Diffusion equation coupled to Burgers' equation

Fluid Dynamics North-Holland

Research

2 (1987) 25-33

25

Diffusion equation coupled to Burgers’ equation Takeshi

MIYAZAKI

The National Institute for Environmental Received

25 August

Studies, Tsukuba, Ibaraki 305, Japan

1986

Abstract. Scalar diffusion in one-dimensional Burgers’ flow is considered. When the Prandtl number diffusion equation with convective term is reduced to a simple diffusion equation by a generalized transformation. An exact solution of an initial value problem is obtained in a closed form. When number is arbitrary, a similar analytical treatment is possible for limited classes of Burgers’ flow wave and single shock). The statistics of scalar field are discussed briefly.

is unity, the Cole-Hopf the Prandtl (expansion

1. Introduction It is well known

that Burgers’ equation

z.++ UU, = VU,, can be linearized u= and reduced

by the Cole-Hopf -2v--,

transformation

(Cole 1951, Hopf

1950)

fx f

to the diffusion

equation,

where v denotes the viscosity. Many investigations (Burgers 1974, Tatsumi and Kida 1972, Kida 1979) have made use of this property to explore the nature of Burgers’ turbulence. This paper generalizes the Cole-Hopf transformation and studies scalar diffusion in Burgers’ flow analytically. An interesting mechanism of scalar diffusion is clarified, which will give a new insight in the investigation of the diffusion phenomena in a flow field. The relation between the statistics of the scalar field and those of the velocity field is established for a high Reynolds number Burgers’ flow. Although Burgers’ turbulence is not a realistic model of the “Navier-Stokes turbulence”, it has been a good test case to check the validity of the closure theories. The results in this paper will offer another crucial check, since many of these theories have given poorer predictions on the turbulent scalar diffusion phenomena, in contrast to their success in predicting the statistical properties of the turbulent velocity field. The Section 2 deals with the case where the Prandtl number is unity, and the problem is solved exactly for arbitrary initial values. Cases of arbitrary Prandtl number are considered in Section 3. Summary and statistical discussions are given in Section 4. 0169-5983/87/$3.25

0 1987, The Japan

Society of Fluid Mechanics

T. Miyazaki / Diffwron equation coupled to Burgers’ equation

26

2. Generalized Cole-Hopf

transformation

Substituting the formal expansion of velocity u, U = U(J+ EUr+ E*u2 + . . .

(4)

into (1) and equating the like powers of E, we have the following set of equations: Uot+ uouox = VUDXX,

(5a)

%t+

b%4=~~lxx~

(5b)

u2t+

Gw2L+

%%x=~~2xx'

(5c)

The 0th order equation (5a) is just Bergers’ equation of motion. The first order equation (5b) describes the scalar diffusion in the Burgers’ flow field uo, where the Prandtl number is unity. The higher order equations, which contain source terms, can be thought to represent some diffusion-reaction phenomena, but are of little interest now. On the other hand, substitution of the similar expansion of the auxiliary function f=fo+&f,+E2f2+

...

(6)

into (2) gives the following relations:

ug= -2v+, u,=

f fo fl -

-2v i

(74

1

U’b)

fo /

-2v[(g9,(~)($j].

u2=

(74

It is evident that each term f, satisfies the same diffusion equation: (8)

f,, = VAX.

Inspection of (5a), (5b), (7a), (7b) suggests and proves that the diffusion equation coupled to Burgers’ equation (9)

@,+(~9)X=v~XX is reduced to the simple diffusion equation,

(loa)

!?f = V&X) by the generalized Cole-Hopf

cp=

i

transformation.

1

$ . x

Let uo(x) and (p,(x) denote the initial velocity and scalar values, respectively; value problem can be solved as

then an initial

(11)

T. Miyazaki / Diffusion equation coupled to Burgers’ equation

The Green

function

G(t,

t;x,)

G(x,

x;x,)

for (9) is given by substituting

G,,(x) = 8(x - x0) into (11):

I) . 02) II,x

x1

m J

=

27

dx,u,(x,)

/

x1 I

-CO

dx,u,(x,)

Since a high Reynolds number Burgers’ flow consists of a random array of expansion waves and shocks, the Green functions in these two cases are very important to clarify the scalar diffusion mechanisms in Burgers’ turbulence. 2.1. Expansion

wave

f(x, t> =

l

2/w

exp

i

x2

034

1’

4v(t+t,)

(I3b)

An enhanced Gaussian observed in this case.

diffusion

with the time-dependent

effective

diffusivity

v(t + to)/to

is

2.2. Shock f(x,

t) = 1 + exp( -ax

U(X,

t)

G(x,

t;

=

+ va2t),

~a(1 - tanh $(ax

x0) = -

- va*t)),

(14b)

1 + e-+

1

2&z

(14a)

(x-x0)* 4vt

1 + e- aX+Y(121

+ fasech2

+(ax - va2t) [ Erf ( -lg’)-Erf(

-x+c$2Vat)], (14c)

where the function Erf(x)

Erf denotes = $j” IT x

In this case, scalar gathers

the error function:

i.e.,

eVy2dy. to the shock front forming

G, = $a sech* i( ax - va’t),

the stationary

profile, (15)

which moves with the shock (Fig. 1).When two shocks collide, two humps of scalar carried by them coalesce into a new stationary shape (Fig. 2). We can conclude from the above observations (see Section 4) (a) that the scalar field in Burgers’ turbulence diffuse quickly to the shock fronts to form a series of sech* profiles, and

T. Miyazaki / Diffusion equation coupled to Burgers’ equation

Fig. 1. Scalar diffusion flow with a single shock;

in the Burgers’ Pr = 1.

Fig. 2. Scalar diffusion in the Burgers’ flow with two colliding shocks; Pr = 1.

(b) that a new sech2 profile is recovered soon after a collision between two shocks, total scalar value being conserved. Thus the scalar diffusion in a high Reynolds number Burgers’ flow is completely determined by the shock dynamics, and the statistical properties of scalar field can be related to those of velocity field. The statistics of Burgers’ turbulence have been investigated extensively, and these results are of fundamental importance in the statistical studies of scalar diffusion in Burgers’ turbulence. A brief discussion on this point will be given in Section 4.

3. Arbitrary Prandtl number If the diffusion coefficient K differs from the viscosity, i.e., if the Prandtl number (V/K) is not unity, the problem becomes much more difficult. The governing equations in this case are u, + UU, = VU,,)

@a)

‘t’t + (@‘>x = ‘@xx-

(16b)

We introduce a similar transformation

of the dependent variable, (17)

T. Miyazaki / Diffusion equation coupled to Burgers’ equation

which reduces

eq. (16b) to a more tractable

g, - K&x = :(I

29

one:

- Pr) uxg.

(18)

It is not easy to solve the full initial value problem because of the potential term in the right-hand side of eq. (18). Nevertheless, analytical solutions are possible for the expansion wave and the single shock, which are again important in clarifying the diffusion mechanisms. 3.1. Expansion

wave

It is interesting

to note that we have only to replace

G(x,

t;x,) =

2/z

1

v in eq. (13~) by

K:

09)

exp[-~Kt(::,,,(x-~xoi;]~

3.2. Single shock A Galilean transformation reduces eq. (18) for a single stationary shock, expressed u = - va tanh iax, to the following from containing a stationary potential term, & - Kg,, = - :va2(1

- Pr) sech2 iaxg.

by

(20)

This equation is familiar to physicists as a solvable potential problem in elementary quantum mechanics (Landau and Lifshitz 1965). The solution can be given in terms of hypergeometric functions. As the actual expressions are rather complicated, we content ourselves with giving concrete results for the case of integer Prandtl number, where the potential is reflectionless (Lamb 1980): G(x,

x0) =

t;

sech2 Pr : ax 28(;, + 2

Pr) [sechP’ +axLp,_l

* *. L2L,h(x,

(21)

t; x0)],

where h(x,

t; x0) =

(-l)p’ 2Pr+i(pr _ I)!

ern2t

$Pr&%

and B(a,

b) denotes L,=m

In deriving

the Beta function. tanh $ax-

L,

- 3 24Z

is the differential

+ x 2&i

, 11 (22)

operator

f&.

eq. (21) we have used the fact that the eigen function Gpr = CPrLPr_i

e Ox Erf :Prm i

. . . L,L,

GPr of eq. (20) is expressed

as

eikax,

C,, being the normalization constant. Figure 3 shows the Green function G(x, t;x,) of Pr = 2. The time and space variations are qualitatively the same as those of Pr = 1 in Fig. 1. We have the stationary profile as t --, co: G

= 00

ar(2Pr)sech2Pr 22Prr2

1 Tax.

(Pr) Note the modified

power 2Pr (see eq. (15)).

(23)

T. ~~,ya~ukj / Dijfiiion

30

equation coupled to Burgers

equation

Fig. 3. Scalar diffusion flow with a single shock;

in the Burgers’ Pr = 2.

The diffusion mechanisms of the case Pr f 1 (moderate and large) are qualitatively the same as those of Pr = 1, and we can repeat the conclusion in the previous section after replacing sech’ by sech2Pr. We should be careful, however, in the limiting case Pr --) 0, since the molecular diffusion is so strong that the scalar may not be trapped by the shock fronts and our picture of diffusion mechanisms would fail to be valid. The case of Pr = 0 (i.e., Y = 0, K = finite) can be treated separately:

G(x,

f; x,,) = <

(for x0 > 0)

Fig. 4. Scalar diffusion flow with a single shock;

(24)

in the Burgers’ Pr = 0.

T. Miyazaki / Diffusion equation coupled to Burgers’ equation

Fig. 5. Stationary numbers.

31

profiles

for various

Pr

A cusped stationary profile is formed as t + co (see Fig. 4) which seems to support the above conclusion even in this limit (Pr -+ 0). However, the x-scale in the figure is somewhat misleading (scaled using K instead of v). Figure 5 shows the stationary profiles for various Prandtl numbers with a fixed velocity field and a fixed x-scale. It is easy to see that scalar exchange between two adjacent shocks will take place as Pr + 0, when the flow field consists of multi-shocks. Then our picture of scalar diffusion must be modified.

4. Summary and discussion Analytical studies in the previous sections have revealed the structure and mechanisms of scalar diffusion (large and moderate Prandtl number) in Burgers’ flow. We can deduce here, directly, more conclusive results for the case of unit Prandtl number. Following the procedure in the analysis of Burgers’ turbulence (Burgers 1974, Tatsumi and Kida 1972. Kida 1979) we obtain the asymptotic expression of eq. (11) in the limit of large Reynolds number (Y << 1, t % 1):

(25) where i_= I

77i+l+Vi

+

vt

s(lli+l)-s(?i)t+

log(

2

Vi+1

-

vi+1

771

-

“;!;;;y,

vi

with

S(x) =

$*u,,(x’)dx’,

(27)

and nI’s denote the parameters which characterize the shock positions. Equation (25) represents a random array of sech2 profiles clearly, as anticipated in Section 2. Further assuming vt -=-c 1, we have,

with ei =

dx,

J?#+,(X)

(29)

Tr and 5

=

I

Vi+1

+

2

4i

+

s(llt+l) Vi+1

-

s(77i)

-

Vi

5.

(30)

T. Miyazaki / Diffusion equation coupled to Burgers’ equation

32

It is interesting to evaluate statistical quantities such as the mean Green function and scalar variance based on eq. (28) with eqs. (29), (30). For example, the mean Green function can be given as

if lx-xO-{l

l,

fj=

i 0,

<$p,

otherwise,

where I(t) is the characteristic shock spacing and the functions f(p) and h(I) are the distribution functions of shock strength and velocity, respectively, whose actual expressions have been determined theoretically (Burgers 1974, Tatsumi and Kida 1972, Kida 1979) and numerically (Tokunaga 1983). It is known that the integral scale J = jOW(~(x, t)u(x + r, t))dt is an invariant of Burgers’ turbulence. The case of J + 0 was considered by Burgers (1974), and Tasumi and Kida (1972). The distribution functions are given as

f(p)

=

+J

epp”(r),

l

h(l)=

I(t)&%

e

-$(+f

2 = 0.817, ’

(32b)

u

where the characteristic length 1(r) is proportional to t 2/3. These results are in good agreement with the numerical studies by Tokunaga (1983). Substituting (32a, b) into (31a), we have

(G(x, w,))

= (33)

Note the “non-Gaussian

turbulent diffusion”. In fact, the Fourier transform of eq. (33)

1 1+ ;/W(t)

-;&V(t) e

’

shows that the diffusion is much faster than a Gaussian diffusion. This large diffusion rate is due to the large-scale motion of the shock fronts, which are allowed in the case of J # 0. The case of J = 0 was considered by Kida (1979) theoretically. He gave concrete expressions of the functions f and g; however, the situation is quite subtle (Mizushima 1978, Mizushima and Saito 1985) and detailed discussions on this case will appear in a separate paper. Equation (28) does not give a good asymptotic evaluation for the case of Pr + 0, where the molecular diffusion causes scalar exchange between b-peaks. Numerical simulations, which are now in progress, will be fruitful in these conditions. Finally, we shall consider the scalar field spectrum briefly. Since the scalar field consists of a sequence of sech’” humps, the spectrum at high wave number will be dominated by that of the stationary profile in eq. (23). Taking the Fourier transform, we have ikx

=

F(Pr+

sjF(PrF(Pr2)

$j

dx

1 F(Pr + ivk)F(Pr r(Pr*)

- ivk) ’

(34)

T. Miyazaki

The last equality

/ Diffusion equation coupled to Burgers’ equation

is based on the assumption

vu C 1. If v + 0, keeping

33

Pr finite,

we have

(35) which leads to the inertial (&$)

subrange

spectrum

-k”.

This result is the most remarkable feature have in the limit v + 0 (Pr --) 0) with finite

of scalar

diffusion

in Burgers’

flow. However,

we

K,

(36)

which leads to

(ii)

-k-4.

This apparent discrepancy will be resolved by numerical studies. In this paper we have investigated the scalar diffusion in one-dimensional Burgers’ flow analytically. When the Prandtl number is unity, the generalized Cole-Hopf transformation gives the exact solution of the full initial value problem (Section 2). Several analytical results are given for the case of an arbitrary Prandtl number (Section 3); these clarify the mechanisms of scalar diffusion. Detailed considerations on the statistical quantities with their Pr number dependence are left for future work.

Acknowledgment

The author encouragement

would like to thank Professor throughout this work.

H. Hasimoto

for his continual

suggestions

and

References

Burgers, J.M. (1974) The Nonlinear Diffuon Equations (Reidel, Dordrecht). Cole, J.D. (1951) On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9, 225-236. Hopf, E. (1950) The partial differential equation II, + au, = u,,, Commun. Pure. Appl. Mech. 3, 201-230. Kida, S. (1979) Asymptotic properties of Burgers turbulence, J. Fluid Mech. 93, 359-377. Lamb Jr., G.L. (1980) Elements of Soliton Theory (Wiley, New York) 2.5, 2.6. Landau, L.D. and Lifshitz, E.M. (1965) Quantum Mechanics (Pergamon Press, Oxford) 23. Mizushima, J. (1978) Similarity law and renormalization for Burgers’ turbulence, Phys. Fluid 21, 512-514. Mizushima, J. and Saito, Y. (1985) Numerical study of the initial condition dependence of Burgers’ turbulence, Phys. Fluid 28, 1294-1298. Tatsumi, T. and Kida, S. (1972) Statistical mechanics of the Burgers’ model of turbulence, J. Ffuid Mech. 55, 659-675. Tokunaga, H. (1983) A numerical study of the Burgers turbulence at large Reynolds numbers, J. Phys. Sot. Jpn. 52, 827-833.