tvd scheme new flux limiter.pdf

Evaluation of TVD high resolution schemes for unsteadyviscous shocked ¯ows

Virginie Darua, Christian Tenaudb,*

aENSAM, Lab. Sinumef, 151 Bd de l'Hopital, 75013 Paris, FrancebLIMSI-UPR CNRS 3251, BP 133, Campus d'Orsay, F-91403 Orsay Cedex, France

Received 14 August 1998; received in revised form 25 November 1999; accepted 1 February 2000

Abstract

The goal of this study is to evaluate the accuracy of several high resolution total variation diminishingschemes in solving complex unsteady viscous shocked ¯ows. Two types of discretization, namely acombined time and space discretization, and an independent time and space discretization areconsidered. Both methods are associated with several limiters, among which a more accurate new familyof limiters depending on the local wave velocity. The accuracy properties of each scheme are ®rstreviewed on inviscid 1D and 2D test cases, in order to establish a ranking with respect to theirdissipative error. We then study the ¯ow produced by the interaction of a re¯ected shock wave with theincident boundary layer in a shock tube. The calculations are performed for two values of the Reynoldsnumber. At Re � 200, convergence is attained and it is shown that the combined time and spacediscretization method converges faster. Good classical limiters do almost the same job as the new familyof limiters. When the Reynolds number is increased to the value of 1000, the ¯ow becomes much morecomplex. Although convergence is hard to reach, the close examination of the results leads us toconclude that the combined time and space discretization method associated with the new limiter givesfrom far the best results. 7 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Shock boundary-layer interaction; Shock tube; Re¯ected shock; Computational ¯uid dynamics

Computers & Fluids 30 (2001) 89±113

0045-7930/01/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.PII: S0045-7930(00)00006-2

www.elsevier.com/locate/compfluid

* Corresponding author. Tel.: +33-1-6985-8130; fax: +33-1-6985-8088.E-mail address: [email protected] (C. Tenaud).

1. Introduction

Total variation diminishing (TVD) high resolution schemes have proven to be very e�ectivefor computing inviscid ¯ows. However, for high Reynolds number viscous ¯ows, the numericaldi�usion which is introduced by the limiter functions can have undesirable e�ects, generallyleading to misrepresentation of the viscous e�ects. There exists now a large number of limiters,among them the Minmod limiter which is known to be very di�usive, the Superbee limiter ofRoe, well adapted to the computation of discontinuities but which tends to ``square'' smoothpro®les, or the Van Leer limiter which gives intermediate results and can be taken as areference. All of these classical limiters do not depend on the Courant number, and retain onlya simpli®ed condition to satisfy the TVD constraints. We must underline that these limiters areassociated with schemes which can eventually depend on the Courant number.More recently, in the inviscid case, several authors [1±3] have used a family of limiters with

some success which, if not really new (the ®rst work in this way was done by Roe and Baines[4]), did not receive much attention for ¯ow computations until now. The main interest of thisfamily of limiters lies in the fact that, in addition to prevent numerical oscillations, it canincrease the order of accuracy of the scheme from 2 to 3 in smooth regions, at least in thelinear scalar case. As these limiters depend on the local Courant number, they are well adaptedonly for unsteady ¯ows.For the computation of unsteady ¯ows, not only the limiter, but also the scheme which is

used is of importance. Our aim here is to evaluate the accuracy of a number of TVD highresolution schemes to solve complex unsteady viscous ¯ows. We will focus our study on thedi�cult problem of the viscous interaction between the boundary layer generated behind ashock wave travelling in a shock tube and the re¯ected shock produced after its re¯ection atthe end wall. This ¯ow con®guration can be encountered in high enthalpy ¯ow facilities or inindustrial equipments for the distribution of pressurized gas for instance, where a goodknowledge of the phenomena involved is of particular interest in terms of safety purposes.Within an incident shock Mach number range, the interaction results in a lambda-shape likebifurcated shock wave pattern, ®rst described in 1958 by H. Mark in the laminar case. Thisphenomenon can be explained by the fact that the stagnation pressure of the boundary-layerbecomes lower than the pressure behind the re¯ected shock. Therefore, the ¯uid cannot passunder the re¯ected wave and a separated ¯ow region appears. This separated region forms a``bubble'' which is dragged upstream with the re¯ected shock (Fig. 1). The analysis of recentexperimental data [5] suggests that the ¯ow is turbulent in the bubble carried along under there¯ected shock.To our knowledge, very few numerical investigations of this problem have been reported in

the literature (see [6] and references therein), until the recent work of Weber et al. [7], whohave carried out detailed studies on this problem, using the FCT algorithm. They havereviewed the results of sensitivity to the Reynolds number, the Mach number and the walltemperature. But, as it is mentioned in their paper, it is not clear whether or not convergencewith respect to the grid re®nement was attained. In this paper, we highlight the di�culty toobtain a converged solution when the Reynolds number value exceeds a few hundred. In themeantime the above new family of limiters, associated with a Mac-Cormack type scheme, isclearly demonstrated to be the more accurate for solving this kind of problem. Finally, we

V. Daru, C. Tenaud / Computers & Fluids 30 (2001) 89±11390

think that the re¯ected shock-boundary layer interaction problem could constitute aninteresting benchmark test case to evaluate the accuracy of numerical schemes for thecomputation of unsteady viscous ¯ows.The paper is organized as follows: after a brief recall of the governing equations (Section 1),

we present in Section 2 the so-called ``O3'' and ``O3Sup'' limiters, and the di�erent numericalschemes we have used (a predictor±corrector Mac-Cormack scheme, a Harten-Yee second-order ®nite di�erence upwind scheme, and a MUSCL scheme). In Sections 3 and 4, we reportthe results obtained for inviscid 1D and 2D test cases. The rest of the paper is devoted to theevaluation and the comparison of the di�erent numerical approaches for solving the re¯ectedshock-boundary layer interaction problem. The Reynolds number is naturally a key parameterin this study, and we limit ourselves to relatively low values (below 1000) in order to limit thegrid size requirements.

2. Governing equations

Our study is limited to the two-dimensional case (it could represent the ¯ow in the middlesection of a cubic shock tube). The ¯ow is treated as laminar, so no turbulence model is used.We solve the Navier±Stokes equations written in cartesian coordinates, expressed using non-dimensional variables as:

wt �ÿf�w� ÿ f v�w, wx, wy�

�x�ÿg�w� ÿ gv�w, wx, wy�

�y� 0 �1�

where w is the vector of conservative variables �r, ru, rv, rE), r being the density, u and v the¯uid velocity components, and E the total energy. f and g are the inviscid ¯uxes, f v and gv arethe viscous part of the ¯uxes. We have:

f�w� �

0BB@ruru2 � pruv�rE� p�u

1CCA g�w� �

0BB@rvruvrv2 � p�rE� p�v

1CCA

Fig. 1. Bifurcated shock pattern.

V. Daru, C. Tenaud / Computers & Fluids 30 (2001) 89±113 91

f v�w� � 1

Re

0BBBB@0txx � l�ux � vy� � 2muxtxy � m�uy � vx�utxx � vtxy � gm

Prex

1CCCCA gv�w� � 1

Re

0BBBB@0tyx � txytyy � l�ux � vy� � 2mvy

utyx � vtyy � gmPr

ey

1CCCCAAs usual, p denotes the pressure, given by the perfect gas law p � �gÿ 1�re, where e is thespeci®c internal energy, and Re is the Reynolds number. For simplicity we restrict ourattention to an ideal gas with constant speci®c heat ratio g � 1:4, constant viscosity coe�cientsl and m and Prandtl number Pr � 0:73:

3. Numerical schemes

We now present the numerical schemes we have used in the computations.

3.1. Combined time and space discretization: the MCO3 scheme

In such an approach, we use the Mac-Cormack scheme, to which is added a limitingcorrection term in order to render the scheme TVD. Beside classical limiters, we have usedrecent speci®c limiters which particularity is that they hybridize a third order scheme and a ®rstorder scheme in the scalar linear case. When implemented in a correction term added to asecond order scheme, they can act either by increasing the order of the scheme to three insmooth regions, either by diminishing it to one to satisfy TVD constraints. Let us describe thislimiter by considering the 1D linear model equation:

@u

@t� a

@u

@x� 0

where a is a positive constant, and u � u�x, t�: If we denote by fnj the discrete quantity f

estimated at a grid point xj � jdx and at a time t � n dt, dt and dx being the time and spacesteps, a limited Lax-Wendro� scheme can be written:

un�1j � unj ÿdtdxÿFj�1=2 ÿ Fjÿ1=2

�where n � adt=dx is the Courant number, dunj�1=2 � unj�1 ÿ unj , and Fj�1=2 � aunj � 1

2a�1ÿn�dunj�1=2Fj�1=2 with the usual de®nition of rj: rj � dun

jÿ1=2dun

j�1=2: Now, if the limiting function F is

equal to 1ÿ 1�n3 �1ÿ rj �, we get the third order upwind scheme. This leads to the de®nition of a

new family of limiters F�r, n�, giving the third order scheme in smooth regions, and restrictedelsewhere such as to satisfy the TVD constraints established by Harten. Such a function wasproposed by the authors at the ICFD conference at Oxford (1995, unpublished) and in [3]. It iswritten as:


FO3�r, n� �

8>>>>>>><>>>>>>>:

0 rR01ÿ �1ÿ r��1� �nÿ 2�r� 0 < rR1=3

1ÿ �1ÿ r�1� n3

1=3 < rR3

1ÿ �1ÿ r�1� nr

r > 3

Arora et al. [2] and Jeng et al. [1] proposed to take the upper bound of the TVD constraints. Itis given by:

FO3Sup�r, n� � max

�0, min

�2r

n, 1ÿ 1� n

3�1ÿ r�, 2

1ÿ n

��The above limiter functions have proven to be very well behaved, as well in capturingdiscontinuities as for computing smooth pro®les. In particular, the latter are not ``squared o�''as it is the case when using the SuperBee limiter. Arora et al. [2] have found that FO3Sup is toocompressive for non-linear systems. Our experience using FO3 did not show such a behaviour.We must underline that this limiter family has been built in a combined time and space

approach, using a ¯ux limiter formulation. It cannot be easily transposed to a slope limiterapproach. This is the reason why the FO3 and FO3Sup limiters will only be associated to acombined time and space discretization. To solve the Euler equations, we make use of the Roe-averaged matrices. In the multi-dimensional case, the correction is applied separately in eachspace direction. To solve the system of conservation laws:

@w

@t� @f

@x� @g@y� 0

the scheme reads, denoting by fni, j the discrete quantity f estimated at a grid point xi � i dx,

yj � j dy and at time t � ndt:

wn�1i, j � wMC

i, j � C xi�1=2, j ÿ C x

iÿ1=2, j � Cyi, j�1=2 ÿ C

yi, jÿ1=2

where wMCi, j stands for the solution given by the Mac-Cormack scheme. The correction term

C xi�1=2, j is given by:

C xi�1=2, j �

Xl

�jnlj2

ÿ1ÿ jnlj

�ÿ1ÿ F

ÿrsl, jnlj

��dal � d l

�i�1=2, j

where, as usual, al and d l are the eigenvalues and eigenvectors of the Roe-averaged jacobianmatrix A � df

dw , dal is the contribution of the l-wave to the variation dw, s � sign�al� and nl �

dtdxa

l: The ratios r�l and rÿl are given by:

r�l � daliÿ1=2, jdali�1=2, j

, rÿl � dali�3=2, jdali�1=2, j

The term C y is analogous.


In the system case, the underlying high order scheme is no more third order but it should beemphasized that the correction term, beside being a TVD correction of a second order scheme,should also reduce the truncation error in smooth regions.The extension to the Navier±Stokes equation is made by using central second order accurate

di�erences to approximate the viscous ¯uxes. Following Abarbanel et al. [8], since the Mac-Cormack di�erencing method cannot be applied straightforwardly to the case of mixedderivatives present in the viscous ¯uxes, we rewrite the Navier±Stokes equations in the way:

wt ��f�w� ÿ f vP�w, wx� ÿ f vM�w, wy�

�x�ÿg�w� ÿ gvP�w, wx� ÿ gvM�w, wx�

�y� 0

where the notations P and M stand for the parabolic or mixed derivatives parts of the viscous¯uxes, ie:

f vP�w� � 1

Re

0BBBBB@0tPxx � lux � 2muxtPxy � mvx

utPxx � vtP

xy �gmPr

ex

1CCCCCA f vM�w� � 1

Re

0BBBB@0tMxx � lvytMxy � muy

utMxx � vtMxy

1CCCCAand similarly in the y direction. The scheme is then implemented as follows:

w�i, j � wni, j ÿ

dtdx

�ÿfÿ f vP

�ni�1, jÿ

ÿfÿ f vP

�ni, j

�ÿ dt

dy

�ÿgÿ gvP

�ni, j�1ÿ

ÿgÿ gvP

�ni, j

�� dt

2:dx

�f vMn

i�1, j ÿ f vMn

iÿ1, j�� dt

2:dy

�gvMn

i, j�1 ÿ gvMn

i, jÿ1�

where the derivatives terms appearing in the parabolic ¯uxes are expressed by backwarddi�erencing, while the derivative terms appearing in the mixed derivative ¯uxes are expressedby central di�erencing.

w��i, j � w�i, j ÿdtdx

�ÿfÿ f vP

��i, jÿ

ÿfÿ f vP

��iÿ1, j

�ÿ dt

dy

�ÿgÿ gvP

��i, jÿ

ÿgÿ gvP

��i, jÿ1

�� dt

2:dx

�f vM�i�1, j ÿ f vM�

iÿ1, j�� dt

2:dy

�gvM�i, j�1 ÿ gvM

�i, jÿ1

�here derivative terms appearing in the parabolic ¯uxes are expressed by forward di�erencing,while the derivative terms appearing in the mixed derivative ¯uxes are still expressed by centraldi�erencing.

wn�1i, j �

1

2

ÿwni, j � w��i, j

�� C xi�1=2, j ÿ C x

iÿ1=2, j � Cyi, j�1=2 ÿ C

yi, jÿ1=2

As the scheme is explicit, the stability condition on the time step dt is expressed by:

dt � CFL �minÿdtE, dtV

�, CFLR0:5


where

dtE � min�dx, dy��u2 � v2�1=2�c

, dtV � 1

2�min�dx, dy��2Pr

grmRe

c being the sound velocity. In our case, there is no need to implicit the viscous terms, as theEuler time step dtE is generally smaller than the viscous time step dtV, at least after re¯exion ofthe shock wave on the end wall, due to large values of the sound velocity.

3.2. Independent time and space discretization methods: the HY and MUSCL schemes

The resolution of the governing Eq. (1) has also been performed by means of a ®nite volumemethod. The discrete equations read as follows:�

@w

@t

�i, j

� ÿ 1

dx

hfn

i�1=2, j ÿ fn

iÿ1=2, j � f vn

i�1=2, j ÿ f vn

iÿ1=2, jiÿ 1

dy

hgni, j�1=2 ÿ gni, jÿ1=2

� gvni, j�1=2 ÿ gv

ni, jÿ1=2

i�2�

3.2.1. Temporal integrationThe time integration of the previous Eq. (2) represented as

@w

@t�L�w� �3�

is performed by means of a third order Runge-Kutta method [9]

w0 � wn

w1 � w0 � dtL�w0�

w2 � 3

4w0 � 1

4w1 � 1

4dtL�w1�

w�n�1� � 1

3w0 � 2

3w2 � 2

3dtL�w2� �4�

where dt is the time step and wn stands for the conservative variable vector evaluated at thetime n � dt: This method has been choosen since it does not increase the total variation of theR.H.S. of Eq (2). The stability limits of this temporal scheme are the ones of an explicitscheme, that means, the Courant number (CFL) and the di�usion number (D) must be lessthan, 1. and 0.5. respectively. In the present calculations, the time step �dt� is prescribed toensure that the CFL and D numbers satisfy the stability limits. Let us mention that the localCFL number is calculated using the spectral radius of the Jacobian matrices of the Euler¯uxes.


3.2.2. Space discretizationThe integration of the convective terms is based on a Roe's approximate Riemann solver

[10]. Two shock capturing schemes have been implemented:

. A second-order upwind total variation diminishing (TVD) scheme, developed by Harten [11]and Yee et al. [12]. The numerical ¯ux fi�1=2, j is expressed as

fi�1=2, j �1

2

ÿfi�1, j � fi, j ÿ Rxi�1=2, j

Hx i�1=2, j

�Hx i�1=2, j

is a 4� 4 matrix; to ensure that the scheme is a second order upwind TVD scheme,the Hl

x i�1=2, jelements must be written [11,12]:

Hlx i�1=2, j

� 1

2

��alx i�1=2, j

��Flx i�1, j� Fl

x i, j

�ÿ��alx i�1=2, j

� zlx i�1=2, j

��dalx i�1=2, j�5�

with dax i�1=2; j� Rÿ1x i�1=2; j

�wi�1; j ÿ wi; j � is the forward di�erence of the local characteristicvariables in the x direction, alx i�1=2, j

represent the eigenvalues of the Jacobian of the Euler¯ux �df=dw�, Rxi�1=2; j

is the matrix whose columns are composed of the eigenvectors ofdf=dw: The ``additional eigenvalues'' are expressed as:

zx i�1=2, j� 1

2

��alx i�1=2, j

��8>>><>>>:�Fl

x i�1, jÿ Fl

x i, j

�dax i�1=2, j

if dax 6�0:

0: if dax � 0:

We must mention that, contrary to the original scheme, no entropic parameter is used as welook for unsteady solutions of the equations. Two limiter functions, which give a second-order TVD scheme, have been implemented:* The ®rst one is the classic Super-Bee function [13]

Flx i, j� S max

n0, min

�2 ��dalx i�1=2, j

��, S � dalx iÿ1=2, j

�, min

��dalx i�1=2, j

��, 2 � S � dalx iÿ1=2, j

�o�6�

where S is the sign of dalx i�1=2, j:

* The second limiter is the Van-Leer Harmonic function

Flx i, j� S max

�dalx i�1=2, j

� dalx iÿ1=2, j��dalx i�1=2, j

� dalx iÿ1=2, j

��dalx i�1=2, j

� dalx iÿ1=2, j

� �7�

To estimate the values at the mid-point �i� 1=2, j� and to be sure that the numerical ¯ux�f� is consistent with f, we have used the classic mass-weighted average introduced by Roe[10] between states wi�1, j and wi, j: A similar formulation is used to express the numerical¯uxes �g� in the y direction. While the integration is third order accurate in time, thenumerical scheme is only second order in both time and space due to the spacediscretization.


. A MUSCL-TVD scheme using the local characteristic approach [14]. the convectivenumerical ¯ux functions �fi�1=2,j and gi, j�1=2� are written as follows:

fi�1=2, j �1

2

�fÿwri�1=2, j

�� f�wli�1=2, j

�ÿ Rxi�1=2, j

Hx i�1=2, j

��8�

Spurious oscillations are eliminated using ``slope'' limiter applied on the conservativevariables (w ):

wri�1=2, j � wi�1, j ÿ 1

4

h�1ÿ Z�dgi�3=2 � �1� Z�dggi�1=2i �9�

wli�1=2, j � wi, j � 1

4

h�1ÿ Z�dggiÿ1=2 � �1� Z�dgi�1=2i �10�

In the following calculations, a minmod ``slope'' limiter has been used with Z � 1=3, whichgives a third order upwind-biased scheme:

dgi�1=2 � minmodÿdi�1=2, bdiÿ1=2

� �11�

dggi�1=2 � minmodÿdi�1=2, bdi�3=2

� �12�where di�1=2�wi�1; jÿwi; j: Several values of the b coe�cient have been checked. The valueb � 4 has been retained since it gives the less di�usive results. The minmod function isexpressed following:

minmod�p, q� � sign�p� �maxf0, min�jpj, q � sign � p��The elements of the matrix Hx i�1=2, j

are written

Hlx i�1=2, j

��alx i�1=2, j

��Rÿ1x i�1=2, j

�wri�1=2, j ÿ wl

i�1=2, j�

�13�

where alx i�1=2, jrepresents the eigenvalues and Rxi�1=2, j

are the right eigenvector matrix ofdf=dw evaluated using the Roe average between the states wr

i�1=2, j and wli�1=2, j:

Concerning the di�usive ¯uxes, a central di�erencing scheme is applied, giving a secondorder accuracy in space.

4. A 1D inviscid test case

The O3 limiter is compared with the Van Leer limiter on the 1D test case proposed in [15]concerning a moving Mach = 3 shock interacting with sine waves in density. The 1D Eulerequations are solved on the spatial domain x 2 �0, 10�: Absorbing boundary conditions areused. The solution is initially prescribed as:

r � 3:857; u � 2:629; p � 10:333 when x < 1


r � 1� 0:2 sin�5x�; u � 0; p � 1 when xr1 �14�A Lax-Wendro�-type discretization is used as described in Section 3.1. Figs. 2 and 3 show thedensity distribution obtained at a dimensionless time t � 1:8, using the two limiters, for 400and 800 grid points and a CFL number equal to 0.8 and 0.1, respectively. The thin solid linesare numerical solutions with 3200 grid points using the same algorithm, they can be regardedas the exact solution. It is clear that the O3 limiter improves the solution whatever the CFLnumber is, compared to the Van Leer limiter which is more di�usive. The comparison with theresults obtained using the O3Sup limiter (see [2]) shows that O3 is slightly more di�usive thanO3Sup. However, these authors mentioned that O3Sup is too compressive for non linearsystems, a behaviour that was not encountered using O3. Let us notice that the O3 and O3Suplimiters using 800 points are comparable to ENO schemes using 400 points (see results in [15]),but the latter is more expensive in CPU cost and storage.Now, it is worth to compare the above results with the results obtained using the HYVL

scheme. The latter are represented in Fig. 4, for a CFL number equal to 0.8 (quite similarresults are obtained for CFL = 0.1). One can see that the use of a Mac-Cormack timediscretization give much better results for this CFL number. This can be explained by the factthat in the separate time±space discretization approach, the limiting procedure is applied inspace at each step of the Runge-Kutta algorithm. By doing so, the underlying scheme which isused around extrema is ®rst order in space, but still third order in time. This imply that theleading term of the truncation error becomes 1

2dxauxx around extrema in the scalar linear case(so it is independent of the CFL number). In the case where a combined time-spacediscretization is used, we rather have 1

2dxa�1ÿ n�uxx as we get the upwind (®rst order in timeand space) scheme. These errors are equivalent if one uses a small CFL number, but thedi�usive error generated by the Runge-Kutta procedure is much higher if the CFL number isclose to 1. This is attenuated in the system case, as the local CFL number is close to 1 only forthe higher velocity wave. Anyway, it is to be expected that the combined time-spacediscretization will give better results for unsteady ¯ows, whatever the limiter is.

Fig. 2. Density distribution for the Shu-Osher test case at t � 1:8: comparison of the O3 and Van Leer limiters, for400 (left) and 800 (right) grid points, CFL = 0.8.


5. A 2D inviscid test case: advection of a vortex

We compare here the results obtained on the inviscid test case of the advection of a vortexat low Mach number. This test case is treated in [16]. The vortex is described initially by ananalytical form of Sculley. The angular velocity of the initial vortex is given as:

vyVmax

�

8>>><>>>:r

a0if rRa0

exp

�ÿ �rÿ a0�2

O

�if r > a0

where Vmax � 0:3, a0 � 0:06 and O � 0:065: The pressure and density are obtained from the

Fig. 3. Density distribution for the Shu-Osher test case at t = 1.8: comparison of the O3 and Van Leer limiters, for

400 (left) and 800 (right) grid points, CFL = 0.1.

Fig. 4. Density distribution for the Shu-Osher test case at t � 1:8, HYVL scheme, for 400 (left) and 800 (right) gridpoints, CFL = 0.8.


relations:

dp

dr� rv2y

r

gp�gÿ 1�r �

v2y2� gp1�gÿ 1�r1

The vortex is initially superimposed on a uniform ¯ow with Mach number M1 � 0:1: Thecomputational domain considered is �ÿ1, 4� � �ÿ1, 1�:We compare the results obtained using the three following numerical schemes described

above:

1. the Mac-Cormack scheme, using the O3 and Van Leer limiters hereafter noted MCO3 andMCVL;

2. the Harten-Yee scheme equipped with the Van Leer limiter, called HYVL;3. and the MUSCL scheme with a Minmod(b � 4� limiter, called MUMI4.

The mesh is cartesian uniform, composed of 251 � 101 cells. In Fig. 5 is represented theminimum pressure (core pressure ), reached at the center of the vortex, as a function of thelength of vortex travel Xcore/a0, where Xcore is the abscissa of the center of the vortex. The¯ow being inviscid and low-speed, the true solution is almost a pure advection of the initialvortex, i.e. the change of the core pressure is mainly due to the numerical di�usion of thenumerical scheme and provides then a good evaluation of the dissipative properties of theschemes. The following comments can be done:

Fig. 5. Evolution of pressure at the center of the vortex (O3 and VanLeer limiters) vs. the length of the vortextravel.


1. the best results, which are nearly equivalent, are obtained using the MCO3 or MUMI4schemes;

2. the Van Leer limiter, which is used in the two other results, is more di�usive. We notice thatthe Harten-Yee scheme is more di�usive than the Mac Cormack scheme when using thesame limiter (Van Leer). This again should be attributed to the fact that the di�usion errorof the RK3 algorithm around extrema is higher than for the Mac Cormack algorithm. Thedi�erence is weak in this case because the Mach number is low, implying that the local CFLnumber for the material wave which transports the vortex is small (around 0.05).

The next ®gure, Fig. 6, shows the longitudinal distribution of the static pressure at adimensionless time t � 30 for the four schemes we used. Let us mention that the O3 andO3Sup limiters give almost the same results for this test case, so only the O3 results are shown.We should also mention that the same computation has been conducted with the SuperBee

limiter, using both the combined and independent time±space discretization methods. It is wellknown that this limiter tend to ``square o�'' smooth pro®les. In the test case considered here,this behavior leads to a very surprising result (see Fig. 7): the core pressure in the vortexdecreases as time goes on, which is very unphysical. Although we expect that this behavior willbe counteracted in the viscous case by the viscous e�ects, numerical problems can arise as thecore pressure can become too low or negative in the center of vortices.Simulations have also been performed using the MUSCL approach with other values of the

b coe�cient, showing that the smaller the value of b is, the more di�usive the results are.As a conclusion following the previous results, we will use three schemes to compute the

shock-boundary layer interaction:

. the TVD Harten-Yee scheme with the Van-Leer Harmonic limiter (HYVL);

. the MUSCL-TVD scheme using the Minmod limiter with b � 4 (MUMI4);

Fig. 6. Longitudinal distribution of the static pressure at y � 0 and for a dimensionless time t � 30:


. the Mack-Cormack scheme using the O3 or O3Sup limiter functions (MCO3 andMCO3Sup). The latter did not show the compressive behavior mentioned in [2] in theinviscid test cases, so we will use it as well in the viscous case.

The other schemes we have tested will not be used in the following, as either they giveequivalent results, or they are not su�ciently ``safe''.

6. Re¯ected shock-boundary layer interaction in a shock tube

6.1. Description of the ¯ow

We consider a unit side length square shock tube with insulated walls (Fig. 8). Thediaphragm is initially located in the middle of the tube �x � 0:5). The initial state, in terms ofdimensionless quantities, is on the left of the diaphragm: rL � 120, pL � rL=g, uL � vL � 0, andon the right: rR � 1:2, pR � rR=g, uR � vR � 0: At the initial time, the diaphragm is broken.The inviscid solution is represented in Fig. 9, showing the evolution of the density in the xÿ tplane. A shock wave, followed by a contact discontinuity, moves to the right (the shock Machnumber is equal to 2.37). The incident shock wave is weak, and re¯ects at the right end wallapproximately at time t � 0:2: After this re¯ection, it interacts with the contact discontinuity.Then, complex interactions occur. The contact discontinuity stays stationary, close to the rightend wall. Afterwards, the re¯ected shock wave begins to interact with the rarefaction wave attime t � 0:4: On the other side, the rarefaction wave re¯ects on the left wall, at a dimensionlesstime t � 0:5, modifying the propagation of the incident rarefaction wave.In the viscous case, the incident contact discontinuity and shock wave, during their

propagation, interact with the horizontal wall, creating a thin boundary layer. After its

Fig. 7. Evolution of pressure at the center of the vortex (SuperBee limiter) vs. the length of the vortex travel.


re¯ection on the right wall, the shock wave interacts with this boundary layer, modifying the¯ow pattern near the horizontal wall. This is illustrated in Fig. 10, showing the densitycontours at t � 0:6: The shock-boundary layer interaction results in a lambda-shape like shockpattern. As the stagnation pressure in the boundary layer is lower than that of the out¯owregion, a separated boundary layer ``bubble'' takes place under this shock pattern. The bubbleis delimited by a supersonic shear layer in which a lot of instabilities occur. The triple pointemerging from the lambda-shape like shock pattern generates a slip line which rolls up around

Fig. 8. Representation of the shock tube.

Fig. 9. x±t diagram of the density.


the vortex downstream of the boundary layer bubble. The bubble is dragged upstream with there¯ected shock and separations and vortices are generated near the wall inside the bubble.The solution being supposed to be symmetric in the vertical direction (i.e. parallel to the

diaphragm), the computations have been performed on half of the shock tube, using the threealgorithms presented above. Two Reynolds numbers have been considered �Re � 200,Re � 1000). The results will be discussed on the density ®eld for the dimensionless time t � 1:In all the computations, we used equally spacing grid such as that dx � dy:

6.2. Wall boundary conditions

On the solid walls, we impose a no-slip condition, i.e. the velocity is set to zero. We treat thecase of an adiabatic wall, so the normal derivative of the temperature is set to zero using asecond order forward di�erence formula, giving the temperature at the wall. The last valueneeded at the wall can then be obtained from two di�erent procedures:

. the ®rst one allows the density to be computed from the continuity equation, where thetangential derivatives are set to zero. A second order forward di�erence formula is used forthe space discretization. The time discretization uses either a Crank-Nicolson or a Runge-Kutta method. The pressure is then deduced from the equation of state for the perfect gas;

. the second procedure is more classical, using the equation of moment normal to the wallwhere the velocity is set to zero at the wall, leading to an equation giving the normalderivative of the pressure at the wall, which is discretized using second order di�erenceformulae.

The ®rst procedure is simple to implement, as there are no viscous term in the equation for thedensity. Nevertheless, the time derivative term, not present in the second procedure, doesintroduce additional discretization errors. We have studied the in¯uence of these boundaryconditions. At low Reynolds numbers, these two procedures lead to very similar results. Thedi�erences increase with the Reynolds number. No evident conclusion emerges to choose which

Fig. 10. Re � 200, t � 0:6, density contours and streamlines.


procedure is the more accurate. In the following paragraphs, all the presented results have beenobtained using boundary conditions based on the solution of the continuity equation.

6.3. Re � 200

For this low Reynolds number value, the computations have been performed using the threealgorithms MUMI4, HYVL and MCO3 on three meshes �dx � 4� 10ÿ3, dx � 2� 10ÿ3 anddx � 10ÿ3). The in¯uence of the grid re®nement is presented on the distribution of the densityalong the horizontal wall (Fig. 11) at a dimensionless time t � 1 and for the HYVL andMCO3 schemes. It can be stated that the coarsest grid is insu�cient for this ¯ow. The middlegrid is not yet su�cient for convergence, but all the structures that are seen on the ®nest gridare correctly represented. We can also observe, when comparing the results for the two shemes,that there still exists some di�erences even in the ®nest grid, notably on the peak levels. Fig. 12shows a comparison of the density ®eld obtained on the ``middle grid'' (501 � 251) with thethree schemes at a dimensionless time t � 1: We see that large vortices are generated within theboundary layer bubble, and are ejected downstream the lambda-shape like pattern. Theposition of these large scale patterns are in agreement with the peaks and valleys recorded inthe longitudinal evolution of the density along the wall. Comparing the results obtained withthe studied schemes, some di�erences can be pointed out. In the MUSCL result, we can noticesmall oscillations behind the re¯ected shock, which do not appear for the other schemes. Thisis probably a numerical artifact due to the compressive behavior of the Minmod �b � 4�limiter. There are also some di�erences in the orientation of the large vortex and in the patternof the contact discontinuity which rolls up in the lower right corner of the tube. The MCO3and HYVL schemes give very similar results, but the HYVL results are more di�usive. Thiscan be seen by examining more closely some details of the ¯ow close to the bottom wall insidethe boundary layer bubble, for example around x � 0:6 or x � 0:76:

Fig. 11. Distribution of the density along the bottom wall at a time t � 1: Left: HYVL scheme, right: MCO3scheme.


When re®ning the mesh, setting dx � 10ÿ3, the HYVL and MCO3 results show fewdi�erences with the precedent mesh (see Fig. 13). The main di�erence lies in the large vortex,which is slightly more rotated in the MCO3 result. The MCO3 scheme again shows a lessdi�usive behavior than the HYVL scheme. This can also be seen on the longitudinaldistribution of the density in the ``inviscid'' region �y � 0:3), at time t � 1 (Fig. 14), in thecontact discontinuity region close to the right-end wall.Let us also say that we have run the MCVL and MCO3Sup schemes for this case and found

that the results are very close to the MCO3 results in the middle mesh. In the ®ne mesh, the

Fig. 12. Re � 200, t � 1, density contours. Top left: MUMI4 scheme. top right: HYVL scheme, bottom MCO3scheme. Mesh size dx � 2� 10ÿ3.

Fig. 13. Re � 200, t � 1, density contours. Left: HYVL scheme, right: MCO3 scheme, Mesh size dx � 10ÿ3.


results are almost indistinguishable. This leads us to state that we have attained convergence inthe ®ne mesh when using the Mac-Cormack time discretization and that for this low Reynoldsnumber value the O3 limiters are almost equivalent to the Van Leer limiter. In comparison, theHYVL algorithm seems to converge rather slowly, as there still exists some signi®cantdi�erences with the MCO3 results in the ®ne mesh. This is probably explained by thearguments we developed in the 1D case and the fact that this is a high Mach number ¯ow.Even in the boundary layer bubble, we ®nd high values of the Mach number, as can be seen inFig. 15 where is represented the Mach number distribution along y � 0:01:

Fig. 14. Comparison of the longitudenal distribution of the density at y � 0:3 given by the HYVL and MCO3schemes on the ®ne grid (1000� 500) at time t � 1 �Re � 200).

Fig. 15. Mach distribution along y � 0:01 at a time t � 1: MCO3 scheme, dx � 10ÿ3:


6.4. Re � 1000

The ®rst computation for this Reynolds number has been performed on the middle mesh wepreviously used �dx � 2� 10ÿ3). We compare the density ®elds obtained at t � 1 (Fig. 16)using the MUMI4, HYVL and MCO3Sup schemes (we have used the O3Sup limiter ratherthan the O3 limiter in order to have better chances to reach grid convergence, as the O3Suplimiter is slightly less di�usive; let us say, however, that the results obtained using the MCO3or MCO3Sup schemes are very similar). At this Reynolds number, the discrepancies betweenthe results are very important. This is attributed to a lack of resolution in the boundary layer,leading to a misrepresentation of the separation points at the wall. The e�ect of the numericalerror speci®c to each scheme is here very marked. The MUMI4 scheme reveals, in this case, astrange behavior since the lambda-shape like shock pattern has almost completely disappeared(Fig. 16). The boundary-layer bubble and especially the large vortex at the bottom of thebubble are ¯attened. This result seems to be completely unrealistic. It is probably due to thecompressive nature of the limiter (Minmod with b � 4). A conclusion which can be drawnfrom this result is that compressive limiters, although suitable in some cases as they counteractthe numerical di�usion, should not be used to compute viscous ¯ows. This is the reason why,in the following, the MUMI4 scheme will not be used.The results obtained using the MCO3 and HYVL schemes are notably di�erent concerning

Fig. 16. Re � 1000, t � 1, density contours. Mesh size dx � 2� 10ÿ3: Top left: MUMI4 scheme, top right: HYVLscheme, bottom: MCO3Sup scheme.


the vortical structures inside the boundary layer bubble (Fig. 16). The triple point is higher inthe MCO3Sup case. One can also remark that the ``inviscid'' zone, above the slip line initiatedat the triple point, is now perturbed by acoustic waves generated by the growth of the vorticeswithin the boundary layer.The ®rst mesh we used is in fact very insu�cient to give a correct representation of the

solution, whatever the scheme is. We then have made a series of computations on successivelyre®ned meshes, corresponding to dx � 10ÿ3, dx � 6:7� 10ÿ4 and dx � 5� 10ÿ4: The resultsare represented in Fig. 17. We can see that small structures are generated just downstream ofthe boundary layer separation. These structures produce distortions of the unstable linedelimiting the separation under the lambda-shape like shock pattern. These warpings induceoblique shock waves moving along this slip line. We can also notice that shocklets are

Fig. 17. Re � 1000, t � 1, density contours. Left: HYVL scheme, right: MCO3Sup scheme. Mesh size dx � 10ÿ3

(top), dx � 6:7� 10ÿ4 (middle), dx � 5� 10ÿ4 (bottom).


generated inside the large vortices at many locations. All these ¯ow characteristics are verysensitive to small perturbations. As a consequence, it is extremely di�cult to ascertain thequality of the schemes, because there does not probably exist a uniquely de®ned convergenceprocess due to the instability of the ¯ow. However, as a general comment, we can observe thatthe HYVL results are smoother and the variation of the density ®eld when the mesh is re®nedseems weak. On the contrary, the MCO3Sup results are very sensitive to the mesh size. Also,they are more chaotic, especially inside the boundary layer bubble where there is no regularstructures as in the HYVL results. Nevertheless, the last two MCO3Sup results seemsu�ciently close one to the other to think that convergence is not far from the last mesh.In the case of the HYVL scheme, the ®ner result shows that the regularity of the solution

Fig. 18. Re � 1000, t � 1, velocity vector ®eld and instantaneous streamlines. Left: HYVL scheme, right: MCO3Supscheme. Mesh size dx � 10ÿ3 (top), dx � 6:7� 10ÿ4 (middle), dx � 5� 10ÿ4 (bottom).


begins to be broken. More complex structures appear, like for example the structure on the leftside of the big vortex, which is very similar to the one in the MCO3Sup result. Theconvergence of this scheme is slower, but we can expect that if we re®ne the mesh again wewill obtain a solution close to the one given by the MCO3Sup scheme.It is interesting, in order to give some more justi®cations to our previous conclusions about

the convergence of the schemes, to get a closer insight in the right bottom corner. We haverepresented in Fig. 18, the velocity vector ®eld obtained using the two schemes in the three®ner meshes. Here, we clearly see that the MCO3Sup scheme converges much more faster thanthe HYVL scheme: the results on the two ®ner meshes are almost the same, indicating that thesolution is almost converged in this area on the intermediate mesh. Moreover, the four vorticesare already captured in the coarsest mesh, though the small one in the corner is barelyoutlined. In contrast, we can see that three vortices are hardly captured using the HYVLscheme on the intermediate mesh. The four vortices appear only on the ®ner mesh. Thecoarsest mesh solution reproduces only one vortex. The examination of this ®gure also showsthat the convergence process of the two schemes are quite di�erent. This is probably due to thefact that the truncation error of the HYVL scheme is less dependent on the local CFL number.Finally, in order to illustrate the contribution of the limiter, we show in Fig. 19, the results

obtained using the MCVL scheme on the mesh de®ned by dx � 10ÿ3: The density ®eld is seento be more regular than the MCO3Sup result (it looks more like the HYVL result). If now weinspect the corner ¯ow, we can again say that the combined time±space discretization iswithout doubt the more accurate, but it is also seen that the O3Sup limiter gives better results,as the small vortex in the corner is not present in the MCVL solution, and the vortex at theleft is not as well developed as in the MCO3Sup result. This shows that the combination of aMac-Cormack type scheme with a O3-type limiter is the best choice to do.

Fig. 19. Re � 1000, t � 1, density contours. MCVL scheme. Left: density contours, right: velocity vectors andinstantaneous streamlines. Mesh size dx � 10ÿ3:


6.5. Comments

When we observe the dependence of the lambda-shape like pattern on the Reynolds number,our results show that the slope of the leading oblique shock wave stays constant in theReynolds number range when we studied. This is in agreement with Mark's theoreticalpredictions based on an inviscid assumption. However, a tendency of the triple point to raiseas Re increases, is noticeable on our results. This is in contradiction with the conclusion drawnby Weber et al. [7], who observed that the asymptotic bifurcation height is larger for lowerReynolds numbers. This could be related to the grid resolution, which have been shown in ourresults to have a very important in¯uence on the ¯ow pattern. We have also noticed that theresults are sensitive to the cell mesh ratios dx=dy: As there exists a lot of small eddy structuresin the viscous ``bubble'', it seems more appropriate to use a uniform mesh inside this area.

7. Conclusions

Our goal in this study was to evaluate the capability of some high resolution TVD schemesto solve complex unsteady viscous shocked ¯ows. We have considered two types ofdiscretization, namely a combined time and space (CTS) discretization and an independenttime and space (ITS) discretization. Both methods are associated with several limiters. In thecase of the combined time and space discretization, a new family of limiters have beenconsidered.The accuracy properties of each scheme have ®rst been reviewed on inviscid 1D and 2D

computations. When the same limiter is used, the best results are obtained using the CTSmethod. This is explained by showing that around extrema, the recovered ®rst order scheme isless di�usive for this method than for the ITS method. The so-called O3 family of limiters isshown to be more accurate than the classical ones.We have then studied the application of the di�erent schemes to the ¯ow produced by the

interaction of a re¯ected shock wave with the incident boundary layer in a shock tube. Thecalculations have been performed for two values of the Reynolds number. At Re � 200, wehave shown that the CTS method converges much more faster than the ITS method. The O3limiter does not improve the results signi®cantly compared to the Van Leer limiter.When the Reynolds number is increased to the value of 1000, the ¯ow becomes much more

complex. The boundary layer separates in several places, giving rise to the development of a lotof vortices with shocklets and large compressibility e�ects. For this ¯ow, we can only exhibitan ``almost converged'' numerical solution due to the excessive grid requirements. However,the close examination of the results leads us to conclude that the CTS method associated witha O3 limiter is from far the best scheme in terms of accuracy and convergence properties. Asensible improvement is notably due to the limiter himself, when compared to good qualityclassical limiters.To conclude, we think that this viscous shock tube problem could constitute an interesting

test case to compare high resolution numerical schemes in the case of compressible viscousunsteady ¯ows, at least for low values of the Reynolds number. We have considered in thiswork only TVD schemes. There exists other approaches, particularly the ENO and WENO


schemes which are gaining popularity as they allow a higher order of accuracy, generally at theexpense of an increased cost. It should be interesting, in a future work, to test these schemeson this ¯ow case.

Acknowledgements

Part of the computations have been carried out on the Cray C98 of the I.D.R.I.S./C.N.R.S.The authors greatly acknowledge the support of these institutions.

References

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Publishing & Printing, 1997. p. 1091±7.[4] Roe PL, Baines MJ. Algorithms for advection and shock problems. In: Viviand H, editor. Proceedings of the

Fourth GAMM Conference on Numerical Methods in Fluid Mechanics. Braunschweig: Vieweg, 1981. p. 281±90 Notes on Numerical Fluid Mechanics, 5.

[5] Kleine H, Lyakhov LG, Gvozdeva LG, Gronig H. Bifurcation of a re¯ected shock wave in a shock tube. In:Proceedings of the 18th International Symposium on Shock Waves, Sendai. Berlin: Springer-Verlag, 1991. p.261±6.

[6] Wilson GJ, Sharma SP, Gillespie WD. Time-dependent simulation of re¯ected-shock/boundary layerinteraction in shock tubes. In: Proceedings of the 19th International Symposium on Shock Waves, Marseille.Berlin: Springer-Verlag, 1993. p. 439±44.

[7] Weber YS, Oran ES, Boris JP, Anderson JD. The numerical simulation of shock bifurcation near the end wallof a shock tube. Physics of Fluids 1995;7:2475±88.

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mixed derivatives. Journal of Computational Physics 1981;41:1±33.[9] Shu CW, Zang TA, Erlebacher G, Whitaker D, Osher S. High-order ENO schemes applied to two- and three-

dimensional compressible ¯ow. Applied Numerical Mathematics: Transactions of IMACS 1992;9:45±71.[10] Roe PL. Approximate Riemann solvers, parameter vectors and di�erence schemes. Journal of Computational

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88325, (1986).[14] Yee, H.C., On the implementation of a class of upwind schemes for system of hyperbolic conservation laws.

NASA TM-86839, (1985).[15] Shu CW, Osher S. E�cient implementation of essentially non-oscillatory shock-capturing schemes. II, Journal

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