A Preliminary Study of the Burgers Equation with Symbolic ...

SYMBOLIC COMPUTATION OF THE BURGERS EQUATION 221
limitations in computer power and memory, the initial effort was necessarily limited in
scope. With accrued experience in use of the symbolic approach, and access to greater
computing power, future efforts can greatly expand the preliminary results. What is now
viewed as classic work by Feigenbaum [12] in the study of chaos in one-dimension (1-D),
numerical computations revealed patterns and universal numbers that became the backbone
of chaos theory. Analogous insights, patterns, and universal relationships in fluid turbulence
potentially may be discovered by expanded use of symbolic computation. What is
particularly encouraging about advances in symbolic software programs and in computer
hardware since Feigenbaum’s studies of chaos in the early 1970s is the potential to explore
2-D and 3-D cases.
2. THE BURGERS EQUATION
2.1. The viscous Burgers equation. The viscous Burgers equation for velocity U =
U0 + u, where U0 is a constant and ν is viscosity, is
∂u
∂t
=−(U0+u) ∂u
∂x +ν∂2 u
∂x 2.
The equation includes the local time derivative of momentum, the nonlinear convection term
(i.e., a quadratic nonlinearity), and the second order viscous diffusion term. The equation
does not contain a pressure derivative term, unlike the general Navier–Stokes equations.
Solutions to (1) experience a decay of momentum (hence, decay of kinetic energy) with
time due to the effects of viscosity.
Casting the equation in non-dimensional form yields
∂u ′
∂t ′ =−(1+u′ ) ∂u′
∂x ′ +Re−1 ∂2u ′
, (2)
∂x ′2
where the various non-dimensional variables are u ′ = u/U0, U ′ 0 = 1, x ′ = x/L0, t ′ = tU0/
L0, and Reynolds number, Re = U0L0/ν. L0 is a characteristic scale length, which corresponds
to a periodic spatial domain in the current study.
2.2. The inviscid Burgers equation. The inviscid Burgers equation, in non-dimensional
form, is simply
∂u ′
∂t ′ =−(1+u′ ) ∂u′
∂x ′,
(3)
in which the second order viscous term in (2) has been excluded. This is equivalent to
Re =∞. The inviscid equation contains the essential quadratic nonlinearity, but no viscous
damping. Thus any solution to this equation must conserve momentum and kinetic energy
with evolution in time.
2.3. Analytic solutions of the Burgers equation. Several closed-form analytic solutions
have been found for the Burgers equation for a wide range of initial and boundary conditions
(Whitham [22], Hopf [17], Cole [9], Fletcher [13, 14]). Fletcher [14] provided the following
solution for the non-dimensional viscous Burgers equation,
u(x, t) =
∞
−∞
x − ξ
t
exp{−0.5ReF}dξ
∞
(1)
exp{−0.5ReF}dξ, (4)
−∞