The interaction of turbulence with shock waves
G.P. Zank∗ , Ye Zhou† , W.H. Matthaeus∗∗ and W.K.M. Rice‡
∗
Institute of Geophysics and Planetary Physics, University of California, Riverside
†
Lawrence Livermore National Laboratory, University of California, Livermore
∗∗
Institute of Geophysics and Planetary Physics, University of California, Riverside, and // Bartol Research
Institute, The University of Delaware, Newark
‡
The University of St Andrews, St Andrews, Fife
Abstract. The interaction of turbulence and shock waves is considered self-consistently so that the back-reaction of the
turbulence and its associated reaction on the turbulence is addressed. Upstream turbulence interacting with a shock wave is
found to mediate the shock by 1) increasing the mean shock speed, and 2) decreasing the efficiency of turbulence amplification
at the shock as the upstream turbulence energy density is increased. The implication of these results is that the energy in
upstream turbulent fluctuations, while being amplified at the shock, is also being converted into mean flow energy downstream.
The variance in both the shock speed and position is computed, leading to the suggestion that, in an ensemble-averaged sense,
the turbulence-mediated shock will acquire a characteristic thickness given by the standard deviation of the shock position.
Lax’s geometric entropy condition is used to show that as the upstream turbulent energy density increases, the shock is
eventually destabilized, and may emit one or more shocks to produce a system of multiple shock waves. Finally, turbulence
downstream of the shock is shown to decay in time t according to t −2/3 .
INTRODUCTION
MATHEMATICAL FORMULATION
The solar wind is intrinsically turbulent and shock waves,
either propagating or formed upstream of obstacles such
as planets and comets, must interact with low-frequency
turbulent fluctuations. Many shocks, especially quasiparallel shocks, generate turbulence in their very extended foreshocks, and this interacts eventually with the
shock ramp. In this paper, we take the first steps towards
addressing the interaction of turbulence and shock waves
at a self-consistent level. In this, we adopt a perspective rather different from previous studies in that we employ a statistical description for the fluctuations from the
outset, rather than attempting to use a linearized perturbation analysis of individual waves and fluctuations. A
schematic of the problem that we address is given in Fig.
1. Figure 1 illustrates a shock distorted by its interaction
with upstream turbulence, and the subsequent transmission of the turbulence into the downstream subsonic region. We wish to compute the mean location and speed
of the shock as a function of the upstream energy density of turbulence and of course the variance of the shock
position and speed. As a consequence, the amplification
of the turbulence at the shock is also addressed, together
with its subsequent dissipation downstream.
For hypersonic flows in which ram pressure dominates
thermal quantities, the gas equations reduce to the inviscid Burgers’ equation,
∂u ∂ 1 2
∂u
∂u
+
u = 0 ⇐⇒
+u
= 0.
(1)
∂t ∂x 2
∂t
∂x
Solutions to equation (1) require the insertion of a shock
to counteract wave steepening and breaking. If we let
x = φ (t) describe the shock location, then the shock
normal is given by n = (−φt , 1), and the R-H condition
is given classically by
1 2
−φt [u] + u = 0.
(2)
2
To solve (2), we may prescribe the initial data
u1
u(t = 0) =
,
u2
and to ensure that the geometric entropy condition [1]
holds, we must have u2 < u1 . One obtains immediately
from (2) and (3) the well-known classical result for the
shock speed
1 2
u
1
= (u1 + u2 ) ,
φt = 2
(4)
[u]
2
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
417
(3)
UPSTREAM
1
shock wave by a mean and fluctuating part, so that u =
ū + u and u = ū after ensemble averaging to eliminate
fast time scales. Of course, the R-H condition (2) continues to hold exactly for u, and x = φ (t) as before. However, we shall assume that a detailed solution is either
inaccessible or undesirable and instead seek a statistical
formulation of the problem. For the 1D problem (1) and
(2), the shock position fluctuates in response to variation
in u, i.e., with respect to u , so that φ (t) = Φ(t) + φ (t),
φ = Φ(t). The mean field form of the R-H condition is
thus given by
DOWNSTREAM
2
U1
U2
UPSTREAM
TURBULENCE
SHOCK WAVE
u′2
1
−Φt [ū] − φt u + ū2 + u u = 0.
2
DOWNSTREAM
TURBULENCE
(5)
The boundary condition associated with the fluctuating
component is
−φt [ū] − Φt u − φt u − φt u + ūu
1 u u − u u = 0.
(6)
+
2
x
FIGURE 1. Schematic of the problem in the stationary shock
frame. The top figure depicts a shock wave that is highly distorted by the repeated random interaction of upstream turbulence with the shock. Quantities upstream of the shock, where
the flow is supersonic, carry the subscript 1 and downstream
quantities have the subscript 2. The flow velocity is denoted
by U. The problem is to compute the mean shock velocity,
position, and variance of these quantities in the presence of
upstream turbulence. In addition, as illustrated in both the top
and bottom figures, we wish to compute the transmission characteristics of the upstream turbulence (the amplification it may
experience) and its subsequent decay as it is dissipated into the
downstream flow. The bottom figure illustrates schematically
the spatial evolution of the turbulent fluctuation energy from
upstream of the shock to downstream.
which is constant and yields x = φ (t) = 12 (u1 + u2 )t for
the shock position.
In the following section, we reformulate the classical problem (1)-(3) to include turbulent fluctuations upstream of the shock. Subject to the restrictions imposed
by our assumption of a hypersonic flow, the basic questions that we address are 1) what is the propagation speed
of a shock in a turbulent medium? 2) Can upstream turbulence destabilize an apparently stable shock wave? 3)
What is the effective or "averaged" thickness of the shock
as a result of the presence of turbulent upstream fluctuations? 4) How strongly is turbulence amplified by a
shock? And 5) What is the decay law for turbulence
downstream of a shock?
Consider again the classical inviscid Burgers’ equation
problem (1)-(3), but now assume that the upstream flow
is turbulent. We may express the flow incident on the
418
By comparing the time scale for spectral transfer in
the fluctuating quantities to the transmission time scale
for fluctuations across the shock, we reach the important conclusion [2] that spectral transfer is unimportant
across narrow shocks for most fluctuations in the turbulence spectrum, and derive a lower bound on the scale
size of upstream turbulent fluctuations that can interact
“linearly” with the shock wave. Since spectral transfer
is unimportant across the shock, the quantities u u and
φt u are independent of the fast nonlinear coupling time
scale in the transition from upstream to downstream of
the shock. We therefore have the remarkable result that,
across the thin shock, the boundary condition for the fluctuating components in a mean-field decomposition of the
variables satisfies the linear equation
−φt [ū] − Φt u + ūu = 0.
(7)
Equation (7) possesses the formal structure of a linearized equation by virtue only of the cancellation of certain terms. We emphasize that (7) does not result from the
linearization of the exact boundary condition (6) and we
have not assumed that the fluctuations are of small amplitude. The argument leading to (7) is a form of rapid
distortion theory.
By employing (7), we can determine various correlations across the shock. Details can be found in [2].
The Rankine-Hugoniot conditions for hypersonic gas dynamic shocks in a turbulent flow number four and are
conveniently collected together as
[u ū] [u ]
[u ]2 1 ū2 + u u Φt = 1 −
−
; (8)
2
[ū]
[ū]2
[ū]2
[ū] τ
2
2
2
−Φt u + ūu = [ū] u exp
; (9)
ν̄
2
u = − [ū] u1 u2 τ ; (10)
1
. (11)
ū1
In the above, 2ν̄ is the viscous scale length for the shock,
denotes the correlation length, and τ is a characteristic interaction time of fluctuations with the shock. The
Rankine-Hugoniot conditions relate the upstream and
downstream states across a shock, but now generalized
to include the turbulent energy density u 2 , the correlation length , and the cross-correlation across the shock
u1 u2 . In this regard, equations (8) - (11) describe statistically the interaction of turbulence with shocks, with
the back-reaction of the turbulence accounted for selfconsistently. This distinguishes our approach from the
linearized approach of [3].
Before solving the R-H conditions explicitly, let us
consider the free decay of turbulence downstream of the
shock (see Fig. 1, bottom). Immediately downstream of
the shock, the turbulence may be characterized by u2 2 and 2 . By means of an energy-containing model, using
1-point correlations, we can derive the relation u 2 ∼
t −2/3 (see [4] for a related approach).
RESULTS AND CONCLUSIONS
The Rankine-Hugoniot conditions yield a quadratic
equation in the mean shock speed Φt , and the remaining
downstream quantities and correlations can be derived
[2]. Two other quantities of interest are the variance in
the shock speed and is the variance in shock position.
The latter quantity may be interpreted as the “turbulent
shock thickness.”
In Figures 2 - 4, we plot various solutions of the R-H
conditions (8) - (11). A solution to the quadratic equation for the mean shock speed Φt is plotted in Fig. 2 as a
function of normalized upstream energy density u1 2 .
From the quadratic equation for Φt , two solutions exist for each u1 2 , and, the classical result is recovered
in the absence of turbulent fluctuations. The second solution (not shown) begins from the classical solution and
decreases with increasing u1 2 . However, not all solutions are admissible. For a shock to exist, Lax’s geometric entropy condition [1] must be satisfied. This requires
that the forward and backward characteristics which intersect at the shock wave can both be traced back to the
initial data. The curve plotted in Fig. 2 can be shown to
be admissible [2]. By contrast, the decreasing solution
is never intermediate to the forward and backward characteristics. In fact, the positive root solutions illustrated
in Fig. 2 continue to increase with increasing u1 2 to
above Φt = 1, and these larger solutions are also inad-
419
1
0.9
0.8
Φt
[] = [ū]
0.7
0.6
0.5
0
0.001
0.002
0.003
0.004
0.005
<u′21 >
FIGURE 2. A plot of the mean shock speed Φt as a function
of the upstream turbulence energy density u1 2 normalized to
the square of the upstream mean flow speed. In the absence
of upstream turbulence, the classical result is recovered for
u1 2 = 0.
missible. Thus, unique stable shock solutions exist for
only a limited range of upstream turbulence levels. Outside these solutions,the shock becomes unstable and we
cannot assume that either a mean speed or position for
the shock is possible. Thus, upstream turbulence, at least
for the simple hypersonic 1D case considered here, leads
to an increase in shock speed as the turbulent fluctuations
merge with and transmit through the shock, but as the intensity increases, upstream turbulence renders the shock
unsteady.
Figure 3 plots the corresponding amplifications of the
upstream turbulence by the shock, and Figure 4 shows
the variance of the shock speed φt 2 (essentially proportional to the variance in shock position, i.e., “turbulent
shock thickness”) and the cross-correlation u1 u2 .
Our results may be summarized as follows.
1. Although based on a simple energy-containing
model for an idealized hypersonic fluid, our particular results emphasized the importance of a selfconsistent coupling of turbulence and the mean
shock variables. Specifically, the mean shock speed
was found to increase with increasing levels of upstream turbulence. Correspondingly, the efficiency
of upstream turbulence amplification by the shock
decreased i.e., u2 2 /u1 2 was a decreasing function of increasing u1 2 . The implication of this result is that the energy in upstream turbulent fluctuations, while being amplified at the shock, is also being converted into mean flow energy downstream.
Thus, models which consider the amplification of
turbulence at a shock wave without considering the
subsequent "turbulent mediation" of the shock will
tend to over-estimate the levels of downstream turbulence.
2. The variance in the shock speed increases with increasing values of the upstream turbulent energy
density. The variance in the shock position can be
used as a measure of the "shock thickness", and we
find that the "ensemble averaged" shock is no longer
infinitesimally thin but instead has a
shock thick-
0.009
2.25
0.008
0.007
0.006
<u′22>/<u′21>
2
<u′22 >
0.005
0.004
0.003
1.75
0.002
0.001
0
0
0.001
0.002
0.003
0.004
1.5
0.005
2
1
<u′ >
FIGURE 3. Two plots of the energy density in downstream
turbulent fluctuations u2 2 . The solid line (left axis) plots
the downstream energy density normalized to the upstream
ram energy and the dashed line (right axis) is the ratio of
the downstream to upstream turbulent energy densities and
is therefore a measure of the efficiency with which upstream
fluctuations are amplified by the shock wave. Thus, although
the amplitude of the energy density in transmitted fluctuations
increases with an increasing upstream turbulent energy density,
the corresponding efficiency decreases.
0
0.008
0.007
-0.001
0.006
2
<φ′t >
-0.002
0.004
<u′ u′ >
ACKNOWLEDGMENTS
0.005
0.003
This work was supported in part by an NSF grant ATM0296113.
-0.003
0.002
REFERENCES
0.001
0
ness length scale given by Lshock ∼ φ 2 . This
suggests the possibility of introducing a “turbulent
viscosity” to model shock structure in the presence
of upstream turbulence.
3. It was found that surprisingly low levels of upstream turbulence energy density could destabilize
the shock. As the mean shock speed increased with
increasing u1 2 , Lax’s geometric entropy condition
[1] was eventually violated and an upstream state
could no longer be traced back to the initial data. In
this case, a steady shock wave is no longer possible
and instead a combination of shocks is needed to
satisfy the Riemann problem. Thus, high levels of
upstream turbulence would drive the original shock
to eventually emit one or more shock waves, so producing a system of multiple shocks, the detailed
study of which is beyond the scope of the present
approach.
4. Finally, we found that in the case of a steady shock
mediated self-consistently by turbulence, the emitted amplified turbulence decayed with a t −2/3 dependence in the downstream region.
0
0.001
0.002
0.003
0.004
-0.004
<u′21>
FIGURE 4. Plot of the shock speed variance φt 2 (solid
curve, left axis) as a function of the upstream turbulent energy
density. The shock speed variance is proportional to the variance in shock position. The dashed curve (right axis) is a plot
of the cross-correlation of the upstream and downstream turbulent fluctuations u1 u2 as a function of the upstream turbulent
energy density. Note that u1 u2 increases from zero and then
decreases becoming negative with larger values of u1 2 .
420
1. Lax, P.D., Hyperbolic Systems of Conservation and
the Mathematical Theory of Shock Waves, SIAM,
Philadelphia (1990).
2. Zank, G.P., Zhou, Ye, Matthaeus, W.H., and Rice,
W.K.M., Phys. Fluids, in press (2002).
3. McKenzie, J.F., and Westphal, K.O., Phys. Fluids,
11, 2350 (1968).
4. Zank, G.P., Matthaeus, W.H., and Smith, C.W., J.
Geophys. Res., 101, 17,093 (1996).