pred-corr-t

Influence of numerical and viscous dissipation on shockwave reflections in supersonic steady flows

Marie-Claude Druguet, David E. Zeitoun *

Institut Universitaire des Syst�eemes Thermiques Industriels, UMR 6595 CNRS/Universit�ee de Provence,

Technopoole de Chaateau-Gombert, 5, rue Enrico Fermi, 13453 Marseille Cedex 13, France

Received 29 January 2001; received in revised form 24 June 2001; accepted 19 September 2001

Abstract

Numerical simulations are investigated to describe precisely the shock wave reflections in supersonic

steady air flow field. The main objectives are to study the influence of the wedge trailing edge corner angle,of the numerical methods and of the viscous effects on the shock wave reflections and on the hysteresis

behavior. The computations are done with different MUSCL–TVD finite volume schemes and the corre-

sponding results are compared. The flow viscosity is also taken into account and comparisons are made

between inviscid and viscous flow simulations. The results display the non-negligible influence of the nu-

merical scheme accuracy on the results, mainly on the position and height of the Mach stem, and the

relatively weak influence of the flow viscosity on these parameters. Comparisons between numerical results

and experimental data have also been done and a good agreement is only observed for small wedge angles

mainly due to the three-dimensional effects in the experimental setup.� 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Regular reflection; Mach reflection; Mach stem and triple point configuration; Finite volumes method; Roe

flux splitting; H-correction

1. Introduction

A couple of two-dimensional (2D) wedges placed symmetrically with regard to a symmetry planein a supersonic steady flow generate two wedge-induced oblique shock waves that reflect on thesymmetry plane. Depending on the wedge angle hw the shock wave reflects in a regular reflection(RR) configuration or in a Mach reflection (MR) configuration. For wedge-induced shock angles

Computers & Fluids 32 (2003) 515–533www.elsevier.com/locate/compfluid

* Corresponding author.

E-mail addresses: [email protected] (M.-C. Druguet), [email protected] (D.E. Zeitoun).

0045-7930/03/$ - see front matter � 2003 Elsevier Science Ltd. All rights reserved.

PII: S0045-7930(02)00002-6

mail to: [email protected]

/w smaller than Von-Neumann criterion angle /N, the shock wave necessarily reflects on thesymmetry plane into a RR. For angles larger than the detachment criterion angle /D, the shockwave reflects into a MR configuration (see Fig. 1). In the range between these two criteria namedthe ‘‘dual solution domain’’, both types of reflections (RR and MR) are theoretically possible,which leads to an hysteresis phenomenon.The transition from regular to irregular reflection of shock waves in steady flows which is very

relevant to better understand supersonic flows through intakes and nozzles, has been widelystudied for the last decades. In the recent past considerable advances have been achieved in un-derstanding the transition and the corresponding hysteresis phenomenon, by means of analytical[1–3], numerical [4–8] and experimental [9,10] investigations that confirmed earlier predictions ofHornung et al. in 1979 [11]. Recently, a general synthesis on shock reflections and hysteresisphenomena has been given by Ben-Dor [12]. The experiments conducted by Chpoun and Leclerc[10] were intended to show the non-influence of the trailing edge corner angle on the MR, fol-lowing a conclusion proposed by Li and Ben-Dor [3]. The first goal of the present study is tonumerically verify that point in order to confirm or not the conclusion given by Ben-Dor et al.[13]. As for the numerical contribution [7], it has displayed the existence of an instability of the slipsurface (shear layer), emanating from the triple point and separating the flow in two streams withdifferent velocities and entropy, and that has not been seen experimentally. This instability,however, is unsteady and disappears long after the Mach stem gets its steady position. Othernumerical simulations have shown the influence of the real gas effects on the transition [14–17].Nonetheless, many studies have been conducted with the assumption of inviscid flow, mainlybecause the dissipation effects are supposed to be negligible. The present study also aims to verifythis assumption and to show the eventual influence of the viscous effects both on the hysteresisand on the instability of the slip line. Moreover, the numerical simulations mentioned in the abovereferences have been conducted with different numerical schemes, and it appears important toshow the influence of their intrinsic numerical dissipation on the transition and on the instabilityalong the shear layer. That is another aim of the present study.The different goals proposed here are fulfilled by means of numerical simulations conducted

with the numerical code laminar non-equilibrium Navier–Stokes (LANENS) which has been usedto solve the 2D Navier–Stokes (or Euler) equations by means of a cell-centred finite volumemethod on structured grids. The Navier–Stokes equations are presented in the following section

Fig. 1. Schematic view of MR configuration.

516 M.-C. Druguet, D.E. Zeitoun / Computers & Fluids 32 (2003) 515–533

and the numerical method is developed in Section 3. Results are displayed and discussed inSection 4, including comparisons between results obtained with several trailing edge corner angles,with different approximate Riemann solvers and different slope limiters, as well as comparisonsbetween inviscid and viscous flow simulations.

2. Governing equations

The principles of conservation of mass, momentum and total energy, together with the physicalproperties of a 2D compressible unsteady viscous—respectively inviscid—gas flow are expressedin the Navier–Stokes—respectively Euler-equation formulation:

oU

otþr � F ðUÞ ¼ 0 ð1Þ

where U is the vector of conservative variables and F the matrix of the flux vectors:

U ¼

qquqvqe

0BB@

1CCA F ¼ ðF;GÞ ¼ ðFc þ Fd;Gc þGdÞ ð2Þ

where the following expressions describe the convective Fc, Gc and dissipative Fd, Gd fluxes:

Fc ¼

ququ2 þ P

quvðqeþ PÞu

2664

3775; Fd ¼ �

0

sxxsxy

usxx þ vsxy þ qx

2664

3775 ð3Þ

Gc ¼

qvquv

qv2 þ Pðqeþ P Þv

2664

3775; Gd ¼ �

0

syxsyy

usyx þ vsyy þ qy

2664

3775 ð4Þ

where q, u, v, qe, P respectively denote the density, the flow velocity components in the x- and y-directions, the total energy per unit volume and the pressure. In Euler equations, there are nodissipative fluxes. The components of the viscous stress tensor in the dissipative fluxes of NSequations are:

sxx ¼ 2louox

� 2

3l

ouox

þ ovoy

�

syy ¼ 2lovoy

� 2

3l

ouox

þ ovoy

� ð5Þ

sxy ¼ syx ¼ louoy

þ ovox

�ð6Þ

and the thermal fluxes

M.-C. Druguet, D.E. Zeitoun / Computers & Fluids 32 (2003) 515–533 517

qx ¼ � lCp

ProTox

qy ¼ � lCp

ProToy

ð7Þ

where l is the air viscosity, Cp the specific heat at constant pressure, Pr Prandtl number, and T thetemperature. The equation of state linking the pressure and the total energy closes the system andreads:

P ¼ ðc � 1Þ qe�

� 12q u2�

þ v2��

ð8Þ

where c is the specific heat ratio.

3. Numerical technique

The Navier–Stokes or Euler equations are discretized according to a cell-centred finite volumetechnique on a structured grid. The steady state is obtained after convergence of the unsteadyformulation of the discretized equations, using a predictor–corrector explicit time scheme. Thedissipative fluxes Fd and Gd are replaced with central differences [18] and the convective fluxes Fc

and Gc at each cell interface are computed by solving a Riemann problem with left and right statesreconstructed from cell averaged variables. The algorithm is second order accurate in space andtime. The numerical code so developed solves as well NS equations as Euler equations, by meansof a switch that allows to compute or not the dissipative fluxes Fd and Gd.

3.1. Finite volume method and approximate Riemann solvers

The finite volume formulation of the 2D NS/Euler equations reads:

o

ot

ZCi; j

UdxþZoCi; j

FðUÞ � ndr ¼ 0 ð9Þ

where the vector of conservative variables U is cell-wise constant on Ci; j and n is the unit normalto the cell boundary oCi; j. The cell boundary convective fluxes FcðUÞ � n are replaced with a one-dimensional (1D), two point, numerical flux function hFc�nðUL;URÞ, where UL and UR are thevariable vectors on the left and right sides, of the interface between two cells where the flux iscomputed. In the present study three approximate Riemann solvers have been used: the Harten–Lax–van Leer solver (HLL), the HLLC solver (a modification of the HLL scheme where themissing contact and shear waves are restored) and the Roe solver. The HLL solver is robust butvery dissipative, while the other two schemes are less dissipative. The expressions of these solversmay be found in [19].

3.2. Multidimensional dissipation: H-correction

The problem of solving the multidimensional Euler equations on structured grids by finitevolume schemes based on strictly upwind numerical flux function (for example Godunov’s fluxand Roe’s flux) is well known. When applied to the 1D Euler equations, these schemes havethe desirable property of accurately resolving shock waves as well as contact discontinuities.


Expansion waves, however, can generally be well resolved either by use of entropic schemes suchas Godunov’s or Osher’s, or by suitable modifications to non-entropic schemes as Roe’s. How-ever, naively inserting a 1D strictly upwind numerical flux function into a multidimensional finitevolume formulation may often lead to a seriously flawed numerical algorithm. It is often observedthat slowly moving or stationary strong shocks aligned with the spatial grid break down in atotally non-physical manner. The carbuncle phenomenon is the name often used to refer to astationary bow shock that contains spurious bump. This failure is an instability which is the resultof inadequate cross-flow dissipation offered by strictly upwind schemes. The usual procedure tocure the carbuncle flaw is based on some ad hoc, parameter-based switch to an extremely dissi-pative convective flux such as Lax–Friedrichs (or some other non-strictly upwind flux) in regionsdeemed as susceptible to the carbuncle phenomenon. Beside introducing a great deal of com-plexity to the computer program, this approach generally leads to a scheme that has excessivenumerical dissipation in regions where it is particularly harmful, such as boundary layers/shockinteractions, slip surfaces, and regions of multicomponent non-equilibria.Sanders et al. [20] introduced and analyzed a multidimensional dissipation which is shown to

eliminate the carbuncle flaw. This dissipation is supported almost entirely within shock layers andat the same time leaves perfectly grid aligned shock resolved exactly the same as would come fromthe 1D calculation. It is a new parameter-free and easy to implement multidimensional upwinddissipation modification that provides sufficient cross-flow dissipation to eliminate the instability.It corresponds to a very simple multidimensional modification to the Roe’s scheme entropycorrection techniques and it essentially results in additional cross-flow dissipation solely in thedirection tangent to the shock layer:Consider a typical cell interface Iiþ1=2; j. First compute

gliþ1=2; jðUL;URÞ ¼ 1

2kl UR; niþ1=2; j

� �� kl UL; niþ1=2; j

� �� ð10Þ

for every cell interface and for every equation (l ¼ 1; 4). Note that gliþ1=2; j is the 1D entropy

correction. Then, symmetrically calculate

gl;H ;Sandersiþ1=2; j ¼ max gl

iþ1=2; j; gli; jþ1=2; g

li; j�1=2; g

liþ1; jþ1=2; g

liþ1; j�1=2

� �ð11Þ

and use gl;H ;Sandersiþ1=2; j to determine j~kklj according to the following entropy correction:

j~kklj ¼ jklj þ gl;H ;Sanders ð12Þ

where jklj are the elements of the diagonal part of Roe’s upwind dissipation matrixjAðUL;UR; nÞj ¼ RðU; nÞjKðUL;UR; nÞjR�1ðU; nÞ. These formulations consist of introducing thelargest 1D entropy correction from all neighboring cell interfaces. It is called the H-correctionfrom its appearance as a lying down H (see Fig. 2). This approach leaves perfectly 1D profilesabsolutely unaffected. This entropy correction is easy to implement; moreover, there is no pa-rameter to adjust, no switch, no test to make. The extra dissipation is automatically adjusted inshock regions where necessary, and therefore does not pollute the numerical results elsewhere.Additionally, Pandolfi and D’ambrosio [21,22] successfully adapted the H-correction to an

Osher-like scheme. Recently, they have proposed a new version of the H-scheme, where the en-tropy correction consists of dissipation in the direction tangent to the shock layer only, and isapplied to the vorticy and entropy waves only (kl ¼ u, l ¼ 2; 3):


gl;H ;Pandolfiiþ1=2; j ¼ maxðgl

i; jþ1=2; gli; j�1=2; g

liþ1; jþ1=2; g

liþ1; j�1=2Þ ð13Þ

3.3. Space and time second order accuracy

3.3.1. Space discretizationWith the following notations Uþ

i; j ¼ ðULÞiþ1=2; j and U�iþ1; j ¼ ðURÞiþ1=2; j, the Turkel and van Leer

(j ¼ 1=3) reconstruction used in the present study reads:

U�i; j ¼ Ui; j �

1

3S�i; j �

1

6Sþi; j and Uþ

i; j ¼ Ui; j þ1

3Sþi; j þ

1

6S�i; j ð14Þ

where

S�i; j ¼ R minmod R�1DUi�1=2; j;-R�1DUiþ1=2; j

� �Sþi; j ¼ R minmod R�1DUiþ1=2; j;-R�1DUi�1=2; j

� � ð15Þ

with

minmodða; bÞ ¼a if jaj < jbj and a � b > 0

b if jbj < jaj and a � b > 0

0 if a � b < 0

8<: ð16Þ

and with

DUi�1=2; j ¼ Ui; j �Ui�1; j and DUiþ1=2; j ¼ Uiþ1; j �Ui; j ð17Þ

Two values of - have been used: - ¼ 1 corresponding to the Minmod limiter, and - ¼ 2 cor-responding to the Superbee-type limiter. Another slope limiter function, the van Leer slope limiter[19], is also used for comparison of the results in Section 4.4.2.

3.3.2. Time discretization

A predictor–corrector explicit time scheme is used:

• predictor step: eUU ¼ Un þ DtP

F ðUnÞ• corrector step: Unþ1 ¼ Un þ 1

2Dt

PF ðUnÞ þ

PF ðeUUÞ

� �where the time step Dt depends on the CFL number and is computed as:

Fig. 2. Interface Iiþ1=2; j and surrounding interfaces.


Dt ¼ CFL1

DtE

þ 1

DtNþ 1

DtViscous

��1

ð18Þ

with

1

DtE¼ jvE;tangent þ cj

LE

and1

DtN¼ jvN;tangent þ cj

LN

ð19Þ

while the viscous time limit is approximated as

1

DtViscous¼ 4lc

Prq1

L2E

þ 1

L2N

�ð20Þ

L is the interface length, the index E refers to East interface (i.e. iþ 12; j), and the index N to North

interface (i.e. i; jþ 12) of each cell Ci; j; c is the speed of sound; CFL ¼ 0:4 for every computation

with a space and time second order accurate scheme.

4. Results and discussion

All the following numerical simulations have been done with the LANENS code that solves theNavier–Stokes/Euler equations by means of the method described above.

4.1. Choice of the computational domain: influence of the trailing edge shape

For time saving reasons in numerical simulations, it is important to adjust the computationaldomain as closely to the flow-field main features as possible. The main issue is the shape of therear part of the wedge—the wedge trailing edge corner angle (see Fig. 3)—and the main questionis: do the downstream flow conditions have an influence on the Mach stem? This question hasbeen the subject of interesting discussions and several papers: first, the model developed by Schotzet al. [23] predicted a strong dependence on the trailing edge corner angle, whereas the Li andBen-Dor’s one [3] did not reveal any dependence. From a numerical study, Ben-Dor et al. [13]showed that this angle does not have any noticeable effect of the location of the Mach stem andof its height. Experimental investigations were then conducted to clear up the situation where

Fig. 3. Computational domain.


Chpoun and Leclerc [10] experimentally showed that the Mach stem height is quasi-constant onaverage whatever the trailing edge corner angle.In the present study, we first aim to numerically verify the non-influence of the trailing edge

corner angle on the Mach stem height by computing a flow on a wedge with different trailing edgecorner angles. We chose to simulate the test case proposed by Chpoun and Leclerc et al. [10]where the wedge angle is given and equal to 32� and where the trailing edge corner angle variesfrom 45� to 148�. The upstream and reservoir conditions of air flow are as follows: M1 ¼ 4:96,T0 ¼ 453 K, P0 ¼ 8:5 105 Pa, Re1 ¼ 12:74 106 m�1. For the numerical simulations, we choseseveral trailing edge corner angles among those proposed by Chpoun. The inviscid flow com-putations were performed with ht ¼ 160�, 148�, 140�, 130�, 120� and for NS ones, three trailingedge corner angles were chosen: ht ¼ 148�, 120�, 100�.Mach number contours for viscous flow simulations with three of those different trailing edge

angles are drawn in Fig. 4. It can be noticed that some differences between contours position arevisible but not very important. For a better comparison the corresponding Mach stem heightsdeduced from all the computations are plotted in Fig. 5 with the experimental data given byChpoun and Leclerc [10], where the experimental uncertainty is about �2%. One has to first pointout that the Mach stem heights obtained numerically (h=w � 0:4) are higher than the experimentalvalues (h=w � 0:1). As shown by Ivanov et al. [24], it appears that the main reason for the dif-ferences comes from the three-dimensional (3D) effects in Chpoun’s setup. But the Mach stempositions and heights obtained numerically for the different trailing edge angles are similar onaverage among themselves as it is also seen experimentally. Numerical simulations, however, tendto show that the Mach stem height slightly increases as the trailing edge corner angle increases,may be because of differences between the grids, whereas the experiments show a non-continuousevolution of the Mach stem height (see Fig. 5, left) which could be attributed to the experimentaluncertainties. As for the evolution of the normalized subsonic pocket length Lsp=w between theMach stem and the sonic throat for different trailing edge corner angles (Fig. 5, right), it is alsoquasi-constant, as observed experimentally [10]. Here also the numerical values are higher than

Fig. 4. MR for different trailing edge angles: ht ¼ 100�, 120�, 148�; hw ¼ 32�; H=w ¼ 0:66; Grid (300 217); NS

simulations with Roeþ H -correction and Minmod limiter; Mach number contours; Chpoun test case: see [10] for

upstream conditions.


the experimental ones. Therefore, these results show the very weak influence of the downstreamflow conditions on Mach stem position and height, and on the subsonic pocket length. Theseobservations apply to both the Euler and NS computations. Hence, in order to reduce compu-tational time, the computational domain for all the following simulations is as depicted in Fig. 3with ht so that the rear part of the computational domain is horizontal.

4.2. Hysteresis behavior of RR�MR transition

In shock wave reflection studies, the main objective is to display the hysteresis phenomenon andthe corresponding curve as depicted in Section 1. From a numerical point of view, it is well knownthat to obtain the RR, numerical simulations are performed for a wedge angle hw smaller than theVon-Neumann criterion hN and by increasing the angle step by step until the detachment occurs athw ¼ hD where a MR is obtained. Then, the wedge angle is decreased step by step while the shockreflection remains a MR, until the Von-Neumann criterion is attained and the shock reflectionbecomes again regular (see arrows in Fig. 6). This loop depicts the hysteresis phenomenon.For that study, the experimental test case proposed by Ivanov for FlowNet workshop [25] is

chosen. The corresponding geometry and the upstream (inlet) conditions are given in Table 1. Fora perfect gas flow with the present inlet conditions, the theory gives the following boundariesof the dual solution domain: Von-Neumann criterion /N ¼ 30:9� (corresponding wedge anglehN ¼ 20:88�) and detachment angle /D ¼ 39:3� (hD ¼ 27:63�). The numerical simulations fordifferent values of wedge angle hw between and about hN and hD are done on a (560 144) gridand the corresponding hysteresis loop is shown in Fig. 6, where the normalized Mach stem heighth=w is plotted vs. the wedge angle hw. A zero value of the Mach stem height h=w means thatthe reflection is regular and a non-zero value means that the reflection is of Mach type. Fromthe hysteresis curve obtained numerically, it can be noted that the detachment is slightly shifted

Fig. 5. Mach stem height (left) and subsonic pocket length (right) vs. trailing edge angle: hw ¼ 32�; H=w ¼ 0:66;computations with Roeþ H -correction and Minmod limiter; Grid (300 190) for Euler and Grid (300 217) for NS;

comparisons between experimental data [10] and numerical results.


towards the larger wedge angles and its value is hw ¼ 27:97� instead of hD ¼ 27:63� as theoreticallypredicted. The transition from MR to RR is obtained at hw lower than 22.66� while the theorygives the value hN ¼ 20:88�. It appears that Von-Neumann criterion as calculated theoreticallyis very difficult to obtain numerically, mainly because of the numerical dissipation. The grid re-finement and the numerical scheme accuracy may indeed influence the Mach stem position andheight; that issue will be the subject of Sections 4.3 and 4.4.Concerning the numerical results, one would like to point out the following remark: when

looking at the Mach number contours for hw ¼ 27:94� the reflection is still regular in the nu-merical results but one notes the presence of a quasi-triangular subsonic pocket between theregularly reflected shock and the symmetry plane. In its vicinity the reflected shock curvature islocally concave (in respect to the symmetry plane), whereas the reflected shock is normally straightor slightly convex in a RR configuration. This subsonic pocket is seen for wedge angles hw equalto and slightly smaller than 27.94�. Beyond this value, the detachment occurs.

Table 1

Geometry, upstream conditions, Von-Neumann and detachment angles for perfect gas airflow [25]

w 8:05 10�2 m

H=w 0:42 (or 0.37 where specified)

M1 4.96

P1 3050 Pa

q1 0.1724 kgm�3

Re1 (unit length) 3:27 107 m�1

/N (hN) 30.9� (20.88�)/D (hD) 39.3� (27.63�)

Fig. 6. Normalized Mach stem height h=w vs. wedge angle hw (hysteresis curve) comparisons: experimental data and

numerical results (Euler and NS). H=w ¼ 0:42; Grid (560 144); experimental see [25].


For the numerical simulations in the dual solution domain, in order to avoid increasing thewedge angle through the whole hysteresis loop to get the detachment, the computation can bestarted with an incident normal shock wave coming from the upstream and moving downstreamin the computational domain where the flow is initially a rest. As the shock moves downstreamand encounters the wedge surface, complex interactions occur and create triple points along thewedge surface, until the Mach stem is formed on the symmetry plane and the steady state is at-tained: the shock reflection is of MR type (see Fig. 7, right). In that configuration, it is importantto note the development of the instability—Kelvin–Helmoltz-type rollups—along the slip linebehind the triple point. This instability remains long after the Mach stem has reached its steadyposition. These rollups are seen only with numerical simulations on fine enough grids and with

Fig. 7. Time evolution—Initial data: uniform flow (left) and incident normal shock wave (right). Euler simulations with

Roeþ H -correction and Superbee-type limiter: H=w ¼ 0:42; hw ¼ 22:66�; Grid (560 144); density contours.


accurate schemes. As for the RR it is obtained with starting the computation with uniform initialconditions constant throughout the whole computational domain. In this case, a growing shockwave slowly detaches from the wedge surface and an expansion fan develops at the trailing edge,until the shock reflects on the symmetry plane. The steady state is of RR type (see Fig. 7, left).Note that an analytical relation links the wedge-induced shock wave angle to the wedge angle.

To verify that, the shock and wedge angles obtained numerically are plotted in Fig. 8 where agood agreement between the numerical results and the analytical data is observed.The computed values of the Mach stem height h=w vs. wedge angle hw are compared to ex-

perimental data given by Ivanov [25] in Fig. 6. One can observe a good agreement for the wedgeangles near the Von-Neumann criterion hN, but some discrepancies appear for the large wedgeangles: the numerical Mach stems are higher than the experimental ones. The reason for this isthat the 3D effects present in the experiments and not in the numerical simulations [24] appear tobe more significant as the wedge angle increases.The experimental data given by Hornung and Robinson [26], where the experimental uncer-

tainties are about 0.25% on the angle and �1% on the Mach stem height, are in agreement withthe numerical results (see Fig. 9) for wedge angles near hN. Numerical results obtained with adifferent numerical approach (LCP-FCT [4]) are also plotted, showing a very good agreementbetween the two numerical methods.All these comparisons allow us to verify the numerical results with respect to experimental data

but no code validation can be formally made because of the 2D model is unable to account for the3D effects in the experiments.

4.3. Grid refinement

In the previous section we have seen that the Von-Neumann criterion was difficult to obtainnumerically, probably because of the numerical dissipation and of the coarseness of the grid.

Fig. 8. Shock angle / (deg) vs. wedge angle h.


Therefore, an analysis of the influence of grid refinement is conducted with computations done ondifferent grid refinement levels. The number of cells and the maximum cell size in the x- and y-directions are reported in Table 2. For viscous flow computations, the grid is refined in the di-rection normal to the boundary layer so that Dymin ¼ 10�5 m.The influence of the grid refinement has been examined on the two types of reflections:

1. In the RR: the angle and the impact point of the wedge-induced oblique shock wave on thesymmetry plane are well predicted by the three grids Grid1, Grid2, Grid4, but the pressurejump slightly differs (Fig. 10). The pressure plateau downstream of the incident shock indicatesthe uniform zone not yet altered by the expansion fan.

2. In the MR: the Mach stem position along the x-axis, its height h normalized by the wedgelength w and the normalized subsonic pocket length Lsp=w corresponding to each grid are re-ported in Table 3 and comparisons between Mach number contours obtained on the differentgrids are seen in Fig. 11.

Fig. 9. Normalized Mach stem height h=w vs. wedge angle hw (hysteresis curve) comparisons: experimental data and

numerical results: H=w ¼ 0:37; Grid (560 130); experiments see [26]; LCP-FCT: see [4].

Table 2

Different grids and average cell size: H=w ¼ 0:42

Number of cells Maximum cell size (m)

x-direction y-direction

Grid1 140 072 10�3 10�3

Grid2 280 072 5 10�4 10�3

Grid3 280 144 5 10�4 5 10�4

Grid4 560 144 2:5 10�4 5 10�4

Grid5 560 288 2:5 10�4 2:5 10�4

Grid6 1120 288 1:25 10�4 2:5 10�4


Fig. 10. Pressure along the symmetry plane for different grid refinement levels in a RR configuration. Euler simulations

with Roeþ H -correction and Superbee-type limiter. H=w ¼ 0:42; hw ¼ 22:66�.

Table 3

Normalized Mach stem height h=w and subsonic pocket length Lsp=w for different grid refinement levelsa

Mach stem Subsonic pocket length (Lsp=w)(normalized)Position (m) Height (h=w) (normalized)

Grid1 RR

Grid2 RR

Grid3 0.0949 0.036 0.337

Grid4 0.0952 0.035 0.341

Grid5 0.0944 0.042 0.342

Grid6 0.0947 0.041 0.347aH=w ¼ 0:42; hw ¼ 22:66�; Euler simulations with Roeþ H -correction and Minmod limiter.

Fig. 11. Shock reflections for different grid refinement levels, Euler simulations with Roeþ H -correction and Superbee-

type limiter. H=w ¼ 0:42; hw ¼ 22:66�; Mach number contours.


First, computations with grid cell size of the order of 1 mm give poor results in the present case:with Grid1 and Grid2, an RR occurs instead of an MR. This shows that to get closer to the Von-Neumann criterion, it is necessary to refine the grid. With finer grids, the MR is obtained asexpected. Then, two types of comments can be made because the grid refinement has been donealternatively in x- and y-directions. The refinement in the x-direction tends to move the Machstem position whereas the refinement in y-direction tends to modify the Mach stem height. Thefiner the grid is, the higher the Mach stem is, until grid convergence. Results on Grid5 and Grid6are rather similar except the width of Mach stem. Therefore, we consider that grid convergencehas been obtained with Grid5. However, Grid4 appears to be a good compromise betweencomputational time requirements and accurate results. In view of these comparisons, all the re-sults given in the next sections have been computed on Grid4.

4.4. Influence of the numerical schemes

To get a numerical hysteresis curve as close to the theoretical one as possible, it is also im-portant to use accurate schemes. Comparisons between results obtained with different schemes aredisplayed in the following subsections. All the following figures display zoomed-out contours nearthe triple point.

4.4.1. Influence of the approximate Riemann solvers: HLL, HLLC, Roeþ H -correctionThe numerical simulations have been done with several upwind schemes: HLL, HLLC and Roe

with H-correction. Comparisons between the results obtained with these schemes are shown inFig. 12. The HLLC scheme gives poor results in the present configuration: there seems to be ashock instability, leading to weird contours just behind the Mach stem. This is typical of upwindschemes that are not very dissipative or not corrected by an entropy correction, as shown bySanders et al. [20]. Roe scheme without H-correction would give similar results. HLL schemethat is more dissipative gives a Mach stem without instability but whose height is smaller than theone given by Roeþ H-correction scheme. Moreover, the Kelvin–Helmholtz-type rollups that

Fig. 12. MR for different solvers: Euler simulations with Minmod limiter: H=w ¼ 0:37; hw ¼ 24�; Grid (560 130);

temperature contours.


generally develop behind the triple point for large enough wedge angles until the steady state isreached do not develop as much with HLL solver as with Roeþ H-scheme. The influence of thesolvers on the subsonic pocket length is rather weak: the subsonic pocket with HLL is less than2% shorter than with Roeþ H -scheme (Lsp=w ¼ 0:417). These comparisons show that dissipativesolvers do not provide sufficiently precise results. In the present study, the RR–MR transition iscloser to Von-Neumann criterion with Roe solverþ H -correction than with the other schemesused here.It can be noted that the two versions of entropy correction proposed by Sanders et al. [20] on

one hand and by Pandolfi and D’ambrosio [22] on the other hand have been implemented and thecorresponding results compared. In the present computational test case, there is no noticeabledifference between the results—Mach stem height or slip surface—obtained with the two entropycorrections.

4.4.2. Influence of the slope limiters: Minmod, Superbee-type, van LeerSuperbee-type and van Leer slope limiter results are very close to each other as well for theMach

stem—position and height—as for the subsonic pocket length, whereas numerical simulations withMinmod limiter give different results (see Fig. 13). Here again, one sees that the numerical dissi-pation tends to decrease the Mach stem height. In particular, with hw ¼ 22� corresponding to datagiven by Hornung and Robinson [26], an RR configuration is obtained with Minmod limitor,whereas an MR is expected compared to experimental data, and obtained with the other slopelimiters that are less dissipative. Moreover, the Kelvin–Helmholtz-type instability of the slip sur-face does not develop behind the triple point when the simulation is done with Minmod limiterbecause Minmod limiter is too dissipative.

4.5. Influence of the viscous effects

In Fig. 14 one shows a comparison between results with and without viscous and thermaldissipation. As expected when the flow viscosity is taken into account, the development of the

Fig. 13. MR for different slope limiters: Euler simulations with Roeþ H -correction: H=w ¼ 0:37; hw ¼ 22�; Grid

(560 130); temperature contours.


boundary layer along the wedge surface leads to increase the wedge angle that is seen by theincoming flow; therefore, the wedge-induced oblique shock angle is larger and the Mach stemmoves backward and is higher than without flow viscosity. The difference of shock angles betweenan Euler simulation and a NS one is lower than 0.5� (for hw ¼ 22:66� and 27.46�). The differencebetween the Mach stem positions from an Euler computation and from a NS computation isabout 1%. The Mach stem height is about 4% higher and the subsonic pocket is about 1% longerin a viscous flow than in an inviscid flow. As for the type of wall boundary condition (adiabatic orisothermal), it has almost no effect on the wedge-induced shock angle, and very little effect on theMach stem. However, in order to better measure the influence of the sole viscosity on the Machstem position and height, a NS computation with slip conditions on the wall has been conductedand the corresponding results have been compared to Euler results. For that test case, theunit Reynolds number is 10 times smaller than the one given in Table 1—by decreasing q1 by a

Fig. 14. MR in inviscid and viscous flows: simulations with Roeþ H -correction and Superbee-type limiter:

H=w ¼ 0:42; hw ¼ 22:66�; Grid (560 144); temperature contours.

Fig. 15. MR in inviscid and viscous flows with slip boundary conditions for NS computations: simulations with

Roeþ H -correction and Superbee-type limiter: H=w ¼ 0:42; hw ¼ 22:66�; Grid (560 144); temperature contours.


factor of 10—leading to Rew¼0:0805m ¼ 2:6 105. Looking at results in Fig. 15, it appears that,in the present test case and in these conditions, the viscosity has almost no effect on the Machstem.

5. Conclusions

Numerical simulations of shock reflections in steady supersonic flows have been conducted witha code that solves Navier–Stokes and Euler equations by means of a finite volume method wherethe fluxes are discretized according to second order accurate TVD upwind schemes. The com-putations have displayed the hysteresis phenomenon encountered in shock reflections. A partic-ular emphasis has been given to show that the theoretical hysteresis curve could be well predictednumerically provided the grid mesh is fine enough and the numerical schemes accurate enough. Ithas been shown that the choice of the numerical method has a significant impact on the quality ofthe predicted shock reflections and in particular on the predicted Mach stem position and height.Our work indicates that the Roe scheme with the H-correction and Superbee-type or van Leerslope limiter gives the most accurate results. The Kevin–Helmholtz-type instability of the slip lineemanating from the triple point is visible only with accurate enough solvers and slope limiters, andfor large enough wedge angles, and vanishes after the steady position of the Mach stem is reached.Moreover, comparisons between Euler and Navier–Stokes computations have shown the negli-gible influence of the flow viscosity on the Mach stem height and on the scale of the phenomenon.Comparisons between numerical results and experimental data have shown a good agreement forthe small wedge angles.

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