a wing with a sharp trailing edge. More precisely, it will be shown that at least for attached flows around a wing, the limit of the solution of the Navier-Stokes equations as the Reynolds number tends to infinity is correctly captured by the Euler equations, provided that the trailing-edge condition introduced in Section 3 is satisfied. As mentioned above, this condition is physically analogous to the classical hypothesis introduced by Kutta (1902) and Joukowski (1907), but its mathematical necessity for the current formulation is unrelated to that of the steady two-dimensional flows covered by Kutta and Joukowski. For this reason, this condition will be referred to as the 'trailing-edge condition for the Euler equations'.

In order to discuss the relationship between the Euler and the Navier-Stokes equations, it is convenient to begin with the case of an impulsive start of a wing from a rest configuration. In this case, as we have seen in Section 4 and 5, for both inviscid and viscous flows, the vorticity is obtained by solving Eq. (4.6) with toy = tos = 0. Therefore, at time t = 0 +, the vorticity fields (and hence the velocity fields) are equal, for inviscid and viscous flows. The difference starts immediately after t = 0 ÷ : for the Navier-Stokes equations, the vorticity immediately diffuses in the surrounding and then is transported, according to the vorticity evolution equation, by diffusion and convection. On the other hand, for the Euler equations, the vorticity does not diffuse, and therefore it remains attached to the surface of the boundary: possible exceptions are the points where the fluid leaves the boundary (separation lines) and the trailing edge.

L. Morino: Helmhottz decomposition revisited: Vorticity generation and trailing edge condition. Part 1 79

In order to discuss this point further, let us consider again the case of a potential flow around a wing with a sharp trailing edge. In this case, as mentioned in Section 3, if the velocity field is continuous (this implies absence of wake), the solution to the Euler equations is unique. However, such a solution is not in agreement with the physical problem (for instance, the lift is zero according to D'Alembert paradox). Therefore, we have to assume that v is not necessarily continuous. In this case, the solution is not unique: in addit ion to the solution with v continuous, a second solution may be built according to the formulat ion of Section 3. The assumption used there is that if a vortex is created at the trailing edge, such a vortex is immediately shed. This assumption appears necessary if we want our solution to be the limit of the solution of the Navier-Stokes equations. For, even if the viscosity is infinitesimally small, but not zero, the vortex would be diffused into its surroundings, and then convected downstream away from the trailing edge. We therefore may conclude, on the basis of physical reasoning, that in the case of a wing with a sharp trailing edge, the trailing-edge condition (of vortex shedding at the trailing edge) must be satisfied if we want the solution to the Euler equations to be the limit of the solution to the Navier-Stokes equations. A rigorous p roof is desirable here.

Next, it will be shown that, if Reynolds number, QUL/I~, is very high, the potential-flow solution (i.e., the solution to the Euler equations for an initially irrotational flow, e.g., for v--0 at t = 0 ) is actually a good approximation for the solution of the Navier-Stokes equations (except, of course, within the boundary-layer and wake regions), provided that the flow is attached. In order to show this, consider again the limiting process introduced in Section 5, with the continuous motion being replaced by a discontinuous one with instants of discontinuity at t = "q. For the specific case of a wing starting from rest, at t = rl > 0, the vorticity generated at t = 0 + (same for both the Navier-Stokes and the Euler equations) is now diffused and convected for the viscous case, whereas for the inviscid one, the vorticity is still attached to the wing, with some of it already shed from the trailing edge. At this point, Eq. (4.6) should be used for the viscous flow (since the vorticity is now in the flow region), whereas Eq. (3.22) should be used for the inviscid flow (for simplicity, the velocity in the solid region is assumed to be solenoidal, i. e., 0s = 0). It is apparent that y on ob has a different physical meaning in the two cases. For viscous flows, Eq. (4.6), ~, represents the new vorticity being generated: since the flow is assumed to be attached, the vorticity already generated is distributed in a thin but finite-thickness layer (boundary layer) near the surface of the wing.

On the other hand, for inviscid flows, Eq. (3.22), ~ includes both the vorticity already generated and the new vorticity being generated. It is apparent that the mot ion of the vorticity over ab is irrelevant, since no matter where the vorticity moves, the new vorticity y (sum of the old one, convected to the new appropriate location on O'b, and of the one being generated) must satisfy Eq. (3.22). However, if the two terms were separated, we would see that each term in Eq. (4.6) would have a corresponding one in Eq. (3.22), and that the corresponding terms would be numerically close to each other.

In order to show this note that if the vorticity is distributed within a thin layer L (bounded by the surface a and having thickness 6), and if x is not within L, one obtains

to(y) d V ( y ) - ~ j ~'(Y) f[S 4Ttlx_y I 4rclx_y t da(y) (6.1)

with

6 = j" e~d~ (6.2)

0 where ~ is in the direction normal to cr. Therefore the velocity fields induced by the existing vorticity are very close to each other. The implication is that, in the two cases, the distributions of the vorticity newly generated are very close to each other. [An improvement would be obtained by choosing a to be the midsurface of the layer L (where the midsurface is located at the centroid of the vorticity distribution as a function of~); as shown by Lighthill (1958, Section 3.4, 'mean vorticity approach'), this is equivalent to the well known displacement thickness correction. The relationship between Lighthill's work and the Helmholtz decomposit ion is further developed in Appendix C, where a new approach to the concept of transpiration velocity is presented.]

In summary, for viscous flows, immediately after t = 0 +, the vorticity is completely in the fluid region, but its integrated value (integrated along the boundary layer thickness) is very close to the


intensity, 7, of the vortex layer on o- b for the inviscid case. Therefore, the vorticity emanating from the trailing edge into the wake is similar in the two cases. Since the transport process is similar in the two cases (Morino 1986), we may expect that at t = zl, the vorticity distributions in the wake are close to each other. The reasoning presented above may be repeated to show that, immediately after z~, the vorticity in the boundary layer of the viscous flow is very "close" to that on the surface o- for the inviscid flow. Therefore, the process may be repeated and the result is valid for all times (provided that the wing does not intercept its own wake, as in the case of a loop). We may conclude that the process of vorticity generation and transport yields similar vorticity distributions for inviscid and viscous problems, and therefore, very similar velocity distributions.

It should be emphasized that the issue of the limit as the Reynolds number goes to infinity has been carefully avoided: the comparison has been presented more in terms of the closeness of the two solutions, provided that the flow is attached in both cases. The next step is to address the limiting process; it seems physically intuitive that the limit of the Navier-Stokes solution is the Euler solution. However it is preferable to think of this limit as a mathematical limit (as a continuous function tends to a discontinuous one, see, e.g., Bykhovskiy and Smirnov (1960)). The physical limiting process, as the Reynolds number goes to infinity, requires a much more subtle process than that used here : in particular, the issue of turbulence (and related issues of instabilities and chaotic behaviour) needs much attention, considerably beyond the objectives of the present work.

It may be worth noting that Wu (1981) has shown that the forces acting on a body (starting from rest) is given by a term related to the velocity of the body plus a term proportional to the first moment of the vorticity in the fluid and in the solid. The implication of this result in the present context is that (contrary to the commonly held belief) the potential flow formulation may be capable of predicting (at least in theory) not only the lift, but also the drag (and not just the induced drag, but the viscous drag as well !), because the corresponding vorticity distributions for viscous and inviscid flows are indeed close to each other. Additional work is clearly needed on this issue.

7 Concluding remarks

The issue of the nonuniqueness of Helmholtz decomposition for the problem of the three dimensional unsteady incompressible flow around a body has been addressed. A new approach has been introduced, by embedding the solution into an infinite-space solution. Several results were obtained.

The first is that the mechanism of vorticity generation at the surface of the body may be explained in terms of the normal boundary conditions; this is true whether the fluid is viscous or inviscid. If the fluid is viscous, the vorticity immediately diffuses into the surroundings of the surface of the body, whereas, if the fluid is inviscid, it remains attached to the surface. As mentioned above, this general result was anticipated quite clearly by Lighthill (1963) and Batchelor (1967), at least for the vorticity generation at time t = 0, for the case of a body impulsively starting from rest.

The second result pertains to the relationship between the Euler solution and the Navier-Stokes solution for the problem of a wing with a sharp trailing edge. It was shown that in order for the Euler solution to be the limit of the Navier-Stokes solution the condition of vortex shedding at the trailing edge must be added. If this condition is satisfied, then the Euler equations actually give a solution that is very close to that of the Navier-Stokes equations. We may anticipate that (turbulence not withstanding) the limit of the Navier-Stokes solution is equal to the solution of the Euler equations provided that the trailing-edge condition is satisfied. A rigorous proof appears desirable here. [A theorem on the Euler solution being the limit of the Navier-Stokes solution is given by Ladyzhenskaya (1963) who however does not address.the wing problem and the issue of the trailing- edge condition.]

The analysis presented in this paper is limited to simply-connected three-dimensional fluid regions. If-the flow is two dimensional the region is not simply connected. In this case the solution of the integral equation for vorticity generation (two-dimensional equivalent of Eq. 4.6) is not unique. This is the mathematical reason for the condition introduced by Kutta (1902) and Joukowski (1907), which is clearly different from that for the trailing-edge condition introduced here. In the present context (unsteady analysis) the nonuniqueness of the flow around an airfoil is simply resolved by noting that according to Kelvin's theorem (which is applicable for a contour surrounding the airfoil


but so far away from it that the viscous stresses are negligible), the total vorticity (in the whole plane, including the solid region) remains constant. This condition is obviously sufficient to eliminate the nonuniqueness because the eigensolution of the problem has circulation different from zero. The implication is that the Kutta-Joukowski condition is needed only for steady flows in multiply connected regions, in particular for two-dimensional flows. For steady flows aroundsimply connected regions, the velocity for given vorticity and expansion distributions is unique if its normal component is prescribed at the boundary (see end of Section 2). Similarly, if the flow is unsteady, the two-dimensional solution is unique because the circulation connected with the region is determined by the Kelvin theorem and by the initial conditions. [For unsteady three-dimensional flows around multiply-connected regions, the issue may be addressed using the decomposition of Bykhovskiy and Smirnov (1960), who show that if the fluid region is (n + 1)-connected, there exist n eigensolutions to the problem: this nonuniqueness is eliminated in the unsteady problem using a similar process around each irreducible contour.]

A third result obtained here is that the solution may be obtained by studying a sequence of impulsive accelerations, with generation of vorticity obtained by imposing the normal boundary condition followed by intervals of diffusion of the vorticity away from the boundary (with tangential boundary conditions automatically satisfied once the vorticity is diffused). An obvious question at this point is: can the above process be used for a computational methodology? The answer is yes. The actual implementation is given in the work by Del Marco (1986). Preliminary numerical results confirm that the tangential boundary condition is automatically satisfied (within numerical approximations) by the representation proposed here. This approach is closely related to that of Wu's (1976, 1982, 1984); a comparison with Wu's approach is presented in Appendix B.

A spontaneous question at this point is: "How does one reconcile the above formulation with the case of a zero-thickness planar wing at zero angle of attack? In this case, the normal boundary condition is automatically satisfied, and there is no generation of vorticity in the inviscid flow !" The answer stems from the fact that the analysis presented in this paper deals with flow around a closed surface, surrounding a finite-volume region. Therefore what happens in the case of a zero-thickness wing at zero angle of attack is best seen in the limit as the thickness approaches zero. If the thickness is small but finite we have two layers of X, orticity (on the upper and lower surface respectively). The separate identity of the two layers disappears as the thickness of the wing goes to zero. In this case the normal boundary condition generates only the sum of the two layers (as in the case of a zero-thickness wing at an angle of attack): the difference of the vorticity on the two layers disappears as the two layers merge into one. This implies that the intensity of the (equal and opposite) vorticity in the two layers may be evaluated by examining the difference between the velocity in the fluid region and that in the solid region. Similar problems arise in studying the flow of air around an isolated building. Using the method of images the normal boundary condition at the ground is automatically satisfied; again the vortex layer existent at the ground has to be evaluated by examining the slip between the velocity of the air and that of the ground.

Finally, a new approach to the concept of transpiration velocity (introduced by Lighthill (1958) for the analysis of the inviscid-viscous interaction) is presented in Appendix C, where it is shown that the expression for the transpiration velocity (derived by Lighthill under the assumption of incompressible flows) is very general: (.1) it is not limited to thin boundary layers, and (2) it is not limited to incompressible flows, but is indeed applicable to unsteady compressible flows as well (this expression differs from that commonly used for compressible flows, an expression that is limited to steady flows (see, e.g., Lemmerman and Sonnad (1979)).

Issues that need further attention include compressibility and turbulence. Compressibility has been briefly discussed in Appendix C in connection with the discussion on the transpiration velocity. Also, a general theory of the Helmholtz decomposition for compressible viscous flows has been developed by Morino (1985). However, the discussion of the issue of the boundary condition on the vorticity is missing in that work. The integration of that work with the defect-velocity approach of Appendix C (as well as the extension of the infinite-space approach to compressible flows) is the subject of Part II of this work (currently in preparation).

On the other hand, a few comments should be made on the issue of turbulence. As mentioned in Section 6, the limiting process, as the Reynolds number goes to infinity, requires careful analysis of the issue of turbulence, considerably beyond the objectives of the present work. On the other hand, as


pointed out above, the formulation of Section 5 is valid in theory whether the flow is turbulent or laminar. However, the computational implementation requires considerable amount of additional work, on the issues of vorticity dynamics (including turbulence modelling) and on the issue of an efficient formulation for the boundary conditions, even for the limited case of inviscid flows. The solution of the vorticity evolution is partially addressed by Morino (1986) and considerable additional work is anticipated. On the other hand, the formulation for the boundary condition (in particular the diffusion of the vorticity within the laminar sublayer) is in principle identical to that presented here: however, a computational formulation for a grid size larger than the laminar sublayer needs to be developed (along the lines of the work by Wu and Sugavanam 1982), if the efficiency of the method is to be retained.

Acknowledgements

This work was partially supported by the NASA grant NAG-I-564 to Boston University. The author wishes to express his appreciation to Dr. E. Carson Yates, Jr., technical monitor of the grant, for the help received in connection with this work, to Prof. C. Samuel Ventres for his insightful comments and constructive criticism, and to Dr. Stephen Del Marco, for the invaluable discussions, in particular on the work of Bykhovskiy and Smirnov.

References

Batchelor, G.K. (1967): An introduction to fluid dynamics. Cambridge: Cambridge Univers. Press Bykhovskiy, E.B. ; Smirnov, N.V. (1960): On orthogonal expansions of the space of vector functions which are square-

summable over a given domain and the vector analysis operators. Trudy Mat. Inst. Steklova 59, 5. Academy of Sciences USSR Press, pp. 5-36. (Also available as NASA TM-77051, 1983)

Campbell, R.G. (1973): Foundations of fluid flow theory, p. 258. Reading: Addison-Wesley Publ. Del Marco, S.P. (1986): The mathematical foundations of the scalar-vector potential approach to analysing viscous flows.

Ph.D. Thesis, Boston Univers. Dept. of Mathematics E1 Refaee, M.M.; Wu, J.C. ; Lekoudis, S.G. (1982): Solutions of the compressible Navier-Stokes equations using the integral

method. AIAA J. 20, 356-362 Hess, J.L. (1977): A fully automatic combined potential-flow boundary layer procedure for calculation viscous effects on the

lifts and pressure distributions of arbitrary three-dimensional configurations. Long Beach: Douglas Aircraft Co., MDC J7491

Hirasaki, G.J. ; Hellums, J.D. (1968): A general formulation of the boundary conditions of the vector potential in three- dimensional hydrodynamics. Quart. Appl. Math. 26, 331-342

Hirasaki, G.J.; Hellums, J.D. (1970): Boundary conditions on the vector and scalar potentials in viscous three-dimensional hydrodynamics. Quart. Appl. Math. 28, 293-296

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Kutta, J. (1902): Auftriebskr/ifte in str6menden Fliissigkeiten. Illustrierte Aueronautische Mitteilungen, vol. 6, pp. 133-135 Ladyzhenskaya, O.A. (1963): The mathematical theory of viscous incompressible flows. New York: Gordon and Bread Lamb, H. (1932): Hydrodynamics, 6th ed. Cambridge: Cambridge University Press Lemmerman, L.A. ; Sonnad, V.R. (1979) : Three-dimensional viscous-inviscid coupling using surface transpiration. J. Aircraft

16, 353-358 Lighthill, M.J. (1958): On displacement thickness. J. Fluid Mech. 4, 383-392 Lighthill, M.J. (1963): Introduction to boundary layer theory. In: Rosenhead, L. (ed): Laminar boundary layers, part 2,

pp. 46-]13. Oxford: Oxford University Press Maskew, B. (1982) : Prediction of subsonic aerodynamic characteristics: A case for low-order panel methods. J. Aircraft 19,

157-163 Morino, L. (1973): Unsteady compressible flow around lifting bodies: General theory. AIAA Paper No. 73-196, AIAA l l th

Aerospace Sciences Meeting, Washington, D.C. Morino, U (1974): A general theory of unsteady compressible potential aerodynamics. NASA CR-2464 Morino, U (1985): Scalar/vector potential formulation for compressible viscous unsteady flows. NASA CR-3921 Morino, L. (1986): Material contravariant components: Vorticity transport and vortex theorems. AIAA J. 26, No. 3,526-528 Morino, L. ; Bharadvaj, B. (1985): Two methods for viscous and inviscid free-wake analysis of helicopter rotors. Boston:

Boston University, CCAD-TR-85-02 Morino, U ; Kaprielian, Z., Jr. ; Sipcic, S.R. (1985): Free wake analysis of helicopter rotors. Vertica 9, 127-140 Morino, L. ; Kuo, C.C. (1974): Subsonic potential aerodynamics for complex configurations: A general theory. AIAA J. 12,

191-197 Morton, B.R. (1984): The generation and decay of vorticity. Geophys. Astrophys. Fluid Dynamics 28,277-308 Quartapelle, L. ; Valz-Gris, F. (1981): Projection conditions on the vorticity in viscous incompressible flows. Internal. J. for

Numerical Methods in Fluids, 1, 129-144


Richardson, S.M. ; Cornish, A.R.H. (1977) : Solution of three-dimensional incompressible flow problems. J. Fluid Mech. 82, 309-319

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Soohoo, P.; Noll, R.B.; Morino, L.; Ham, N.D. (1978): Rotor wake effects on hub/pylon flow, vol. 1, Theoretical formulation. Appl. Technology Lab., U.S. Army Research and Technology Laboratories (AVRADCOM), Fort Eustis, Va., USARTL-TR-78-1A, p. 108

Sugavanam, A. ; Wu, J.C. (1982) : Numerical study of separated turbulent flow over airfoils. AIAA J. 20, 464-470 Thompson, J.F. ; Shanks, S.P. ; Wu, J.C.(1974): Numerical solution of three-dimensional Navier-Stokes equations showing

trailing edge vortices. AIAA J. 12, 787-794 Weyl, H. (1940): The method of orthogonal projection in potential theory. Duke Math. J. 7, 411-444 Wu, J.C. (1976): Numerical boundary conditions for viscous flow problems. AIAA J. 14, 1042-1049 Wu, J.C. (1981): Aerodynamic force and moment in steady and time-dependent viscous flows. AIAA J. 19, 432-441 Wu, J.C. (1982): Problems of general viscous flows. In: Shaw, R.P., Banerjee, P.K. (eds): Developments in boundary element

methods, vol. 2, Chpt. 4, pp. 69-109. London: Applied Sci. Wu, J.C. (1984): Fundamental solutions and numerical methods for flow problems. Intern. J. Numer. Meth. in Fluid 4,

185-201

Appendix A: Decomposition of Bykhovskiy and Smirnov

In this Appendix the relationship of this work with the decomposition of Bykhovskiy and Smirnov (1960) is discussed. In Section 2 we saw that the Helmholtz decomposition for the infinite space is unique. Now consider the decomposition for a finite region. The simplest results are obtained for interior problems (region inside a surface a); therefore, this case is considered first. More specifically, the volume V is assumed to be simply connected, and without holes, unless otherwise stated (see Bykhovskiy and Smirnov (1960) for the general case). In order to conform with the convention used in the main body of this work (that, for external flows, the normal is outwardly directed) in this Appendix the normal is assumed to be pointing into V even for interior problems.

Following Bykhovskiy and Smirnov (1960), the decomposition for this case may be made unique by arbitrarily choosing the convenient boundary condition

V x A . n = 0 (A.I)

It may be worth noting that Eq. (4.1), introduced by Bykhovskiy and Smirnov (1960), insures that vl = V~b and v2 = V x A belong to orthogonal vector spaces, i.e.,

~J'S v,. v2dV=0 (A.2) v

For, using Eq. (A.1) one obtains

~ Vdp. V x a d V = ~ (oV x A . ndo-=0 (A.3) V a

Equation A.I implies that ¢ satisfies Eq. (2.3) with

8¢ 8n = vb" n (A.4)

For simply connected regions, the solution for ¢ is unique, and therefore the decomposition is unique (for multiply-connected regions, see Bykhovskiy and Smirnov, 1960).

Next, we want to show that if A satisfies Eq. (2.6), then a necessary and sufficient condition for Eq. (2.2) to be satisfied is that

V . A = 0 on a (A.5)

In order to see that, note that Eq. (2.2) implies Eq. (A.5). Viceversa, taking the divergence of Eq. (2.6) one obtains V2(V • A)=0, which, with Eq. (A.5), yields V. A - 0 , i.e., Eq. (2.2).

Equations (2.6, A. 1) and (A. 5) represent a system of three equations for three unknowns With only two boundary conditions. The implication is that, although Eq. (A. 1) makes the decomposition of the velocity field unique (and therefore V × A unique), A is still not unique : for, if A is a solution, then


A' = A + VZ gives the same values for V × A. It may be noted that since both A and A' are solenoidal, X must be harmonic.

In order to obtain a unique solution for A, we may take advantage of this arbitrariness in A and obtain a convenient additional boundary condition for A, as follows. Equation (A. 1) implies that if A is a solution, there exists a function ~ such that

At=V,~ ~ o n ab (A.6)

where V, is the surface gradient operator and At is the portion of the vector A tangential to the surface

a (for, Eq. A. 1 yields ~ A. dx = 0 for any C on a, and hence i A. dx is path independent and defines C xo

a function :~(x) such that V~2= At). Consider the harmonic function Z that is equal to ~ on a. As mentioned above, we can subtract VZ from A without affecting V x A. This implies that we can eliminate the arbitrariness in A by imposing the convenient boundary condition

At=O on a (A.7)

If Eq. (A.7) is satisfied, Eq. A.1 is automatically satisfied, whereas Eq. (A.5) reduces to

~VaA"-o on a (A.8) ~n

(where A, is the normal component of A and a is the determinant of the surface metric tensor for surfaces parallel to a, so that da = ]//ad~ld~2; if ¢ 1 and ~z are orthogonal coordinates then, using

familiar symbols, ]//a-= h l h 2 ) .

Note that the above conditions uniquely define A for a region that is surface-wise simply connected (i. e., a region without holes). For, using the same reasoning used above, two solutions would differ by a factor VZ, where Z is harmonic and (Eq. A.7) constant on a; this yields that Z is constant in V and therefore A is unique. On the other hand if the region has a hole (e. g., in the case of the exterior problem), V,Z = 0 may be satisfied by using a different constant for each surface: therefore, in this case, the solution for A is not unique.

It may be worth noting that using the integral identity

III Va,.Vai+a'V2a dV= - ~ "'~-n da (A.9) V a

one may see that the solution would be unique if At and ~A,/~n were prescribed instead. In conclusion, we have obtained for a simply connected region without holes, that by imposing the

condition (A.1), and conveniently choosing the arbitrariness in A, the field has been decomposed uniquely into two vector fields, V4~ and V x A (which are orthogonal in the sense that they satisfy Eq. 5.4), where q~ and A are both unique; ~b satisfies Eqs. (2.3) and (A.4), whereas A satisfies Eq. (2.6) with boundary condition given by Eqs. (A.7) and (A.8). The function q~ satisfies the equation

Un I CUnn da(Y)+/SIv 4=lx-yl If x is on a Eq. (A. t0) is an integral equation which may be used to obtain q~ on a (Section 3). If the region is not simply connected, ~b may be multivalued, and the solution is not unique: it becomes unique by specifying the circulation around each irreducible path (if ck is continuous, the solution is unique; this corresponds to choosing all the circulations to be zero).

Similarly, the function A satisfies the equation

E(x)A(x)=~ ~nn -4n~--y I A ~n 4n~--rl da(Y)+lIIv 47tlx-y[ Using Eqs. (A.7) and (A.8), and recalling that da=~ /ad~ ld¢ 2, Eq. (A.11) reduces to

~At - 1 ]//-dd~l de 2 E(x)A(x)=~en 4~[x_y I

A.. 47tlx_y I 4 lx_y I


If x is on o- Eq. (A.12) is an integral equation which may be used to obtain A, and OA~/~n (recall however that the solution for A is unique only for regions without holes; see discussion following Eq. A.8).

Next, the relationship between the above decomposition (applied to the exterior problem) and that of Section 4 is discussed. First consider w~" outside a, this term coincides with the term V~b, since they are both potential outside a, and satisfy the same boundary condition. Next consider the term w~" note that w" is identically equal to zero outside a, and therefore need not be included in the decomposition of Bykhovskiy and Smirnov (1960). As shown in Section 6, however, the inclusion of this term is essential for the understanding of the mechanism of vorticity generation.

Finally consider the term w.}. In order to show that w} is a particular solution of Eq. (2.6) with boundary conditions (A. 1) and (A.5) (not necessarily A.7), it is convenient to recast w} in a different form. Note that the velocity w} may be written as

co(y) w>(x) : V~ x y:: 4rclx_Y I with

dV(y)+V x 4rclx_y [ dot(y) (A.13)

~}=Tf+?y (A.14)

where ~y is the intensity of the vorticity in the infinitesimally thin layer L: (note that whereas ?y in L: was defined in a nonunique fashion, ~s compensates for that arbitrariness and therefore ~} is unique).

Next we want to show that Equation (A. 13) may be rewritten as

w'f(x) = Vx x A}(x) (A. 15)

where

SSS 4~]~(Y2~,[,,~ .~ dV(y) (A.16) A}(x)=V~ x D

For, note that, using Gauss Theorem (with n directed into VI),

1 1 A}(x)=SS S 4 [x-y[ × w}(y)dV(y) = -SSS vy 4rc[x-y[ × w'f(y)dV(y)

D D

=SSS Vy x w}(y) dV(y) n x w} v, 4rclx-yl 4rclx-yl de(y) (A.17)

[Alternatively, note that, for any function f and for E given by Eq. (3.13),

~E S~R! VyE(y)f(y)d V(y) = Y~R3 ~nn nf(y) (y)d V(y) = ~ f(y) n (y)da(y) (A. 18)

Then

A~(x)=Vx x ~ E(y)4~[}x(Y2y[ v, x (Ew:)

dV(Y)=Sd: 47fix-Y] dV(y)

EVy X W~f VyE x w' s =S~3 ~ 4rc[x-y] dV(y)+y~f 4rc[x-y] dV(y)

t i co(y) d V( y ) +~ xw) =S~S arc[x-y[ 4rclx_y I da(y) (A.19) v:

in agreement with Eq. A. 17.] Recalling that w)= 0 inside a, we notice that

y = n x w} (A.20)

and therefore Eqs. (A. 15) and (A. 16) are equivalent to Eq. (A. 13). The advantage of Eq. (A. 16) is that it shows clearly that A) is solenoidal, and that Eq. (2.2) (and hence Eq. A.5) is satisfied: thus it is apparent that both A and A) satisfy Eq. (2.6) with the boundary conditions Eqs. (A.I) (see

86 Computational Mechanics I (1986)

Eq. 4.1 1 .a) and (A.5), and therefore V x A and w} coincide. However, A) does not necessarily satisfy Eq. (A.7), and therefore A and A) may differ by the arbitrary term V;~ discussed in introducing Eq. (A.7).

Appendix B: Helmhoitz decomposition in fluid dynamics

This Appendix deals with the use of Helmholtz decomposition as the basis of computational methods for the solution of viscous incompressible flows, with emphasis on the works of Hirasaki and Hellum (1970), Quartapelle and Valz-Gris (1981) and Wu (1976, 1982, 1984). None of the theoretical fluid dynamics issues that are discussed in the main body of this paper are addressed in the above works. Therefore, this Appendix is limited to the computational aspects of the present formulatuion. The same convention used in Appendix A (i.e., that, the normal is pointing into V, even for interior problems) is used in this Appendix.

The problem is governed by Eq. (2.3) (with 0 = 0) for the scalar potential, Eq. (2.6) for the vector potential, and Eq. (5.6) for the vorticity. The boundary condition for the scalar potential is given by Eq. (A.4) (which with Eq. A.1) guarantees the normal boundary condition is satisfied). Also, eliminating 09 between Eq. (5.6) (for the vorticity) and Eq. (2.6) (for the scalar potential) one obtains a fourth order partial differential equation for A, which involves the bi-Laplacian of A. This equation requires six scalar boundary conditions on a. Three of these are given by Eqs. (A.7) and (A.8); two more are obtained by imposing the no-slip conditions which may be written as (Eqs. 5.1 and 2.1)

n x V x A = n x v b - n x V~p (B.1)

and the last one is obtained by imposing that 09 is solenoidal

V.09=0 on rr (B.2)

(Note that the fact that the normal component of the vorticity is continuous at the boundary

09. n =09b" n (B.3)

is a consequence of Eq. (B.1) and therefore, to the contrary of the assumption of Quartapelle and Valz-Gris (1981), it cannot be used as an independent boundary condition.)

The above six conditions plus the initial conditions on 09 appear to be sufficient to determine uniquely ~b, A and 09, at least for a simply connected region without holes (see comments following Eq. A.8). As pointed out by Quartapelle and Valz-Gris (1981) the problem arises only when one wants to solve the two problems (the one of A and that for 09) independently. In this case we have too many (five) conditions on A and not enough on 09 (only one).

Hirasaki and Helium (1970) resolve the problem by finding the vorticity through an iterative process. More elegant is the approach used by Quartapelle and Valz-Gris (1981), who use the equation V2A = -09 (Eq. 2.6) to "shift" two of the boundary conditions on A into conditions (of integral type) on 09. This is obtained by using the integral relation (for n pointing into the volume V)

lJ'J" (a .V x V x h - b . V xV x a )dV= - ~ (a .n x V x h - h . n x V x a)da (B.4) V a

If a = A (satisfying Eqs. 2.2, 2.5, 2.6, A.7 and B.1) and b = B one obtains that the equation

~y B.09dV= - ~ B. n x (vb-V~p)da (B.5) V a

must be satisfied for all B such that V x V x B = 0 (Eq. 38 of Quartapelle and Valz-Gris 1981). This is a constraint on the distribution of the vorticity, which is used to determine the vorticity on the boundary in the finite-difference scheme used by Quartapelle and Valz-Gris (1981): two conditions on A may now be dropped, in order to have three conditions on A and three on 09. It seems natural to drop Eq. (B.1). [It may be worth noting that in this case Eq. (B.3) is now an independent condition, and may be included in the formulation. Therefore only one scalar integral condition is actually needed on o9. The process whereby the three scalar conditions expressed by Eq. (B.5) yield one (or two) conditions o n 09 is not addressed by Quartapelle and Valz-Gris (1981)].


A similar approach, more closely related to that of this paper, is used by Wu (see, e. g., Wu 1976, 1982, 1984). The major difference between Wu's approach and that of this work (as well as that of Quartapelle and Valz-Griz (1981)) is that Wu addresses the issue directly on the discretized formulation and limits himself to 'computational boundary conditions' (lacking, for instance, the interpretation of his condition as the physical generation ofvorticity). In his analysis, the values of the vorticity at the boundary-nodes are unknown; therefore conditions for the evaluation of those values are necessary.

Nonetheless, Wu's computational formulation is closely related to that presented in Section 5 (as well as the numerical implementation by Del Marco (1986). Because of the similarity of these two formulations, a detailed analysis of Wu's work (amongst the best computational work in the field, in this author's opinion) is presented here. Wu uses a decomposition different from those examined thus far. The decomposition used in the main body of this paper was obtained from the infinite-space unique decomposition by arbitrarily choosing the velocity inside a to be equal to that of the solid region. The decomposition used by Wu may be obtained by choosing, also arbitrarily, that the velocity field is equal to zero inside the surface tr (in the frame of reference connected with the undisturbed air). In this case, applying to Eq. (2.12) (with 0 = to = 0 inside a) the same procedure used to derive Eq. (A.17) (or A.19) one obtains (for n pointing into the volume V)

v(x)= -Vx ~!~ Vy.v(y) dV(y)+V~xj'~ Vyxv(y) 4~zlx-Yl v 4rc]x-Yl dV(y)

n(y). v(y) do-(y) +V~ x ~ n(y) x v(y) -Vx 4 [x_y I . 4=]x_y I da(y) (B.6)

Equation B.6 is a decomposition for the exterior problem which is actually used by Wu (1976, 1982, 1984), as the basis for his computational scheme. (By comparing Eq. B.6 with Eq. A. 10 we see that the scalar-potential contribution in Eq. B.6 does not satisfy Eq. A. 10, and therefore this decomposition does not fall within the type of decomposition introduced by Bykhovskiy and Smirnov 1960.)

In analyzing the mathematical meaning of Eq. (B.6), Wu reaches conclusions similar to those of Quartapelle and Valz-Gris, that the conditions on the vorticity must be of integral type: "the specifications of both the normal and tangential components of velocity on a overspecifies the problem. It should be pointed out that both the normal and the tangential velocity components are prescribed by the physics of the problem; and both appear in the boundary integral equation, Eq. (B.6). The use of Eq. (B.6) therefore appears to overspecify the kinematics of the problem. In reality, the prescription of both the normal and the tangential velocity component places a kinematic restriction on the distribution of vorticity in V. This restriction permits the boundary vorticity values to be determined. Detailed discussions of the use of Eq. (B.6) in the computation of the boundary vorticity values are presented in Wu (1976)." (Quotes are from Wu 1984, with minor modifications introduced to conform with the notation and the equation numbering of this paper.) Unfortunately, the detailed discussions (on the computation of the boundary vorticity) presented in Wu (1976) are limited to two-dimensional problem. [Three-dimensional problems are considered by Thompson et al. (1974): however the issue on the boundary conditions is resolved by an iterative scheme similar to that of Hirasaki and Hellums (1970). No recent work by Wu and his associates deals with three- dimensional flows.]

Equation (B.6) is fully equivalent to Eq. (4.3), which may be obtained from Eq. (B.6) by adding a similar expression for the solid region and allowing for a tangential discontinuity between the fluid and the solid on the boundary. The major difference between the approach of this paper and that of Wu is the following. In this paper the phenomenon is studied as a the limit of a sequence of impulsive accelerations (when the classical no-slip condition is violated) followed by intervals when diffusion occurs. Wu, on the other hand, assumes that both normal and no-slip boundary conditions are satisfied. Accordingly, Eq. (B.6) is seen by Wu as a computational condition (expressed directly on the discretized form of Eq. B.6), without the physical interpretation (in terms of generation of vorticity) introduced here. Therefore, in his formulation the normal condition and the tangential condition appear to be interchangeable : typically (although not exclusively) Wu chooses to use the tangential condition (inStead of the normal condition used here) to evaluate the vorticity on the boundary; he


distributes the vorticity over a finite thickness (half the size of the mesh-grid). The normal boundary condition then is automatically satisfied (within numerical approximations).

The advantage of the infinite-space approach used here is that the simplicity (and uniqueness) of the representation facilitates the physical interpretation on the formulation in terms of vorticity generation and diffusion. This interpretation yields almost as a necessity that the normal, not the tangential condition, must be imposed. This result is crucial in this paper because of its consequences on theoretical issues: for instance the applicability of Eq. (4.6) to both viscous and inviscid flows, allows for the analysis of the relationship between the Euler solutions and the Navier-Stokes solutions presented in Section 6.

Besides its theoretical interest, the approach used here has computational implications, for instance the fact that the thickness of the vorticity layer at the boundary is ] / / ~ (a consideration missing in Wu's analysis). Also note that, for the case of a body in a uniform stream, the representation used here reduces to

~o(y) v(x)=Vxxfllv 4rclx-y[ dV(y) (B.7)

which is considerably simpler than Wu's (see Eq. B.6, with 0 = 0 in Vs : in this case, the two surface integrals in Eq. B.6 happen to be equal and opposite, a fact not immediately apparent from Wu's formulation, even in the two-dimensional problem).

Even more important is the simplicity of the present formulation for the case of three dimensional flows. The extension of Wu's formulation to three-dimensional flows (e. g., the number of integral conditions required on co, see discussion following Eq. B.6) is not immediately apparent (for instance, one scalar condition appears to be sufficient if both Eqs. B.3 and B.4 are used; the discussion of this and other subtle issues for the three-dimensional flow analysis is missing in Wu's work).

But the problem with Wu's formulation ultimately lies in the difficulty in extending it to compressible flows. For, Eq. (B.6) is useful if 0 is known. However, in fluid dynamics 0 is equal to ~2(~/~t2 divided by the square of the speed of sound (plus nonlinear or higher order terms, see Morino, 1985). The implication is that the equation governing q~ is actually the wave equation rather than Poisson equation, making Wu's approach not easy to use for compressible flows (see for instance E1 Refaee et al., 1982). The formulation of the problem simplifies considerably using the approach of Morino (1985), i.e., starting directly from the differential equation and using the Green's function for the wave equation, written in a frame of reference connected with the solid region (Morino, 1974). Details of this formulation are presented in Part II of this paper (see also Appendix C).

Appendix C: On transpiration velocity: The defect velocity approach

This Appendix deals with the relationship of the present work with the classical work of Lighthill (1958) on displacement thickness: in this classical paper, Lighthill presents four different methods to study the interaction between the boundary layer region and the outer region for three-dimensional incompressible flows. These include:

(1) flow reduction, (2) equivalent sources, (3) velocity comparison, (4) mean velocity.

The use of the mean-vorticity approach in connection with the present work was discussed in Section 6 (see discussion following Eq. 6.2). In this Appendix, it will be shown how the other approaches in Lighthill's work, in particular the equivalent-source approach (now better known as the transpiration-velocity approach, see, e.g., Hess 1977, and Lemmerman and Sonnad 1979) are related to the Helmholtz decomposition. In particular, using a different type of decomposition, it will be shown that the transpiration velocity formulation may be interpreted directly in terms of Helmholtz decomposition. It will be shown that (contrary to the displacement thickness formulation)


the transpiration-velocity formulat ion is indeed exact (in the sense that it does not contain errors of the order of magnitude of the boundary layer thickness formulation). In addition, it will be shown that the same expression introduced by Lighthill (1958) for incompressible flows is valid even for unsteady compressible flows: the usual expression used for compressible flows (e. g., Lemmerman and Sonnad 1979) is limited to steady compressible flows.

For the sake of clarity, it is convenient to start the discussion with the case of a symmetric wing at zero angle of attack. The vorticity is then limited to a finite region around the wing (a boundary layer region plus a small wake region). Outside this region, the flow is potential and may be expressed as Vq~'. Let q~ be an harmonic function defined in the whole fluid region and equal to ~b' outside the vortical region. This definition involves a process of continuation for harmonic functions (which for the two-dimensional case, reduces to the well known continuation of analytic functions of complex variables); some issue connected with this process are discussed later in connection with the problem of a wing at an angle attack).

According to the results of Section 2, the difference between v and V~b', (i. e., the velocity correction or 'defect velocity' due to the presence of the vorticity in the field) may be expressed in terms of a vector potential. Indicating with re the external velocity (re = Vq~) and with ra the defect velocity due to the vorticity, we may write

V=Ve+Vd (C.1)

Recall that the defect velocity, Vd is the contribution due to the vector potential: therefore Vd is solenoidal and satisfies the equation

1 -hih2h3 ~xl (h2h3vai)+ (hlh3Va2)+~ Ox3 (h~h2va3) = 0 (C.2)

(where xl , x2 and x3 are orthogonal curvilinear coordinates) and

va=0 (C.3)

outside the boundary layer and wake region. For any point x on the surface of the boundary surface o-, one may choose the x3 coordinate to be

the arclength in the direction of the outer normal n (this implies h3 = 1) and x 3 = 0 on o-. Then, indicating with 3 the thickness of the vortical region, and using Eq. (C.3), Eq. (C.2) may be integrated to yield

V a" n= - - (heval) (hlVa2) dx 3 on a (C.4) haoh2o o ~ ~x2

Next using Eq. (C. 1), one obtains that the normal boundary condition for q~ may be expressed as (Eq. 3.3)

8q~ an r b ' n - - r d ' , o n ~ (C.5)

where vb is the velocity of the body. Equation (C.5) is equal to that of a body with flux through the surface a (e. g., the inlet of a nacelle)

with velocity equal to - ra" n; hence, this term is called ' transpiration velocity' and is indicated with vtr. Using Eqs. (C.1) and (C.4), one obtains

vtr= -va 'n -h ioh2o o ~xl [h2(Vei--Vl)]'k- [hl(Ve~-V2)] dx3 (C.6)

which is a slight generalization of Eq. (25) of Lighthill (1958), where Ve~ is not included because of the approximation used in the expression for the boundary conditions (also, Lighthill uses the assumption that ~hl/©x3 = Ohz/OX3 = 0; this assumption, typically not satisfied in actual applications, is not necessary and is not used here).

It may be worth noting that the assumption of the boundary layer thickness being small was not used in the derivation (this is essentially true for Lighthill's formulat ion as well). Hence the derivation is valid even if the vortical region is not small (actually it is valid even if va is not zero outside the

90 C o m p u t a t i o n a l M e c h a n i c s I (1986)

vortical region; in this case the integration should be extended to infinity). Therefore the formulation is applicable, at least in principle, to the general case, even to the case of a wing at an angle of attack with separated flow. In this case, however, a more efficient approach appears desirable and is discussed later in this Appendix. [It should be pointed out that the link between transpiration velocity and displacement thickness is provided by Lighthill's velocity comparison approach, where higher order terms in 6 are neglected. Therefore, the displacement thickness formulation is not necessarily exact.]

It should be noted that the formulation used here to derive Eq. (C.6) is different from that used by Lighthill, and has a major advantage over Lighthill's : it is valid for compressible unsteady flows as well. In order to show this, note that the key in the present formulation consists of recognizing that the defect velocity (difference between the fluid velocity and the continuation of the external velocity) may be expressed in terms of the vector potential and therefore satisfies Eq. (C.2) (Lighthill uses the continuity equation in his derivation). The fact that the defect velocity may be expressed in terms of the vector potential is true for compressible unsteady flows as well (see, e. g., Morino 1985 ; this point is discussed in detail in Part II of this work). Therefore, the above expression for the transpiration velocity is valid even for compressible unsteady flows. [In order to emphasize the use of Eq. C.2 as the key to the above formulation (and to distinguish the decomposition used in the infinite-space approach employed in the main body of this paper from that used in this Appendix), this method is referred to as the 'defect-velocity approach.']

The above result is in contrast to the classical expression for the transpiration velocity (see, e.g., Lemmerman and Sonnad 1979), which contains the density, e, in front ofv and ~e in front of re, and is obtained by using the continuity equation for steady compressible flows. This expression is (1) approximate, and (2) valid only for steady flows (and not easily extendable to unsteady compressible flows). The advantages of Eq. (C.6) are its generality and its simplicity.

Next, the transpiration-velocity formulation is extended to the case in which the vortical region is not limited to a small region around the boundary. In this case, the formulation presented above is still valid, but the numerical implementation is not necessarily efficient, since Eq. (C.6) requires integration over at least the whole vortical region. Also, the harmonic continuation of ~b' into ~b may yield a discontinuous value for 4, akin to the analytic continuation in function of a complex variable; this is true even for a simply connected domain, as is easily seen in the case of a wing with a thin wake (where, in the limit, one obtains the potential flow discussed in Section 2 which has a discontinuity in across the wake). Therefore, for the case in which the vorticity is not limited to a small region around the body, it is convenient to use a different decomposition. Then let Ve indicate the velocity in the external volume Ve (which is now defined as the volume outside the internal region V~, a small region around % that may be arbitrarily defined, e.g., the boundary layer region for attached flows or the viscous sublayer for turbulent flows). The external velocity is given by

13 e = V ¢ -~- IJ v (C.7)

where

vv(x)--Vxx//~ 4~[ta}Y?y l _ dV(y) (C.8) v~

is the velocity "induced" only by the vorticity in V~, and ~b is harmonic with suitable boundary conditions. This expression does not include the contribution of the vorticity inside the boundary layer, and therefore, its use in the internal region V~ (e. g., in the boundary layer region) yields the desired harmonic continuation. Except for this generalization of the definition of the external velocity, the analysis presented above is still valid, and the boundary condition for q~ becomes (Eq. C.5)

an =Vb" n - - vv " v + vtr (C.9)

with v~ given by Eq. (C.8) and vtr given by Eq. (C.6). Extension of this formulation to compressible flows is similar to that discussed above, and is given in detail in Part 2 of this work.

Communicated by S.N. Atluri, December 31, 1985

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morino_1986 Helmholtz decomposition revisited; Vorticity generation and trailing edge condition

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