The interaction of a shock wave with a hot spot, a single vortex and a vortex street is studied within the framework of the two dimensional compressible Euler equations. The numerical results obtained by the pseudospectral method and the finite difference MacCormack method are compared. In both the methods the shock wave is fitted as a boundary of the computational domain

Hussaini, M. Y.

Salas, M. E.

Zang, T. A.

English

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UM-FITUD €dn SOLt nON$ TO SIM-WRTEX INTERACTIONS
(NASA-TH-85411) SHOCK-FITTED RULER
SOLUTIONS TO SHOCK-VORTEX INTERACTIONS
(NASA) 13 p HC A02/fiF 1101	 CSCL 01A
Manuel E. Salas
Thaws A. Zang
M. rousuff Hussaini
Report No. 82-24
N83-33850
Unclas
G3/02 36471
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A prel iini nary version of this report by the same
authors appeared as Tel 84481, flay, 1982.
August 16, 1982
SHOCK-PITTED PULER SOLUTIONS TO SHOCK-VORTEX INTERACTIONS
Manuel D. Salas
NASA, Langley Research Center
Thomas A. Zang
College of William and Mary
M. Yousuff Hussaini
Institute for Computer Applications in Science and Engineering
ABSTRACT
The interaction of a shock wave with a hot spot, a single vortex and a
vortex street is studied within the framework of the two-dimensional
compressible Euler equations. The numerical results obtained by the
pseudospectral method and the finite difference MacCormack method are
compared. In both the methods the shock wave is fitted as a boundary of the
computational domain.
Research of T. A. Zang was supported by NASA Grant No. NAG1-109. Research of
M. Y. Hussaini was partially supported by NASA Contracts No. NAS1-16394 and
NAS1-15810 while in residence at ICASE, NASA Langley Research Center, Hampton,
VA 23665.
Introduction
In their recent paper, Pao and Salas [1] presented a finite difference
solution to the two-dimensional Euler equations governing the phenomena of
shock wave interaction with an isolated vortex. Their study emphasized the
acoustic aspects of the problem. Zang, Hussaini, and Bushnell [21 extended
this numerical approach to the problem of turbulence amplification in shock
wave interactions. In this work it was necessary to resolve rather complex
fine-scale structure in order to draw meaningful conclusions about the
transient processes that dominate turbulence amplification. The present study
is a continuation of these efforts in two directions. First, to develop a
highly accurate pseudo-spectral method capable of resolving the crucial small-
scale structure on a relatively coarse grid. Second, to gain insight into the
nonlinear dynamics of the transient processes involved in the passage of a
shock wave over a single vortex, a vortex street and a hot spot.
Spectral methods have been demonstrated [31, [4] to be powerful
alternatives to finite difference methods for the numerical solution of smooth
flows. Recently, the work of Gottlieb, Lustman, and Orszag [51 and of Zang
and Hussaini [6] have shown their applicability to simple one-dimensional
compressible flows with shocks. The present paper discusses a Chebyshev
pseudo-spectral method that has produced reliable, accurate and efficient
solutions to complex, two-dimensional flows with a strong shock. Although the
main emphasis is on the spectral technique, solutions to the governing
equations obtained by the well known, second-order finite difference method
originated by HacCormack are also given. This is the method that was used in
[1] and [2). The finite difference results for the present set of problems
were calculated on a very fine grid and are used here for comparison with the
solutions obtained with the spectral method.
Statement of the Problem
The physical problem that we model corresponds to an infinite, initially
planar normal shock wave moving from left to right into a downstream region
containing a flow field representative of one or more vortices, or a hot
spot. In order to model the interaction of the shock wave with some given
flow field ahead of it, it is only necessary to compute the flow field
upstream of the shock. The physical domain, therefore, need consist only of
the region between some left boundary h(t), judiciously chosen such that it
will be far from the interaction region, and the shock wave front itself
xs (y,t). It is mapped onto the computational domain by the transformation,
x - h(t)
X ' 
x8(y.t) - h(t)
Y - tanh	 + 1
2
OF POOR QUALITY
T - t.
The computational domain is thus (X,Y) a [0,1) x [0,1].	 Note the
stretching that has been used to handle the infinite extent of the lateral
coordinate y. If the relative shock Mach number Ms is sufficiently high
(Ms
 > 2.08), the flow upstream of the shock remains supersonic. 	 In this
case, the left boundary corresponds to a supersonic inflow, and all dependent
variables can be prescribed on it. However, if the relative shock Mach number
is low, then radiation-type boundary conditions are used at the left
boundary.	 On the right, the computational region is bounded by the shock
wave. Downstream of the shock the flow field is given analytically. The flow
field immediately upstream of the shock, as well as the shape and velocity of
-2-
t
ORMI"M PACs IS
OF POOR QUALITY
the shock, are evaluated such that the Rankine-Hugoniot jump conditions and
the compatibility condition reaching the shock wave from the upstream side are
simultaneously satisfied.
The unsteady, two-dimensional, compressible, F.uler equations in the
computational plane are written in the form,
QT +AQX + y - 0 ,
where Q - [P,u,v,S] T
 and
U	
YXx YXY
	
0	 V	 Y X Y y	 0
a 2X /y	 U	 0	 0	 a2Y /y	 V	 0	 0
A 	 x	 B a	 x
a 2Xy /y	 0	 U	 0	 a2y/y	 0	 V	 0
0	 0	 0	 U	 0	 0	 0	 V
The natural logarithm of the pressure, the speed of sound, and the entropy are
represented by P, a, and S, respectively, and y is the ratio of specific
heats.	 The velocity in the
	 x	 and	 y	 4 iirections are
	 11	 and	 v,
respectively.
	 All variables are normalized with respect to reference
conditions at downstream infinity, as in [1]. 	 The contravariant velocity
components are defined by
U - Xt +uXx+vX
Y
and
V t+uX +vY .
Y
Subscripts denote partial derivatives with respect to the independent
variables.
-3-
I	 s
Solutign Techniques
	
OF POOR QUAL,i i"Y
Let k denote the time level and let At be the time step increment.
The time discretization of eq. ( 12) is then as follows:
Q - 11 - AtLkIQk,
Qk+l 1/2 (Qk + (1 - At^)Q] ,
where the spatial operator L represents an approximation to A6  + Ra Y.	 In
the finite difference KacCormack method, the operators 	 Lk	 and L	 are
evaluated as two points forward and two points backward differences in the
predictor (left) and corrector (right) levels, respectively. In the paeudo-
spectral method studied here, the solution Q is first expanded as a double
Chebyahev series,
M N
Q(X , Y , T) - F	 Y Q (T) t (C) ti (r),
p•0 q.0 Pq	P	 q
where
Ca2X-1,
il- 2Y- 1,
and 
T  
and T  are the Chebyshev polynomials of degrees p and q. The
derivatives appearing in the spatial operators are then evaluated as 	 .
I	 QX . 2 F	 Q(1,0) z z
P qu
o Pq	 P qua1	 •Q
j
	
	 -
where
4
-4-
	(1,0) . 2	 M
Qpq	 C	 I mQmqV
p M-P+l
m+p odd
OF POOR QUAUTY
and
c 0 . 2,
c p - 1, p > 0.
The Qy derivative is evaluated in a similar fashion.
The evaluation of the shock wave shape and velocity followed the same
procedure described in [1), except that in the spectral formulation, the
derivatives that must be evaluated on the upstream side of the shock are
expressed as Chebyshev expansions. At the left boundary, all variables were
specified for supersonic inflow. For the case of subsonic inflow, the two
velocity components and the entropy were specified, while the pressure was
computed from a quasi-one-dimensional characteristic as described in [83.
The pseudo-spectral method has a tendency to develop slowly growing
oscillations. Because of the global nature of this method they are spread
over the entire flow field rather than being confined to the vicinity of sharp
gradients. The underlying smooth solution can be recovered by a variety of
filtering techniques. The results presented here were obtained by applying a
von Hann window filter (see [61 for details) every 160 time steps. Another
practical consideration is the explicit time-step restriction. The Chebyshev
collocation points are clustered near the boundaries. Thus, smaller time-
steps must be used in the pseudo-spectral calculations.
-5-
C,,j  Y 4l
Results
	
nc POOR QUALITY,
Perhaps the simplest interaction to consider is that of a planar shock
wave with a ho- spot, as shown in Fig. 1. The flow field downstream of the
shock wave situated at x - 0 at t - 0 is taken as a quiescent field whose
temperature distribution a, is given. by
Cr - 2% exp(-((x-x 0) 2 + (Y-YO ) 2 Mr 2).
where for this case K - 0.25, r - .125, x0 - 0.5, y0 - 0.0, and Me - 3 at
t - 0. The vorticity distributions obtained by the finite differerce method
and by the spectral method after the shock wave has passed over the hot spot
are shown in Fig. 3. The finite difference solution presented here, and in
all cases that follow, was obtained with 75 mesh points in the X direction and
50 in Y. The spectral solutions were all obtained with 33 collocation points
in the X direction and 17 in Y. There is very little difference between the
two solutions. Both show the two counter-rotating vortices upstream of the
shock, which is typical of this interaction. See (2) for more details on the
physics.
Figure 2 shows the velocity field for a single vortex about to interact
with a shock wave of the same initial strength as in the previous case. The
downstream conditions here are obtained by assuming a constant density field,
calculating the velocity from the stream function,
- 27t log	 r2 + (x-x0 ) 2 + 
(y-y0)2
the pressure from Bernoulli's relation, and the temperature from the equation
of state. For the case shown in Fig. 2, the circulation K - 2 and the
-6-
1
re-...	 - ,	 r,-..
OF P CC R (,°.5  F 
^i
a	 _	
-^
softening scale r - 0.1. This model approaches an idealized incompressible
point vortex at large distances but is much smoother near the center. Figure
4 shows the resulting pressure field after the shcck wave has passed over the
vortex. Overall, the results are qualitatively very similar, although the
spectral method seems to be resolving the pressure field more accurately. See
(1) for more details on the physics.
Finally, Figs. S and 6 show the results for the interaction with the
Karman vortex street that simulates the conditions of the experiment reported
in (7). For this case, the stream function representing the vortex is given
by the difference of 41+ and 4o
-
 where
t^ - K
log[cosh(^^ r2 + (y t 1/Z b) 2 ^ - COS('-' (x *1/2c))]-
To match the experiment, the circulation, core radius, shock Mach number and
vortex separation parameters were determined as K - 0.186, r - 0. 1, Ms
1.3, c - .33, and b - .048. For this calculation, the inflow Mach number was
subsonic and radiation boundary conditions were applied at the left
boundary.	 The results shown in Fig. 6 are in agreement with the
'-	 experimentally observed [7) longitudinal compression and lateral elongation of
the vortex field after passage through the shock. 	 The finite difference
_-
	
	 results are noticeably smoother than the spectral ones. However, it is well
known that the idealized Karman vortex street is unstable "or all but one
, special raLio of the horizontal and vertical. separations. The downstream flow
has the stable ratio. Thus, it is likely that the upstream flow is physically
unstable. It may be that the spectral results have captured this phenomena,
but more computations are needed to settle the issue.
I
ij
Ref erg
s
i (lj Pao, S. P. and Salas, M. J.: A Numerical Study of Tiro-Dimensional Shock
Interaction. AIM Paver 81-1205. Presented at the AIAA 14th Fluids
and Plasma Dynamics Conference, June 23-25, 1981, Palo Alto, CA.
(2j Zang, T. A.; Hussaini, M. Y.; and Bushnell, D. M.: 	 Numerical	 i
Computations of Turbulence	 Amplification in Shock Wave Interactions.
AIAA Pager No. 82- 0293.	 Presented at the AIM 20th Aerospace
Sciences Meeting, January 11-13, 1982, Orlando, FL.	 j
I
131 Orszag, S. A. and Bells, L. C.: Transition to Turbulence in Plain
Poiseuille Flow and Plain Couette Flow, J. Fluid Mech., Vol. 96, 1980,
pp. 159-205.
(4) Wray, A. and Hussaini, M. Y.: Numerical 8xperiments in Boundary-Layer
Stability. AIAA Pager No. 80-0275. Presented at the AIM 18th
Aerospace Sciences Meeting, January 14-16, 1980, Pasadena, CA.
(5) Gottlieb, D.; Lustman, L.; and Orszag, S.: Spectral Calculations of One-
Dimensional Inviscid Compressible Flows, SIAM J. Sci. Statis• Compute,
Vol. 2, 1981, pp. 296-310.
(6) Zang, T. A. and Hussaini, M. Y.: Mixed Spectral-Finite Difference
Approximations for Slightly Viscous Flows, Proc. of the 7th Intl.
Conf. on Numerical Methods in Fluid Dynamics, Springer-Verlag, 19810
pp. 461-466.
s
1965 0 pp. 216-223.
Theoretical and Practical Aspects of Some
obleme in Fluid Dynamics, SIAM J. A001,
L9-446.
-4--
(7) Dosanjh, D. S. and Weeks, T. M.: Interaction of a Starting Vortex an
well as a Vortex Street with a Traveling Shock Wave, AIM J., Vol. 13,
ORIGINAL PACE SIB'
OF POOR QUALITY
%f' rrJ
...---
•	 w_1 ^^^ 1 1 1 1 i• t t
. 1 ^`^ 1^^^ 1 1 1 1 1 1 t t t t
1	 1 1 t i 1/•• t t
}	 ,	 s11?11t11 ► •t•t
.	 a	 ` ^ `.^,,.,// / /III r r • r . t
Fig. 1: Surface plot of entropy	 Fig. 2: Velocity field of a single vortex
distribution for a hot spot about 	 about to interact with a Mach 3 shock wave
to interact with a Mach 3 shock
	
(solid curve). The velocity vectors repre-
wave.	 sent perturbetfon from the mean values.
Fig. 3: Vorticity fields at t 0.20 computed by pseudo-spectral (left) and finite
difference (right) methods for a hot spot after interaction with a Mach 3 shock
wave (solid curves).
0
0
	 0	
i :`
N e• 4: Isobars at t - 0.20 computed by pseudo-spectral (left) and finite
cifference (right) methods for a single vortex after hilerArtion .+ith a
..-cn 3 shock rave.
••
•^ -. •. r rr41 .. ^ ,►rte •- ,-
^...\ ♦. ^^r^\..Ire\^
1	 1/	
1	 1 ••	 1•.^	 t
/	 1
I/' .	 . 1	
'r'.i(^..9.U^t:lit1 ^. iL ^ t ....+
OF POOR QUALITY
Fitz. 5: Velocity field (left) and vorticity contours (right) for a Karman vortex
street about to interact with a Hach 1.3 shock wave (solid curves). The velocity
vectors represent perturbations from mean values. Negative contour levels are
Are indicated by dashed lines.
..........
.
 . . . .. . . . . 	 .-.- 	_.^..._.
...........
...........
. . . . . . . . . . .
	 ,_ r
.........	 .-wr-^.-ww.r
............ - - - - - - -
•^ ^r^'.	 Vii/•	
•fir.'	 ^; 1^.;`	
: ^•i1
.	 J
Fig. 6: Velocity fields (top) and vorticity contours (bottom) at
	 0.36 computed
by pseudo-spectral (left) and finite difference (right) methods for a Kaman vortex
street after interaction with a Mach 1.3 shock wave (solid curves). The velocity
vectors represent perturbations from mean values. Negative contour levels are Indi-
cated by dashed lines.
	 ^_ {
A


Shock-fitted Euler solutions to shock-vortex interactions

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