Exact Solutions of the Unsteady Two-dimensional Navier-Stokes Equations

Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/87/005689-14 $ 5.30/0 Vol. 38, September 1987 �9 Birkhfiuser Verlag Basel, 1987

Exact solutions of the unsteady two-dimensional Navier-Stokes equations

By W. H. Hui *), Aerospace and Mechanical Engineering Dept., University of Arizona, Tucson, Arizona U.S.A.

1. Introduction

Most existing exact analytical solutions of the Navier-Stokes equations have been obtained for various special cases for which these equations can be linearized, e.g., in the case of parallel laminar flow. Taylor [1] first observed that the nonlinear convective terms in the two-dimensional Navier-Stokes equations for incompressible fluid vanish when the vorticity is a function of the stream function alone. For the simple case of vorticity proportional to the stream function, he obtained an exact analytical solution that represented a double infinite array of vortices decaying exponentially with time.

Kovasznay [2] extended Taylor's idea by considering the vorticity to be proportional to the stream function perturbed by a uniform stream. He was also able to linearize the Navier-Stokes equations and obtain an exact solution for steady flow which resembles that downstream of a two-dimensional grid. His method was further modified recently by Lin and Tobak [3] who gave two new steady flow solutions: the first represents the reversed flow about a flat plate with suction and the second represents the wavy flow over a plate with a periodic variation of suction and blowing along the plate.

In this paper we also study the two-dimensional steady and unsteady incompressible viscous flow in which the vorticity is proportional to the stream function perturbed by a uniform stream. We shall prove that there exist only two types of steady flow solutions: Kovasznay grid flow solution and Lin and Tobak reversed flow solution, and that Lin and Tobak wavy flow solution with periodic suction and blowing is incorrect. For unsteady flow, new classes of exact analytical solutions are given representing perturbation over a uniform stream. In the frame of reference moving with the uniform stream they are shown to be pseudo- steady in the sense that the flow pattern is steady but the magnitude of motion decays, or grows, exponentially in time. They include Taylor standing vortex array solution as a special case.

*) On leave from University of Waterloo, Ontario, Canada.

690 W.H. Hui ZAMP

2. Formulation

Following Lin and Tobak [3] we non-dimensionalize the stream function by the kinematic viscosity v, velocity components u and v by some characteristic

velocity V, lengths x and y by v/V, and time t by v / V 2. With u = ~ - and

v - ~x ' the two-dimensional Navier-Stokes equations become

v2 ,)

~2 ~2

(1)

where V z - + - - is the Laplacian operator, and the right hand side of (1) ~X 2 ~y2

is the usual Jacobian. As pointed out by Kovasznay [2], who followed Taylor [1], this nonlinear equation can be linearized for flow having the following particular vorticity distribution

V2~ = K ( q - Uy) (2)

where K and U are real constants*), with K =~ 0 t) and U to be identified later with the non-dimensional uniform stream velocity. We shall take the positive x direction to be in the free stream direction so that U > 0. With this vorticity distribution, the Navier-Stokes equation (1) reduces to

(~ - v 2 q, = - u (3)

It is clear that any stream function ~, that simultaneously satisfies Eqs. (2) and (3) is a solution of the Navier-Stokes equation (1). Conversely, any solution of the Navier-Stokes equation (1) that obeys the vorticity distribution (2) must also satisfy Eq. (3). Our task is then to find analytical solutions to the system of linear Eqs. (2) and (3). To facilitate solution procedure, we render the system homogeneous by introducing 7 j = ~k - Uy, whence

e~-+ U - ~ x = K e

V2~ = K ~ .

(4)

(5)

3. Steady flow

3.1. Derivation o f all possible solutions

~0 = 0. Hence, if U = 0 Eq. (4) has only the trivial solu- For steady flow, ~ -

*) The case of complex K is discussed in Appendix A. *) The special case K = 0 corresponds to the well-known irrotational flow.

Vol. 38, 1987 Unsteady two-dimensional Navier-Stokes equations 691

tion leading to 0 = 0, i.e. no flow. We shall therefore consider U > 0. In that case, U could be taken equal to unity without loss of generality; this would amount to using U as the characteristic velocity in non-dimensionalisation. We shall, however, keep U explicit in order to facilitate comparing the steady flow results with the unsteady flow results in w167 4 and 5.

With U > 0, the solution to Eq. (4) is

~t' = F (y) e ~x/U . (6)

Substituting Eq. (6) into Eq. (5) we get

< ) F"(y) + ~ - K V(y ) = O (7)

which is readily solvable. Accordingly, we have the following general solutions of the stream function

(a) F o r K = - k z < O

~b(x, y) = Uy + ae -k2x/v cos {kx / l + k2/U 2 y + b} (8)

(b) F o r 0 < K = k 2__< U 2

Uy + e k2x/v (ae kvl-k2/v2y + be-kVl-k2/V2Y}, (k 2 < U 2) (ga)

O ( x , y ) = g y + e W ( a y + b ) , (k 2 = u 2) (9b)

(c) F o r K = k 2 > U 2

~b(x, y) = Uy + ae k~x/v cos { k x / k 2 / U 2 - 1 y + b} (10)

where a and b are arbitrary constants.

3.2. Discussions

Solution (8) can be used for the half space x > 0 to represent a uniform stream u = U and v = 0 with a perturbation part which is periodic in y and decays exponentially as x increases. Likewise, solution (10) can be used for the half space x __< 0 to represent a similar flow. On the other hand, solutions (9) represents a uniform stream plus a perturbation which grows for x > 0 and decays for x < 0, exponentially, and which is not periodic in y.

It is seen that Kovasznay [2] grid flow solution belongs to the type of solution (8), whereas Lin and Tobak [3] reversed flow solution about a flat plate with suction belongs to the type of solution (9a). Typical and interesting streamline patterns for these flows are to be found in their papers. Solution (10) in the x __< 0 half plane is qualitatively the same as (8) in the x __> 0 half plane and is thus not fundamentally new. F rom the above analyses we see that there is no other possible solution to the Navier-Stokes equation (1) for two-dimensional incompressible steady flow for which the vorticity distribution obeys Eq. (2).

692 W.H. Hui ZAMP

We note, in passing, that the second solution of Lin and Tobak [3] representing wavy flow with periodic suction and blowing is, apart from notational differences and some obvious misprints, the real part of the solution (9a) with

U 2 a = 0 and a complex K = k 2 = ~ - + iqU, i.e.

= Uy + b exP[�89 - x / U z + 4q2y)] cosqx.

It is shown in Appendix A that although the complex stream function corresponding to a complex K may satisfy both (2) and (3), neither its real part nor its imaginary part is solution of the Navier-Stokes equation (1). The above solution of Lin and Tobak is thus not a solution of the Navier-Stokes equation. This fact is, of course, also verified by direct substitution of above solution into Eq. (1).

4. Unsteady flow: class A

4.1. Derivation of plane wave solutions

For unsteady flow, Eq. (4) is a first order partial differential equation for whose general solution is

7 s = G ( X , y ) e Kt, X = x - U t . (11)

Substitution of (11) into (5) shows that G must satisfy Helmholtz equation (also called the reduced wave equation)

O2G ~2G ~X ~ + --~y2 = KG. (12)

Plane wave solutions to Eq. (12) exist in the form

G = g ( 0 , ~ = X c o s 0 + y s i n 0 (13)

where - rc < 0 < re. Substitution of Eq. (13) into (12) yields

g"(O - K g ( O = 0. (14)

There are two cases:

(a) K = - k 2 < 0.

The solution to Eq. (14) is

= A(O) Cosk + B(O)) (15)

and the solution for the stream function is

O (x, y, t) = U y + A (0) e - k2, cos k {(x - U t) cos 0 + y sin 0 + B (0) } (16)

where A and B are arbitrary real constants depending on 0.


(b) K = k z > O.

The solution to Eq. (14) in this case is

0 (~) = A (0) e k r + B (0) e - k ~_ (17)

and the solution for the stream function is therefore

ip(x, y, t) = Uy + ek2t(A(O) e k~(x-vt) ~o~O+y~ino~ + B(O) e -k~(x-v') r sin0~).

(18)

The special case of K -- 0 corresponds to irrotational flow; indeed, it is the following uniform flow

tp = U y + A ( (x - U t) c o s O + y s inO)+ B. (t9)

4.2. Interpretation o f solutions

Consider the solution (16) for K = -- k z < 0. It represents a uniform stream U in the positive x direction plus a perturbation that decays exponentially in time at a rate equal to k 2 = - - K which is, as seen from Eq. (2), a measure of the strength of the local vorticity or flow rotationality. The perturbation flow is that of a plane wave, of wavenumber k, propagating at the phase velocity c equal to U cos 0 in the direction making an angle 0 with the positive x-axis. This phase velocity U cos 0 is just equal to the component of the free stream velocity U in the direction n, i.e., c = U �9 n, where n = (cos 0, sin 0) is the unit normal of the wave front. This means that signals are carried by the undisturbed uniform flow, regardless of the magnitude of the perturbation wave.

Recalling that the time t in Eq. (16) is scaled by v /V 2, we see that the rate of decay in real time, due to the action of visocity v, is equal to k z v / v 2. Thus, shorter waves, i.e. larger k, die out much faster than longer waves, as expected. Apart from this decay in the magnitude due to effects of viscosity, the perturbation flow is periodic in both spatial directions x and y, and in time t. This is in contrast to the steady flow case for which, as seen from the general solutions (8)-(10), periodic variation of flow in x direction is not possible if the vorticity distribution (2) is to be obeyed.

At a given time t the stream function ~ in Eq. (16) may be written as

tp = Uy + Ae-k~t cosk {~o + (B - Ut cosO)} (20)

where

~ o = X C O S 0 + y s i n 0 . (21)

The streamline ip = 0o at time t is given parametrically by Eqs. (20) and (21) with r as the parameter. Typical streamline patterns are given in

694 W . H . H u i Z A M P

Figure 1 a

Y

2O

o

-I0-

~:20

Streamlines for Eq. (20), U = 1, A = 10, B = 0, k = n/4, 0 = n/6. (a) t = 0.

Figure 1 b (b) t = 3.

Y

20-

I0-

5-

~ : 2 0

~-=0

Figs. 1 a, b. The velocity component s are given by

~r Ake-k2, u - - U - s in0 s ink {{ o + B - - U t cos0} Oy

~r - A k e - k 2 t c o s O s i n k { { o + B - U t c o s 0 } . / ) - - ~ X

(22)

(23)

This so lut ion represents the wavy flow over a plate at y = 0 with a periodic variation of suct ion and blowing a long the plate whose strength decays expo- nential ly in time. The suction and blowing at fixed locat ions also alternate


periodically with time. If the perturbation flow is strong enough, i.e. if

[A k sin 0[ > U (24)

reversed flow occurs at early time (Fig. 1 a). However, such reversed flow must diminish at later time (Fig. i b), as the perturbation dies out exponentially with time. On the other hand, if condition (24) is not satisfied, no reversed flow occurs at any time.

Solution (18) for K = k 2 > 0 also represents plane wave motion in which signals are carried by the undisturbed uniform stream, regardless of the magnitude of the perturbation. However, the flow is not periodic in space or in time. Furthermore, perturbations in the flow grow exponentially in time, which is unphysical for large time. It can therefore have only limited usefulness in that it may be used to describe some region of a flow field over a short time.

4.3. Superposition o f plane waves

As Eq. (12) is linear the plane wave solutions obtained in section 4.1 may be superposed to generate more solutions to (12). Thus, if K = - k 2 < 0~ we get

G = GI(X,y) = i A(O) cosk (X cos0 + y sin0 + B(O)}dO. (25)

This represents combinations of the plane waves discussed in section 4.2, in all directions 0, with variables amplitude A (0) and phase B(O) for each direction.

Similarly, for K = k 2 > 0, we get

G = G2(X, y) = i {A(0) e k(x e o s 0 + y s i n 0 ) _[_ B(O) e -k(x c o s 0 + y s i n 0 ) } dO. (26)

We note that although the above solutions (25) and (26) contain arbitrary functions A(O) and B(O), they are not general enough to include the steady flow solution of section 3 as a special case. This is because, as seen from (11), (25) and (26), for 7' to be independent of time, it is necessary that' U = K and G (X, y) = eXf (y) which is, in general, not the case. However, any steady flow solution of section 3 for any U and K is also a special solution of the unsteady Eqs. (4) and (5), we can therefore enlarge the above solutions (25) and (26) by superposing to them the steady solution (8), (9) or (10). In this way we obtain the following class of unsteady flow solutions - called class A solutions in what follows.

(A) If U > O .

(a) f o r K = - - k 2 < 0

~(x ,y , t ) = Uy + ae-k~/U c o s { k x / l + kZ/U2y + b}

+ e -k2t Gl(x - Ut, y) (27)

6 9 6 W . H . H u i Z A M P

(b) for 0 < K = k 2 = U 2

Uy -t- ae kzx/U {ae k ~ y q- be - k ~ r }

+ e k~t G2(x - Ut , y), (k 2 < U 2) 0 (x, y, t) = U y + etrx(ay + b)

+ e v~t Gz(x - Ut , y), (k 2 = U 2)

(28a)

(28b)

(c) for K = k 2 > U 2

6(x, y, t) = U y + a e k~x/v cos { k x / k 2 / U 2 - 1 y + b}

+ e k~t G 2(X -- Ut, y). (29)

(B) If U = 0

(a) for K = k 2 < 0

~b(x, y, t) = e -k2t G l ( x , y) (30)

(b) for K = k 2 > 0

O (x, y, t) = e k2' G 2 (x, y). (31)

We note that Taylor solution representing a double infinite array of vortices

is obtainable as a special case of Eq. (30). Indeed, with k - n V/2v B(0) = 0, Vd '

( 4)1 A ( 0 ) = ~ - 6 0 - + 6 0 + ,Eq.(30) reduces to

7~ X d e - 2~z2vt/d2 ~k(~, y) = A o cos --d- cos (32)

which is Taylor [1] solution, where 2 and y are dimensional coordinates, and t- dimensional time.

Solution (27) in case (A) for K = - k 2 < 0 and U > 0 is the sum of a uniform stream U and a perturbation flow, the latter consisting of Kovasznay steady flow plus an arbitrary combination of plane waves G1 which decays exponentially in time. The relative importance of the steady flow to the unsteady plane wave combination depends on the relative magnitudes of a and A (0). As time increases the unsteady flow diminishes and the steady part becomes dominant. An analogous discussion can be given to the solutions (28) and (29) for U > 0.

The case (B) with U = 0 requires special attention. It may be treated as the limiting case of (A), as U ~ 0+, only in the appropriate half space. Thus, Eq. (30), which holds for all x and y, is the limit of Eq. (27) as U ~ 0 + only in the half space x > 0 but not in x < 0. Likewise, Eq. (31) is the limit of Eq. (29) as U ~ 0+ only in the half space x < 0 but not in x >___ 0.


4.4. Reduction to pseudo-steady f low

Solutions (30) and (31) for U = 0, being the product of e +kzt multiplied by a time-independent function, represent a kind of pseudo-steady flow in the sense that the flow pattern is steady but the magnitude of the motion decays or grows with time exponentially. It is interesting to note that in the general case when U > 0, the unsteady flow is also a pseudo-steady one when viewed from a frame X y moving with velocity U. Indeed, the stream function 7" of flow relative to this moving frame is obtainable from Eqs. (27) and (29) under the transformation X = x - Ut and 7" = 0 - Uy. The result is as follows:

(a) for K = - k 2 < 0

7"(X, y, t) = e -kh [ae -g2x/v cos {k x / l + k z / u Z y + b} + G 1 (X, y)] (33)

(b) for 0 < K = k 2 ~ U 2

[ e~t[e~2X/V { a f l , ~ y + be-~V1-k~/V~y}

7"(X, y, t) = + G2(X, y)], (k 2 < U 2) (34a)

e vh [eVX(ay + b) + G2(X, y)], (k 2 = U z) (34b)

(c) f o r K = k 2 > U 2

7"(X, y, t) = e k~ [ae k2x/v cos {k x/k2/U 2 - 1 y + b} + G2(X, y)]. (35)

The stream functions 7" in (33) to (35) are all equal to e • multiplied by a function of X and y alone, as such they clearly represent pseudo-steady flows in the sense explained above.

5. Unsteady flow: class B

We now derive another class of solutions to Eqs. (4) and (5) as follows. We first write the general solution to (4) for U > 0 as

7" = t t (X, y) e Kx/v = H (X, y) e rx/v +I~ (36)

where, from (5), H satisfies

O 2 H ~ 2 H 2 K ~ H ( ) - - K K 8 2 2 -[- --~y2 q- --U --OX + ~ 5 - 1 H = 0. (37)

The steady flow discussed in section 3 is now included in the representation (36) as a special case when H is independent of X, whence H = F (y).

Plane wave solutions to (37) exist in the form

H = h ( ~ ) , ~ = X c o s 0 + y s i n 0 (38)

698 w.H. Hui ZAMP

subst i tut ion of (38) in (37) yields

2 K c ~ K ) h"(~) + U ~ 5 - 1 h(~) = 0. (39)

Before proceeding to solve (39) we note that except in the trivial case when 0 = 0, a plane wave solut ion in the form ~ = h (~) e Kx/v cannot be obta ined from any plane wave solut ion of the form 7 / = g (~) err discussed in section 4, and vice versa, as this would require eKX/V/e ~t be a funct ion of ~ = X cos 0 + y sin 0 alone which is impossible except when 0 = 0. Therefore, the plane wave solutions and their superposi t ion to be given below based on (36) form a new and distinct class of solutions - class B - to the unsteady flow Eqs. (4) and (5), and hence to the Navier-Stokes equa t ion (1).

We also note that the steady flow solutions of section 3 are a special case of the plane wave solut ion (38) when 0 = ~ and hence H = h(y) - F(y) .

Solutions to Eq. (39) are of the form

h(~) = const, e me/v (40)

where, f rom (39)

m 2 + 2 i n K cos0 + K ( K - U 2) = 0. (41)

The two roots of (41) are

ml. 2 = - - g cos 0 __ x / K ( U 2 - K sin2 0). (42)

Depending on the sign of K ( U 2 - K sin2 0), we have the following cases"

(a) K = - k 2<0.

In this case

m12 = k 2 cos0 -1- ik N /U 2 "~- k 2 sin20 (43)

and

h = hl(~,O ) = A(O)ek2r176176 + k 2 sinZO/U2~ + B(0)} (44)

(b) 0 < K = k 2 "< U 2.

In this case

k 2 (45) rnx.2 = - c o s O + k x / U Z - k 2 s i n 2 0

and

h = h 2 ( ~ , 0 ) = e - k z~ eosO/U [A(0) e kVx-k2sin2~ + B(O) e -kVl-k2 sinZO/UZ~]. (46a)

Solut ion (46a) holds for any 0 < k 2 ~ U 2 and any - zc < 0 < rc except when k 2 ~-- 0 2 and 0 = ___ re/2. In tha t case m 1 = m 2 = 0 and h -- const. However , a


non-tr ivial solut ion for h exists and can be obta ined directly from (39) which

reduces to h" = 0. Hence

h = a y + b. (46b)

(C) K = k 2 > O 2 ,

In this case, we have either

k 2 COS 0 "~ k % / U 2 k 2 sin 2 0 (47 a) ml, 2 = -- _ --

and hence

h = h2(~, 0) (47b)

for [0[ -5_ 0o or rc - 0 o < l01 __< re, or

- k 2 cos 0 + i k , , /k 2 sin 2 0 U 2 (48 a) m l , 2 = __ - -

and hence

h = h3({,O)= A(O)e -k2~ ~176176 cos {k x/Fk2 sin 2 0 /U 2 - 1 { + B(0)} (48b)

for 0 o < 10[ < rc - 0o, where

0 o = s i n - 1 U k" (49)

As in section 4, superposi t ion of plane wave solutions for all - rc < 0 < rc yields more solutions for H (X, y) and hence, f rom (36), for 7 j. Thus we have

(a) f o r K = - k 2 < 0

H = H I ( X , y ) = i h l ( X cos0 + y s in0 ,0 )d0 (50)

and

7s (X, y, t) = e -k:' H I ( X , y) e -*:x/v. (51)

(b) for 0 < K = k 2 ~ U 2

H = H 2 ( X , y ) = i h2(X cos0 + y sinO, O)dO (52)

and

~t'(X, y, t) = e k2t Hz(X, y) e kax/v. (53)

(c) for K = k 2 > U 2

-7c +0o Oo i H - - H a ( X , Y ) = ~ + ~ + h 2 ( X c o s O + ys inO, O)dO

-r~ - 0 o n - O o

- 0 o n - 0 o

+ ~ + ~ h3(X cos0 + y s in0 ,0 )d0 (54) - ~ + 0 o 0o

700 w.H. Hui ZAMP

and

7t (X, y, t) = e k2t H 3 ( X , y) e k2x/v. (55)

We note that the steady flow solutions of section 3 may be recovered from these more general solutions (51), (53) and (55) for unsteady flow�9 For instance, with A(0 )= ~ 6 ( 0 - ~ ) and B(rc/2)= b we get, from (44) and (50),

H i ( X , y) = e cos {k~/1 + k 2 / U 2 y + b}, whence (51) reduces to 7 ~ = ae -k2x/v

�9 cos {k ~/1 + kZ/U z y + b}, which corresponds to the steady flow solution (8). Equations (51), (53) and (55) show that in all cases 7 ~ is equal to e +-k~t

multiplied by a function of X and y. As 7 ~ is the stream function of the unsteady flow when viewed from the frame of reference moving with velocity U, we conclude that unsteady flow of class B is also pseudo-steady in that frame of reference in the sense that the flow pattern is steady but the magnitude of the motion decays, or grows, exponentially in time.

6. Conclusions

Classes of exact analytical solutions to the two-dimensional incompressible Navier-Stokes equation are obtained for flows in which the local vorticity is proportional to the stream function perturbed by a uniform stream U. They are valid for any Reynolds number and include all existing solutions as special cases.

For steady flow it is shown that there exist only two possible types of such solutions: Kovasznay downstream flow of a two-dimensional grid and Lin and Tobak reversed flow about a flat plate with suction.

For unsteady flow, the two new classes of solutions are shown, when viewed from a frame of reference moving with the undisturbed uniform stream U, to be pseudo-steady in the sense that the flow pattern is steady but the magnitude of motion decays, or grows, exponentially in time.

Acknowledgements

The author wishes to thank K. Y. Fung, M. Tobak and S. P. Lin for their useful comments on the work. He is also grateful to the reviewer for his valuable comments on the first draft of the paper, especially the sections on unsteady flow.

Appendix A. The case of complex K

We first note that the analyses leading to the general solutions (8) (10) hold true for complex K as well as real K, and that a complex solution 7 ~ to (4) and (5) also leads to a complex stream function q, (= ~ + Uy) satisfying the Navier-


Stokes equation (1). However, a complex stream function does not represent a real flow. We shall now further show that neither the real part nor the imaginary part of the complex stream function ~ is a solution to the Navier-Stokes equation, and hence neither can represent a real flow.

Let 7 s = ~ + i ~ be a solution to (4) and (5) corresponding to a complex K = Kr + iKi , K i + O. Then

t-17 2 - U

( 8 ) U ~ (12) - v 2

V 2 ~ = K r ~ - K ~ (A3)

V 2 ~ = K r ~ + K ~ . (A4)

To see if the real part of the stream function, G = ~ + Uy, is a solution of (1) we substitute (A 3) for 172 0r into (1) and use (A 1) and (A 2) to simplify the result to get

~ ( Z ' ~ ) 0. (A5) Ki ~(x, y) -

For K~ 4= 0, condition (A 5) is equivalent to that

(x, y) - 0. (A 6)

Now, the general solution for T in steady flow is given by Eq. (6) with F ( y )

satisfying Eq. (7). Let K - K 2 / U 2 = (m + in) 2, where m and n are real, then m 4= 0 for K~ 4= 0. The solution to Eq. (7) is

F ( y ) = A e (m+i")y + B e -(m+i")y (A7)

Let I = K r / U , k = K i / U , A = a e i4' and B = b e ~z, where a, b, ~ and X are real, the real and imaginary part of 7 j are

= a e lx+"y cos(kx + n y + ~b) + b e tx-my cos(kx - n y + )0

= a e lx+"y sin(kx + n y + c~) + b e l ~ - " y sin(kx - n y + 7.).

(AS)

(A9)

Elementary but tedious calculations show that Eq. (A 6) cannot be satisfied by and ~ in (A8) and (A9) for Ki :t = 0. We therefore conclude that the real part

of the stream function ~ is not a solution of the Navier-Stokes equation (i). Similarly, the imaginary part ~ is not a solution of Eq. (1) either.

702 W.H. Hui ZAMP

References

[1] G. I. Taylor, On the decay of vortices in a viscous fluid. Phil. Mag. 46, 671-674 (1923). [2] L. I. G. Kovasznay, Laminar flow behind a two-dimensional grid. Proc. Cambridge Phil. Soc. 44,

58-62 (1948). [3] S. P. Lin and M. Tobak, Reversed flow above aplate with suction. AIAA J. 24, 334-335 (1986).

Abstract

This paper studies the two-dimensional incompressible viscous flow in which the local vorticity is proportional to the stream function perturbed by a uniform stream. It was known by Taylor and Kovasznay that the Navier-Stokes equations for flow of this kind become linear. From the general solution to the linear equations for steady flow, we show that there exist only two types of steady flow of this kind : Kovasznay downstream flow of a two-dimensional grid and Lin and Tobak reversed flow about a flat plate with suction. In the unsteady flow case, new classes of exact analytical solutions are found which include Taylor vortex array solution as a special case. It is shown that these unsteady flows are, as viewed from a frame of reference moving with the undisturbed uniform stream, pseudo-steady in the sense that the flow pattern is steady but the magnitude of motion decays, or grows, exponentially in time. All these solutions are valid for any Reynolds number.

R6sum6

Dans ce travail nous 6tudions l'6coulement plan d'un fluide visqueux incompressible dans lequel la rotation locale est proportionelle fi la fonction de courant perturb6e par un courant uniforme. Conform6ment aux travaux de Taylor et Kovasznay les 6quations de Navier-Stokes pour cet 6coulement~ deviennent lin6aires. Par consbquent nous utilisons la solution g6n6rale pour d~montrer que seulement deux cat6gories d'6coulement stationnaire peuvent exister: l'6coulement de Kovasznay en aval d'une grille plane, et l'6coulement invers6 de Lin et Tobak pour une plaque plane avec aspiration. Nous ~tudions aussi l'6coulement non stationnaire et nous d6couvrons des classes nouvelles de solutions exactes qui contiennent, en particulier, le r6seau de tourbillons de Taylor. Enfin nous d6montrons que ces 6coulements sont pseudo-stationnaires dans un syst6me de coordonn6es en mouvement avec le courant uniforme non perturb6; ce qui signifie que l'amplitude de l'6coulement stationnaire croit ou d6croit exponentiellment dans le temps. Toutes ces solutions sont valides pour tous les hombres de Reynolds.

(Received: July 21, 1986; revised : December 22, 1986 and March 27, 1987)

Date post:	28-Dec-2015
Category:	Documents
Upload:	vineeth-maxx
View:	29 times
Download:	3 times

Exact Solutions of the Unsteady Two-dimensional Navier-Stokes Equations

Documents