An asymptotic derivation of the initial condition for the incompressible and viscous external unsteady fluid flow problem

International Journal of Engineering Science 38 (2000) 1983±1992

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An asymptotic derivation of the initial condition for the incompressible and viscous external unsteady ¯uid ¯ow problem R.Kh. Zeytounian * 12, Rue Saint ± Fiacre, 75002 Paris, France Received 2 June 1999; accepted 20 December 1999

Abstract This short note is devoted to a consistent asymptotic derivation of a Neumann problem for the determination of the initial condition for the evolution equation (the so-called ÔNavier equationÕ) governing an incompressible, viscous unsteady external ¯uid ¯ow. This Navier equation is, in fact, derived as a limiting form of the full `exact' Navier±Stokes±Fourier equations for a perfect viscous, compressible and thermally conductor gas, when the characteristic Mach number (Ma) tends to zero. Since the initial time region is singular for the above limit process, it is necessary to consider the role of acoustics for a consistent formulation of the initial condition for the Navier velocity vector. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction We neglect gravity and Coriolis forces and we assume that the gas is linearly viscous and heat conducting, while its thermodynamic behavior is of the perfect gas type, the pressure p, density q and temperature T being related by the classical equation of state p ˆ RqT ;

…1†

where R is the so-called gas constant. We assume furthermore that the speci®c heats Cp and Cv are constants, with c ˆ Cp =Cv as their ratio. The coecient of shear viscosity l , the one of bulk

*

Also: University of Lille I, 59655 Villeneuve d'Ascq Cedex, France. Fax: +33-1-40-36-62-07. E-mail address: [email protected] (R.Kh. Zeytounian).

0020-7225/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 0 ) 0 0 0 1 8 - 5

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R.Kh. Zeytounian / International Journal of Engineering Science 38 (2000) 1983±1992

viscosity lv , and the one of heat conductivity k  , depend on the temperature T, but not on the 1=2 pressure p. The speed of sound is known to be, a…T † ˆ …cRT † . The so-called Navier±Stokes±Fourier (NSF) equations (see, for instance, Truesdell [11]), which rule the evolutions of such a gas are well known, and a derivation may be found in Lagerstrom [6, Sections B.3 and B.18]. Here we are concerned, exclusively, with the so-called external problem, for which p, q and T tend to constant values at in®nity, namely p0 ; q0 ; T 0 , and we use those values for getting dimensionless thermodynamic quantities, by setting: p ˆ p0 …1 ‡ p†;

q ˆ q0 …1 ‡ x†;

T ˆ T 0 …1 ‡ h†:

…2†

We make the choice that the velocity is 0 at in®nity, and that the gas is set into motion by the displacement of some body. We put U 0 for a characteristic velocity value of such a displacement. This allows us to introduce the characteristic Mach number …a0  a…T 0 †† Ma ˆ U 0 =a0 :

…3†

We shall be concerned only with situations according to which p, q and h remain small when Ma  1. Guessing that the velocity of the ¯uid is forced by the motion of the body, we set u ˆ u =U 0

…4†

for the dimensionless form of such a velocity. We set x for the dimensionless vector position and t for the dimensionless time, the corresponding dimensional values being x ˆ L0 x and t ˆ t0 t, where L0 is a characteristic length related with the body in motion and t0 a characteristic time such that the Strouhal number: S ˆ L0 =t0 U 0 ˆ 1: For both viscosities and heat conductivity we set: l ˆ l0 l…h†;

lv ˆ l0v lv …h†;

k  ˆ k 0 k…h†

…5†

and we make the convention that: l…0† ˆ lv …0† ˆ k…0† ˆ 1. Of course, the functions l…h†; lv …h†; k…h† are consistent with the third part of (2), and, as a consequence, they are dimensionless, as well as are all their derivatives with respect to h. We are now ready for setting the dominant NSF equations, namely: ox=ot ‡ u
rx ‡ r
u ‡ xr
u ˆ 0;

…6†

ou=ot ‡ u
ru ‡ …1=cMa2 †rp ‡ x‰ou=ot ‡ u
ruŠ ˆ …1=Re†‰r2 u ‡ ‰…1=3† ‡ r0 Šr…r
u† ‡ r
‰N …u†hŠŠ ‡ O…h2 †;

…7†

R.Kh. Zeytounian / International Journal of Engineering Science 38 (2000) 1983±1992

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oh=ot ‡ u
rh ‡ …c ÿ 1†r
u ‡ x‰oh=ot ‡ u
rhŠ ‡ …c ÿ 1†p…r
u† ˆ …c=Pr Re†r2 h ‡ …c ÿ 1†…cMa2 =Re†fD…u† : D…u† ‡ ‰r0 ÿ …2=3†Š…r
u†2 g ‡ …c ÿ 1†  …cMa2 =Re†‰…dl=h† ÿ r0 …dlv =dh†Šhˆ0 …r
u†2 h ‡ O…h2 †; p ˆ x ‡ h ‡ hx:

…8† …9†

We emphasize that these above equations are not the exact ones, as indicated by O…h2 † occurring in (7) and (8). As a matter of fact, we are going to investigate the asymptotic structure of these Eqs. (6)±(9) when Ma is small and, as we shall see under general setting, goes like O…Ma2 † so that the neglected terms in (7) and (8) will have no e€ect on the analysis to come. But it is necessary to note that these above dominant equations are a direct consequence (for small h) of the full exact NSF equations for a perfect gas with constant speci®c heats, when we take into account (2)±(5). We call the readerÕs attention to the following dimensionless ratios: Re ˆ

U 0 L0 ; m0

r0 ˆ l0v =l0 ;

Pr ˆ Cp l0 =k 0

…10†

where Re is the Reynolds number, r0 the viscosities ratio and Pr is the Prandtl number, respectively. The term involving N…u†, arises from the variation of the viscosities with the temperature N…u† ˆ ‰…dl=dh† ÿ r0 …dlv =dh†Šhˆ0 …r
u†I ‡ 2…dl=dh†hˆ0 D…u†:

…11†

A similar e€ect is expected to arise from the variation of the heat conductivity with the temperature but it would be included in the neglected O…h2 † term in (8). Of course, I stands for the unit tensor, while D…u† ˆ …1=2†‰…r
u† ‡ …r
u†T Š

…12†

stands for the rate of strain tensor and the superscript ÔT Õ stands for the transpose of a second rank tensor. 2. The Navier limiting problem 2.1. Formulation of initial and boundary conditions for the NSF equations First, we make the choice that the compressible, viscous and heat conducting ¯uid starts from a state of rest, at constant density q0 and temperature T 0 , so that we get x ˆ 0;

h ˆ 0 and u ˆ 0;

at t ˆ 0

…13†

in the whole of the domain occupied by the ¯uid. We assume that this ¯uid is set into motion by the displacement (and eventually a deformation) of a body, the ¯uid pervading all the domain X, complementary to this body. We set C for the boundary of X, and n for the unit vector normal to

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C, pointing towards the ¯uid. Concerning u we make the usual assumption that the ¯uid adheres to the wall (C) and this amounts to state that at each point of the wall we attach a dimensionless velocity Uw , depending on the time and on the position P on C such that u ˆ Uw …t; P†  1…t† uw …P†; all along C;

…14†

where 1(t) is the so-called Heaviside (or unit) function. This means that we bear interest on situations when the body starts its motion impulsively (indeed, through (14) we are somewhat mimicking a catapulting process). We note consider here the boundary condition for h, since our objective in this present paper is only the derivation of the Neumann problem for the determination of the initial condition for the limit Navier velocity which satis®es the below limit Navier equation (19a). Being concerned with the motion of the ¯uid in a domain extending to a whole neighborhood of in®nity, we need some conditions relative to it. It seems obvious that we require: U  …x; h; p; juj† ! 0;

when

jx ÿ x0 …t†j ! 1;

…15†

where x0 …t† is some point inside the body. Of course for any ®nite time we might simply ask that (15) holds when jxj ! 1. As a matter of fact, (15) raises some diculties which come from the wake trailing behind the body, but we leave aside this peculiarity. We are rather concerned with what arises when the Mach number is small, and we try to solve our problem by an expansion in powers of the Mach number. Then, when we go far from the body, excluding the region within the wake, we know that the e€ects of viscosity and heat conductivity die out more rapidly than inviscid e€ects (indeed, the perturbations are expected to decay like O…jx ÿ x0 …t†j2 †. Then again, we know from the classical low Mach number aerodynamics that the Mach number plays a peculiar game when jx ÿ x0 …t†j ! 1. As a matter of fact, when ‰x ÿ x0 …t†ŠMa ˆ x ˆ O…1†;

…16†

the ¯ow approximates an acoustic ®eld. This is a basic process in the pioneering work by Lighthill [9] on the generation of sound by turbulence, but the singular nature of the Mach number expansion, which requires two matched asymptotic expansions (MAE) was ®rst recognized by Lauvstad [8], and more thoroughly discussed by Crow [1] and Viviand [14]. We want to emphasise that this matter of far ¯ow is not a purely academic one, since any numerical simulation must work on a bounded domain, and is faced with the problem of choosing appropriate boundary conditions to be enforced on the external boundary of the computations grid. The main goal is then to devise the so-called transparent boundary conditions which do not pollute the computations. In the past, this problem has been treated by some ad hoc approaches, but from the middle of the 1990s a good deal of mathematical work has been devoted to it. The reader may ®nd a number of informations, with applications to both inviscid and viscous ¯ows in Givoli and Cohen [2], Grote and Keller [4], and Tsynkov et al. [12]. Halpern [5] has developed a general method for deriving such transparent boundary conditions applied to incompletely parabolic perturbations of hyperbolic systems. She ®nds that the proper boundary conditions have to

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be non-local in time, of the pseudo-di€erential (integral) type, which may, often, be replaced by a local approximation. Recently, Tourrette [10] applied this technique to the (linearized) NSF equations. 2.2. The Navier system We now start investigating the e€ect of letting Ma # 0

with Re; r0 ; t and x fixed

…17†

in the dimensionless dominant equations (6) and (7). In such a case, it is not dicult to guess that the following `principal' expansion is appropriate for the asymptotic investigation of Eqs. (6) and (7): u ˆ uN ‡ O…Ma2 †; …p; q; h† ˆ Ma2 …pN ; qN ; hN † ‡ O…Ma4 †:

…18†

As a matter of fact, from (7), we see that rp is of the order of Ma2 , and observing that p has to go to 0 at in®nity, the only mechanism which could generate a p of an order di€ering from that of Ma2 is through matching with an outer expansion. We forget about this, but notice that we shall be bound to check that this matching will not change this order of magnitude estimate. For the (Navier) couple : …uN ; pN †, the well-known set of equations: ouN =ot ‡ uN
ruN ‡ …1=c†rpN ˆ …1=Re†r2 uN ;

…19a†

r
uN ˆ 0

…19b†

is derived, which are usually, in the mathematical literature, named under Navier±Stokes, but that we prefer to refer (historically justi®ed) as the Navier equations. From (14) we have the boundary condition (since t > 0 is ®xed when Ma # 0): uN ˆ uw ; all along C

…20†

while, from the conditions at in®nity, we may expect that juN j ! 0; as jxj ! 1

…21†

provided this conclusion is not invalided by a di€erent (outer) asymptotic expansion valid near in®nity. Surprisingly enough, we cannot use (13) to set as uN …0; x† ˆ 0; rather, we have to put uN …0; x† ˆ u0N …x†

…22†

and obviously, u0N …x† should be divergence-free r
u0N ˆ 0:

…23†

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It is somewhat puzzling that the initial value of uN is not zero, as one might expect from (13) and on the other hand, we do not know, from what has been said any indication on how u0N …x† might be obtained. Nevertheless, from (14) we have the indication that the body is set impulsively into motion. This problem of impulsive motion has long been known (see, for example, Lamb [7, Section 11]. The derivation of the Navier system (19a) and (19b) is valid, only if we assume that ox=ot is of the same order as x, that is, if it is small. In the case when a discontinuity in the velocity occurs at t ˆ 0, we may suspect that, close to t ˆ 0, for t > 0, the order of magnitude of ox=ot is not the same as the one of x itself. Through the limit process (17) with (18), the derivative ox=ot is lost and from what we know about asymptotic expansions, this is a clue that we need an inner expansion, in the vicinity of t ˆ 0. The divergencefree character of uN is directly tied to this loss of ox=ot. This omnipotence of the incompressibility constraint and its relation with the initial condition, which guarantees the well-posedness of the Navier problem (19a)±(23), is thoroughly discussed in the review paper by Gresho ([3, 47±52]) and scrutinizing what is stated there, we see that there is a close relation with what may be found in Lamb [7, Section 11]. We shall come back, shortly, to this matter in Section 4 of the present paper. 3. The acoustics equations and the adjustment problem 3.1. Acoustics equations Now, our main goal is to derive a limiting initial boundary value problem, issued from Ma # 0 (but for the time near to t ˆ 0 and x ®xed), such that the time derivatives in (6) remains after getting the limiting form of the dominant equations (6)±(9) Due to the impulsive character of the motion of the body, we expect that changes occur within a small interval of time after t ˆ 0. Although that is not fairly obvious, we nevertheless may expect that during this short time interval, x, h; p remain all small. On the other hand, we should not expect such a behavior for the velocity u, because of the smallness of such a velocity, with respect to the speed of sound which has already been taken care of within the non-dimensionalization of starting NSF equations. From inspection, we guess that the following changes: s ˆ t=Ma;

u ˆ ua ;

…x; p; h† ˆ Ma…xa ; pa ; ha †;

…24†

where the `acoustics' functions ua and xa ; pa ha , depend on s; x and Ma, will work. As a matter of fact, it is very easy to check by substituting (24) into (6)±(9), gives, as a consequence of the inner limiting process: Ma # 0;

with s and x fixed;

…25a†

with …ua ; xa ; pa ; ha † ˆ …ua;0 ; xa;0 ; pa;0 ; ha;0 † ‡ O…Ma†

…25b†

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the following set of leading equations: oxa;0 =os ‡ r
ua;0 ˆ 0;

…26a†

oua;0 =os ‡ r…pa;0 =c† ˆ 0;

…26b†

oha;0 =os ‡ …c ÿ 1†r
ua;0 ˆ 0;

…26c†

pa;0 ˆ xa;0 ‡ ha;0 :

…26d†

The ®rst consequence of this is that none of the time derivatives being lost in the inner limiting process (25a), we may apply the initial conditions (13) to the system (26a)±(26d) and get: sˆ0:

ua;0 ˆ 0;

pa;0 ˆ xa;0 ˆ ha;0 ˆ 0;

all outside the body:

…27†

Now, we run into a problem, this time with the boundary conditions. Eqs. (26a)±(26d) are the dimensionless form of the equations of (linear) acoustics, in a homogeneous gas at rest. We know that, for those equations, the only condition that might be applied on the boundary is one of slip of the gas with respect to the wall. We have to come back to (14) and observe that 1…t† ˆ 1…s†, provided s > 0: As a matter of fact, such a statement would necessitate a proof, but we may argue physically and this will be sucient for our purpose. Then we get the desired boundary condition ua;0
n ˆ uw
n  Ww ;

all along C

…28†

and we observe that Ww …P† does not depend on s. 3.2. Adjustment problem Now, we concentrate on the solution of the acoustics problem: (26a)±(28). We observe that, due to (23), substracting u0N from uN does not change anything to (26a)±(26d). It is then very easy to check that the following formulae: ua;0 ˆ u0N …x† ‡ r/a;0 …s; x†;

xa;0 ˆ ÿo/a;0 =os;

pa;0 ˆ ÿco/a;0 =os;

ha;0 ˆ …1 ÿ c†o/a;0 =os …29†

solve system of equations (26a)±(26d) provided /a;0 be a solution for the dimensionless dÕAlembertÕs equation of acoustics, namely o2 /a;0 =os2 ÿ r2 /a;0 ˆ 0

…30†

the speed of the sound being replaced by 1 due to the choice made in the process of getting the NSF equations in a dimensionless form such as (6)±(9).

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From the initial conditions (27) we derive s ˆ 0 : r/a;0 ˆ ÿu0N …x†; o/a;0 =os ˆ 0; all outside the body

…31†

while from the boundary condition (28), we get the condition u0N
n ˆ Ww ; all along C

…32†

to it, so that the boundary condition, on the body wall, for /a is o/a;0 =on ˆ 0; all along C:

…33†

As far as (23) and (32) are the only restrictions put on uN0 , we are somewhat short of a condition at in®nity, for the complete determination of /a;0 . But, we may get rid of this slight diculty by setting u0N ˆ rw0N

…34†

which is allowed by (29) and observing that we may then determine w0N through r2 w0N ˆ 0; all outside the body;

…35a†

ow0N =on ˆ Ww ; all along C;

…35b†

w0N ! 0;

…35c†

when jxj ! 1

which is a straightforward Neumann problem for the Laplace equation. Then, we get, instead of (31) s ˆ 0 : /a;0 ˆ ÿw0N ; o/a;0 =os ˆ 0; all outside the body:

…36†

Now, (30), (33) and (36) lead to a well-posed problem for /a;0 provided we add /a;0 ! 0 as jxj ! 1

…37†

which amounts to an added information, namely that no perturbations are coming from in®nity towards the body. One must consider that such an information is of physical rather than mathematical character. We are not, actually, interested in getting /a;0 and all that we want to know is that s ! 1 : /a;0 ! 0

…38†

which is guaranteed by the mathematical theory of dÕAlembertÕs equation (see, for instance, Wilcox [15]).

R.Kh. Zeytounian / International Journal of Engineering Science 38 (2000) 1983±1992

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This provides us with the consequence s ! 1 : ua;0 ! u0N

…39†

so that, matching the Navier solution with the present (acoustics) one, gives lim uN …t; x†  lim ua;0 …s; x†

t!‡0

s!1

…40†

and in turn, we have found that what was missing for achieving a complete Navier initialboundary value problem, namely (29), where /a;0 is the solution of the acoustics problem (30), (33), (36) and (37) and u0N is completely determined by the Neumann problem (35a)±(35c) with (34). We note that the paper by Ukai [13] is devoted to a rigorous mathematical analysis of the incompressible limit and the initial layer of the compressible Euler equation. In this last Eulerian case we derive again in the initial layer the classical acoustics equations (26a)±(26d) and the Neumann problem (35a)±(35c) for the initialization of Euler incompressible equations. 4. Conclusion The ®rst remark concerns the case when the body is set in movement rapidly, during a lapse of time proportional to Ma, or else progressively during a lapse of time O…1†. In the ®rst case, the above result (problem (35a)±(35c) with (34)) concerning the initial Navier value u0N , derived in Section 3 remains true, since, in fact, in this case: Uw …t; P† ˆ Uw …s; P† and according to matching condition, (40), with (38) and (29), in place of Ww …P† in condition (35b), we can write Uw …1; P†
n ˆ Ww …P†: On the contrary, in the second case, the corresponding Neumann problem, (35a)±(35c) has only trivial zero solution, since in this case the displacement velocity of a material point P of the boundary C, of the body X Uw …t; P† ˆ Uw …Mas; P† ! Uw …0; P†  0 when Ma # 0 with s ®xed (inner (local) limit (25a)) and as consequence Ww  0 in Eq. (35b) ± in this second case, the acoustic region near the t ˆ 0 plays a `passive' role and does not have any in¯uence on the Navier limit problem! Finally, a short comment concerning the formulation of a well-posed initial-boundary value problem for the Navier equations given by Gresho [3, 47±52]. According to Gresho (and with the Gresho notations), if the initial …t ˆ 0† velocity ®eld u0 for the Navier equations is not divergencefree and if the vertical component of this initial velocity is not equal, on the boundary, to vertical component of the speci®ed boundary condition at t ˆ 0, …w0 †, then it this necessary to solve (again!) the following Neumann problem for the unknown k: r2 k ˆ ÿr
u0 in X with ok=on ˆ n
…w0 ÿ u0 † on C: Then, it is necessary to compute v ˆ u0 ‡ rk, such that r
v ˆ 0 in X with

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R.Kh. Zeytounian / International Journal of Engineering Science 38 (2000) 1983±1992

n
v ˆ n
W0 on C and replace u0 by v; such that u…x; 0† ˆ v in X ‡ C: The analogy with our result (obtained in Section 3) is disconcerting; in fact, for u0  0, our w0N is the Gresho k), but our result is only true for the unsteady external aerodynamics, when for the acoustics problems (30), (33) and (36) we have the behavior (38). For internal aerodynamics, this behavior (38) is true only when the deformable wall of the bounded body is set in movement progressively during a lapse of time O…1†, according to Zeytounian and Guiraud papers [16] but in this case the matching condition (40) is not valid! Acknowledgements While this paper expresses the views of the author, it has been in¯uenced by discussion with J.-P. Guiraud, going a little beyond our common work (see, for instance, [17]). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

S.C. Crow, Studies Appl. Math. XLIX 1 (1970) 241±253. D. Givoli, D. Cohen, J. Comput. Phys. 117 (1995) 102. P.M. Gresho, Advances in Applied Mechanics, vol. 28, Academic Press, New York, 1992, pp. 45±140. M.J. Grote, J.B. Keller, J. Comput. Phys. 127 (1996) 52. L. Halpern, SIAM J. Math. Anal. 22 (5) (1991) 1256. P.A. Lagerstrom, in: F.K. Moore (Ed.), Theory of Laminar Flows ± Section B Laminar Flow Theory, Princeton Universty Press, Princeton, 1964, pp. 20±285. H. Lamb, Hydrodynamics, Cambridge University Press, Cambridge, 1932. V.R. Lauvstad, J. Sound Vibration 7 (1968) 90. M.J. Lighthill, Proc Roy. Soc. A 211 (1952) 564. L. Tourrette, J. Comput. Phys. 137 (1997) 1±37. C. Truesdell, A First Course in Rational Continuum Mechanics, vol. 1, Academic Press, New York, 1977. S.V. Tsynkov, E. Turkel, S.S. Abarbanel, AIAA J. 34 (4) (1996) 700. S. Ukai, J. Math. Kyoto Univ. 26 (1986) 323±331. H. Viviand, J. de Mecanique 9 (1970) 573. C. Wilcox, Scattering theory for the dÕAlembert equation in exterior domain, Lecture Notes in Mathematics, vol. 442, Springer, Berlin, 1975. R.Kh. Zeytounian, J.P. Guiraud, C.R. Acad. Sci., Paris 290 B (1980) 75. R.Kh. Zeytounian, J.P. Guiraud, in: J.J.H. Miller (Ed.), Advances in Computational Methods for Boudary and Interior Layers, Boole Press, Dublin, 1984, pp. 95±100.