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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids (2011)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.2701

An upwinded state approximate Riemann solver

B. Srinivasan1, *, , A. Jameson2 and S. Krishnamoorthy1

1Department of Applied Mechanics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India2Department of Aeronautics and Astronautics, Stanford University, Durand Building, 496 Lomita Mall, Stanford,

CA 94305-4035, USA

SUMMARY

Stability is achieved in most approximate Riemann solvers through flux upwinding, where the flux at theinterface is arrived at by adding a dissipative term to the average of the left and right flux. Motivated by theexistence of a collapsed interface state in the gas-kinetic BhatnagarGrossKrook (BGK) method, an alter-native approach to upwinding is attempted here; an interface state is arrived at by taking an upwinded average

of left and right states, and then the flux is calculated as a function of this collapsed interface state. Thisso called state-upwinding approach gives rise to a new scheme called the linearized Riemann solver forthe Euler and NavierStokes equations. The scheme is shown to be closely associated with the Roe scheme.It is, however, computationally less expensive and gives qualitatively comparable results over a wide rangeof problems. Most importantly, this scheme is found to preserve stationary contacts while not exhibiting thecarbuncle phenomenon which plagues the Roe and other contact-preserving schemes. The scheme is there-fore motivated as a new starting point to analyze the origin of the carbuncle phenomenon. Copyright 2011John Wiley & Sons, Ltd.

Received 17 March 2011; Revised 7 September 2011; Accepted 18 September 2011

KEY WORDS: finite volume; hyperbolic; hypersonic; compressible flow; euler flow; NavierStokes;transonic

1. INTRODUCTION

Finite volume schemes have been widely used to derive computational schemes for flow problems,

because of their desirable property of maintaining conservation at the level of discretization. Various

finite volume schemes arise depending on how the flux at the interface between two cells is formu-

lated. A primary use of finite volume schemes has been in solving problems with stationary or

moving discontinuitieswhere the property of being conservative is especially useful in maintain-

ing the right discontinuity speed. Such discontinuities often arise naturally in hyperbolic partial

differential equations describing conservation laws such as the Euler equations.

In addition to conservation, problems involving shocks are also found to require additional dis-

sipative terms for stabilization and in the absence of any explicit shock tracking for shock

capturing. These stabilization or artificial diffusion terms have been motivated and derived in var-

ious ways some schemes add these terms explicitly and others use some feature of the physics ofthe problem equivalent to adding some stabilization terms. In the latter category, one of the most

popular class of finite volume methods employed in CFD problems is the one that employs Riemann

solvers. First suggested by Godunov [1], this method involves a piecewise constant discretization

of the fluid properties under examination and then calculating the evolution by means of solving a

series of Riemann problems. In order to avoid the expense of an exact Riemann solver, Roe pio-

neered the use of cheaper, approximate Riemann solvers [2]. Roes method, which is still widely

*Correspondence to: B. Srinivasan, Department of Applied Mechanics, Indian Institute of Technology Delhi, Hauz Khas,New Delhi-110016, India.

E-mail: [email protected]

Copyright 2011 John Wiley & Sons, Ltd.


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AN UPWINDED STATE APPROXIMATE RIEMANN SOLVER

3. Apply the upwinding constraint on the Riemann invariants from the left and right state as

Jinterfacei D

JLi , ci > 0

JRi , ci < 0.(1)

4. Recompute the primitive (or conservative) variables at the interface, W.WL, WR/ of the

system from the characteristic variables J

interface

i obtained from the previous step.5. Compute the interface flux F.WL, WR/D F .W

/.

2.1.1. Scalar advection. For the scalar linear advection equation ut C cux D 0, upwinding can beachieved by evaluating

ujC1=2 D

uL , c > 0

uR , c < 0.(2)

Here, the states uL and uR represent, respectively the left and right estimates for the interfacestate. These estimates can be obtained from some local interpolation defined in the cells. Then, the

interface flux can be evaluated as f .ujC1=2/D cujC1=2. For constant interpolants within each celland c > 0, this results in the familiar upwind scheme

Duj

DtC c

uj uj1

xjC1=2 xj1=2D 0. (3)

2.2. System of equations

The aforementioned method can be extended to the linear hyperbolic system of equations

Wt CAWx D 0. (4)

State upwinding can be achieved by writing the system in characteristic form and upwinding the

Riemann invariants along the respective characteristics. For example, the linear system of equations

ut C cvx D 0,

vt C cux D 0,(5)

have the characteristic speeds c and Riemann invariants u v. State upwinding can now beperformed using

uC v D

uLC vL , c > 0

uR C vR , c < 0,(6)

u v D

uR vR , c > 0

uL vL , c < 0,(7)

where u and v are the desired upwinded-state variables. These two equations can now be solved forujC1=2 and vjC1=2, and the flux can be evaluated as

f .WjC1=2/D

"cvjC1=2

cujC1=2

#. (8)

It is worth noting that the final result is equivalent to the Roe and exact Riemann solvers for

the linear system of equations. For nonlinear systems in which Riemann invariants can be found, a

state upwinding procedure via Riemann invariants can be used. For example, for the 1-D shallow

water equations, "u

gh

#t

C

"u2

2C gh

ugh

#x

D 0. (9)

Copyright 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2011)

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B. SRINIVASAN, A. JAMESON AND S. KRISHNAMOORTHY

The characteristic speeds are u c and Riemann invariants are u 2c

u 2c D

uL 2cL , u0 c0 > 0

uR 2cR , u0 c0 < 0,(10)

where the state W0 is evaluated from some suitable average ofWL and WR. The conservative fluxes

can now be evaluated as in the linear case. This upwinded-state scheme for the nonlinear system isnow no longer equivalent to the Roe scheme.

2.3. State upwinding for Euler equations

We can now attempt to apply the aforementioned method to the 1-D Euler equations. These can be

written in characteristic form as @

@t

1

c2@p

@t

C u

@

@x

1

c2@p

@x

D 0, (11)

@u

@tC

1

c

@p

@t C .uC c/

@u

@xC

1

c

@p

@xD 0, (12)

@u

@t

1

c

@p

@t

C .u c/

@u

@x

1

c

@p

@x

D 0, (13)

where c Dq

p

. These can be written in the equivalent form

D

Dt

1

c2Dp

DtD 0, (14)

DCu

DtC

1

c

DCp

DtD 0, (15)

Du

Dt

1

c

Dp

DtD 0, (16)

where

D

Dt

@

@tC u

@

@x, (17)

DC

Dt

@

@tC .uC c/

@

@x, (18)

D

Dt

@

@t C .u c/

@

@x , (19)

are the directional derivatives along the respective characteristic lines.

2.3.1. Riemann invariants for Euler equations. The characteristic equations previously written are

not in invariant form, that is, one cannot directly determine from the aforementioned equations

what the conserved quantity along each characteristic is. Equation (14) can be multiplied by the

integrating factor 1

to obtain the invariant form

Dv0

DtD 0, (20)


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where v0 D log.p/, the entropy, is the invariant along this characteristic. In the isentropic

case, the first characteristic equation becomes redundant and the other two equations can be directly

integrated to obtain the invariants

v1 D uC2c

1, (21)

v2 D u2c

1. (22)

Perhaps, because of this special case, it is a common practice to represent the invariants in the

general, anisentropic case as

v1 D uC

Zdp

c, (23)

v2 D u

Zdp

c. (24)

However, the term uC Rdpc

cannot be evaluated as an indefinite integral in the general case as u is

a state function, whereasRdp

c is path dependent.This, of course, does not preclude the possibility of there being an integrating factor for these

characteristics. However, the existence of an integrating factor for differential forms with three

independent variables is not guaranteed [11]. In fact, as will be shown here, an integrating fac-

tor does not exist for either of these characteristics in general, nonbarotropic case when p and areindependent variables.

We begin by assuming that there exists an integrating factor for the third characteristic equation;

that is, the differential form du dpc

. Then, using c2 D p

, we obtain the differential form

cdu 2cdc c2d.

This has an integrating factor if and only if

du 2dc cd

has an integrating factor. Now, the Pfaffian u1.x, y, /dx C u2.x, y, /dy C u3.x, y, /d has anintegrating factor if and only if [11]

!u r !u D 0. (25)

We get for the present Pfaffian

!u r !u Dc2, (26)

and hence, no integrating factor exists along this characteristic. Note that it is not that an analytical

expression cannot be found for the invariants or that they are not well-known properties of the fluid;

there are simply no invariants (in the sense of properties preserved along the characteristics) in the

general case when p and are independent.An alternate interpretation of the characteristics [12] is that they are the path lines along whichsmall perturbations of flow variables travel. For example, for the scalar wave equation utCcux D 0,

perturbations u of the mean flow u0.x, t/, satisfyD.u/Dt

= 0, that is, u D const . along xct D 0.Similarly, for the Euler equations, one has

D

Dt

p

c2

D 0, (27)

Note that this substitution only makes the existence of three independent variables explicit and does not add any furtherconstraints to the original differential form.


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DC

Dt

uC

p

c

D 0, (28)

D

Dt

u

p

c

D 0. (29)

Using the aforementioned expressions, we can now find the relations for the numerical flux for

the LRS solver.

3. LINEARIZED RIEMANN STATE SCHEME

The interpretation in the previous section can now be utilized to derive an upwinded-state scheme.

For example, for the advection equation, we define the mean state

u0 DuLC uR

2. (30)

Unless the flow is extremely well resolved by the grid, in a general region of the flow, the left and

right state and the aforementioned average are all different from each other. Then, we may interpret

the situation at the interface as follows The left cell tries to change this average, interface state u0

to its current state, that is, uL, whereas the right cell similarly attempts to change it to uR. That is,the left cell effects a perturbation uL D uL u0 over the average state, whereas the right effectsthe perturbation uR D uR u0. Then, an upwinded average state at the interface may be obtainedas u D u0C u

, where

u D

uL , c > 0

uR , c < 0,(31)

where the perturbation has been upwinded using the characteristic direction. As can be quickly seen,

this results in exactly the same scheme as the one obtained via Equation (2).

For the Euler equations, one may similarly upwind along the characteristic directions, starting

from Equation (27). It is easiest to find an expression for the upwinded state starting from primitive

variables (we have not found any qualitative difference in using conservative variables for defining

the upwinded state)

W D

264

u

p

375 . (32)

We can define the upwinded state as

W DW0C W, (33)

where

W0 DWLCWR

2(34)

and W is obtained by solving

p

c20D

8 0

R pR

c2, u0 < 0,

(35)

uCp

0c0D

8 0

uRC pR

0c0, u0C c0 < 0

(36)


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u p

0c0D

8 0

uR pR

0c0, u0 c0 < 0.

(37)

The aforementioned process is equivalent to solving the linearized Riemann problem with the states

WL and WR for the state W. This particular upwinded-state scheme has hence been called the

LRS scheme here. It can be shown with a small amount of algebraic manipulation that the solutionto the aforementioned system can be written in the form

W DR0 sgn.0/ R10

WR WL

2

, (38)

where R0 is the matrix of Eigen vectors

RD

2664

1 2c

2c

0 1=2 1=2

0 c2

c2

3775

, (39)

evaluated at the state W0. Similarly,

R1 D

2664

1 0 1c2

0 1 1c

0 1 1c

3775 , (40)

and

D

264

u 0 0

0 uC c 0

0 0 u c

375 . (41)

It can be seen from Equation (38) that W D sgn.u0/WL when ju0j> c0, that is, as with the Roe

scheme, the upwinding is sharp when the flow is locally supersonic. As will be seen, this property

is essential for maintaining stationary shocks.

Significant computational savings can be made by noting that for ju0j< c0, the following simpleexpressions for the effective perturbations result:

p D 0c0uL, (42)

u DpL

0c0, (43)

D sgn.u0/

L

pL

c20

C

p

c20. (44)

In addition to the fact that the LRS uses only one flux evaluation per interface, whereas the Roe

(like all upwinded flux schemes) uses two; the existence of such simple expressions makes the LRS

scheme cheaper than the Roe scheme. As will be seen, this comes without any loss of quality of

shock or contact resolution. In fact, as will be seen later, there seems to be an actual advantage over

Roes scheme when it comes to handling the carbuncle phenomenon.


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4. SOME PROPERTIES OF THE LINEARIZED RIEMANN SOLVER SCHEME

4.1. Relation with Roes scheme

The form (38) is a reminiscent of the following way of expressing Roes scheme

F .WRoejC1=2/DF .WL/CF .WR/

2

jAjRoeWR WL

2

. (45)

To see the exact correspondence between Roe and LRS, we note that

WR DWLCWR

2

WL WR

2, (46)

WL DWLCWR

2C

WL WR

2. (47)

Hence,

F .WR/D F

WLCWR

2

A0

WL WR

2

CH.O.T, (48)

F .WL/D FWLCWR2

CA0 WL WR2

CH.O.T, (49)where H.O.T represents higher order terms. Similarly, for the LRS scheme,

F .WLRSjC1=2/D F

WLCWR

2

A0 sgn.A0/

WL WR

2CH.O.T. (50)

Combining these, we get

F .WLRSjC1=2/DF .WL/CF .WR/

2 jAj0

WR WL

2CH.O.T. (51)

The LRS and Roe scheme are then identical up until the lowest order dissipative term except for

the state at which jAj is evaluated. Surprisingly enough, this difference causes not only the expectedquantitative differences but qualitative differences as well, as will be seen in Section 7.

4.2. Treatment of discontinuities

Because of the way that the Roe state is constructed, that is,

fR fL D ARoe.WR WL/, (52)

it supports a stationary shock as well as a stationary contact. We try and investigate here LRSs

behavior in these situations.

Stationary shock: For a stationary shock, we have

c20 D pLC pR

LC R. (53)

It can be shown on the substitution of the RankineHugoniot jump conditions that for a station-

ary shock juLCuR

2 j > c0, if uL and uR satisfy the shock jump conditions and c0 is computed asmentioned earlier. Hence,

WjC1=2 D

WL u0 > 0

WR u0 < 0,(54)

)

F .WiC1=2/D

F .WL/ u0 > 0

F .WR/ u0 < 0.(55)

and because F .WL/D F .WR/ for a shock, the shock is maintained as is.


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Stationary contact:

Here,

pR D pL D p, (56)

uL D uR D 0, (57)

L R. (58)

Hence,

F .WR/D F .WL/D

0BB@

LuL

Lu2LCpL

LuLHL

1CCAD

0@ 0p

0

1A . (59)

Now, because

uL D pL D 0,

we have,

u D 0,

and

p D 0.

Then, from Equation (44),

D sgn.u0/

L

pL

C20

C 0

uL

C0(60)

D sgn.u0/L, (61)

)

WiC1=2 D

0C L 0 p

T , (62)

)

F .WiC1=2/D

24 0p

0

35 . (63)

Hence, the stationary contact is maintained.

One-point Shock : A more important property of the Roe scheme is its ability to hold an ideal

one point shock (see [13] for a proof). We see if something can be said about the LRS scheme in

this situation.

Consider the situation

WjD

8 is,

(64)

with WL and WR satisfying the normal shock jump conditions (i.e F .WL/ D F .WR/). Then, tomaintain this one point structure, we need

1. F .WL/D F .Wis1=2/ for equilibrium at is 1.2. F .Wis1=2/D F .WisC1=2/ for equilibrium at is .3. F .WisC1=2/D F .WisC 32

/ for equilibrium at is C 1.


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Condition .1/ would be met for any WA with uL > uA > uR, because, then u

RCuL

2< c0,

uLCuA

2< c0. Hence, Wis1=2 DWL and F .WL/D F .Wis1=2/.

Now, if condition .2/ is met, condition .3/ is automatically satisfied because F .WisC32/ D

F .WR/ D F .WL/ D F .Wis1=2/. We require that F .WL/ D F .WisC1=2/. It can be shown thatF .W1/ D F .W2/ supports only the two states given by the normal shock jump conditions. Hence,we need WisC1=2 DWL or WisC1=2 DWR. That is,

WisC1=2 DWACWR

2

1

2sgn.AisC1=2/.WR WA/. (65)

Consider WisC1=2 DWR. This is not possible because it would require that

sgn.AisC1=2/DI, (66)

where I is the identity matrix. This is not possible because from condition(1), uL > uA > uR > 0and hence, at least one eigenvalue ofAisC1=2 is positive.

Now, consider WisC1=2 D WL. This can be split into two further cases. If W0 DWACWR

2is such

that u0 > c0, then,

WisC1=2 DWACWR

2 1=2

WR WA

2DWA WL, (67)

because we want an intermediate state and not a zero-point shock. Hence, a supersonic intermediate

point is ruled out.

IfW0 DWACWR

2is such that u0 < c0, then, from Equations (42) and (43), we have

p DpACpR

2C 1=2.uA uR/0c0 (68)

u DuAC uR

2

C 1=2pA pR

0c0. (69)

Because we require that p D pL, this leads to

pL pACpR

2D 1=2.uA uR/0c0. (70)

Because we also need pL < pA < pR, we are led to

pL pACpR

2< 0. (71)

However, the RHS is > 0, because uAuR < 0. Hence, it is not possible to maintain a nonoscilla-tory one point shock with the LRS scheme. It was not found possible to prove analytically anything

further about what shock structure the LRS scheme does support. In practice, we have found that

for transonic flow, the LRS scheme maintains a shock structure very close to a one point shock, and

any post-shock oscillations are not significant.

It is worth noting here that it may be possible to maintain the one point shock by choosing the

intermediate state to be something different from the arithmetic average. As a quick examination

of the proofs for the zero-point shock and contacts show, the arithmetic average is not by itself

necessary in order to maintain them. However, we have not yet found any simple way to define an

intermediate state that satisfies the one point shock criterion. We leave this, for now, as an avenue

for future work.


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5. EXTENSIONS TO HIGHER DIMENSIONS

The LRS scheme can easily be extended to higher dimensions with little additional cost if one

solves, as is usual with other approximate Riemann solvers, the one-dimensional problem normal

to the face. In three dimensions, the Eigen values are u, u, u, uC c and u c, and the characteristicequations in face normal coordinates are

D

Dt

1

c20

D

DtD 0, (72)

DCu

DtC

1

0c0

DCp

DtD 0, (73)

Du

Dt

1

0c0

Dp

DtD 0, (74)

Dv

DtD 0, (75)

DwDt

D 0, (76)

with DDt

, DC

Dt, D

Dtdefined as before. Notice that the equations are identical to the one-dimensional

case except for the two new decoupled equations. Hence, one can find the interface flux by simply

solving the one-dimensional problem with the additional equationsv w

D sgn.u0/

vL wL

. (77)

If one wishes to avoid the costs associated with local rotation to the face normal coordinates, the

standard methodology of using An DAnxCBnyCC n could also be used where .nx , ny , n/ arethe face normals. Then, it can be shown that

Winterface DWLCWR

2C W, (78)

where W can be found by using the following:

1. For supersonic flow, ju0j> c0' u v w p

TD sgn.u0/

L uL vL wL pL

T. (79)

2. For subsonic flow, ju0j< c0

qn DpL

0c0(80)

p D 0c0.nxuLC nyv

LC nwL/D 0c0.nxq

L/ (81)

D sgn.u0/

pL

pL

c20

C

p

c20(82)

u D sgn.u0/.uL.n2y C n

2/ .nxnyv

LC nxnwL//C nxqn (83)

v D sgn.u0/.nxnyuL vL.n2x C n

2/C nxnw

L/C nyqn (84)

w D sgn.u0/.nxnuL vLnynC w

L.n2x C n2y//C nqn. (85)

Finally, Finterface D F .Winterface/ as earlier.


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6. NUMERICAL RESULTS

The aforementioned scheme was implemented into structured and unstructured Euler solvers

for a number of cases and proved to be a robust and accurate solver. Some of the results are

presented herein.

6.1. One-dimensional results

The LRS scheme was used to simulate two classic one-dimensional test cases Sods shocktube

test and the Woodward and Colella 1-D blast case.

1. Sods test case: In this case, a shock was placed in the middle of a computational domain

extending between x D 5, 5. The initial condition was specified as follows:

WL D .L, uL, pL/D .1, 0, 1/,

WR D .R, uR, pR/D .0.125, 0, 0.1/.

The boundaries are fixed at initial values and Figure 1 plots vs. x at t D 1.8.2. Woodward Collela 1D blast case: First discussed by Woodward and Colella [14], this is a fairly

severe test case involving strong shocks and good test of any schemes robustness. The initial

conditions prescribed are

WL D .L, uL, pL/D .1, 0, 1000/

for the leftmost one-tenth of the computational domain,

WR D .R, uR, pR/D .1, 0, 100/

for the rightmost one-tenth of the computational domain, and

WMD .M, uM, pM/D .1, 0, 0.01/

in between the two extremes. Reflecting boundary conditions are prescribed on the walls of

the domain. Figure 2 displays the result when this case is implemented using 500 points in the

computational domain.

It can be seen that the current scheme performs extremely well on these cases. Note that we have

not used any tunable parameters in the two cases previously mentioned. It is important to mention

here, however, that like Roes scheme, LRS does require some additional tunable dissipation in cer-

tain cases which generate expansion shocks in the Roe scheme. However, as seen in the following

results, in transonic 2D and 3D cases, even this additional dissipation is typically unnecessary.

5 0 50

0.2

0.4

0.6

0.8

1

x

Sods Test Case (t=1.8)

Theoretical Solution

LRS

Figure 1. Sods shocktube test using linearized Riemann solver.


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Figure 2. 1-D blast case using linearized Riemann solver.

6.2. Two dimensional results

The LRS scheme was implemented into the following two transonic solvers

1. Flo82 : This is a structured solver, which along with the LU-SGS (Lower-Upper Symmetric

Gauss-Seidel) scheme [15] and multigrid leads to excellent convergence. Figure 3 shows the

pressure contours for the standard case of a National Advisory Committee for Aeronautics

(NACA) 0012 airfoil at a free stream Mach number M1 D 0.8 and angle of attack D 1.25with this solver.

Figure 3. Pressure contours for a NACA 0012 airfoil with M1 D 0.8 and D 1.25 with the linearizedRiemann solver scheme. The solver used was FLO82-SGS.


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Figure 4 compares the Cp distribution on the surface of the airfoil. The LRS schemecompares well with the Convective Upwind Split-Pressure (CUSP) scheme[13]. (CUSP has

been used here mainly to compare the convergence rate, which in our experience, has been

extremely good for CUSP. The final flow field is very similar for CUSP and Roe.). The struc-

ture of the shock also seems to be close to a one point, ideal shock structure. Figure 5 has the

convergence history for the two schemes. It can be seen that while the CUSP converges much

faster than the LRS scheme, the performance of the latter is still good. In particular, there is nostalling due to insufficient dissipation at the sonic point which can happen for schemes such

as the LRS where there is no dissipation at the sonic point. This is perhaps because of the

inherent extra dissipation that results due to upwinding normal to the face (instead of using

truly multidimensional upwinding).

2. Flo76 : This is an unstructured solver which uses a cell-centered triangular mesh along with

a multistage time-integration scheme integrated with multigrid in order to accelerate conver-

gence. Figure 6 shows the pressure contours for the aforementioned case with LRS. Figure 7

compares the Cp distribution for the LRS scheme with the BGK scheme for the same case.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.5

1

0.5

0

0.5

1

1.5

X

Cp

LRS scheme with SGS

LRSCUSP

Figure 4. Surface Cp distribution for the linearized Riemann solver and CUSP schemes for a NationalAdvisory Committee for Aeronautics 0012 airfoil with M1 D 0.8. The solver used was FLO82-SGS.

Figure 5. Convergence history for L1 norm of the density residual for the SGS scheme.


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X

Y

0 2 4 6 8

-2

0

2

4

Figure 6. Pressure contours for a National Advisory Committee for Aeronautics 0012 airfoil withM1 D 0.8 and D 1.25 with the linearized Riemann solver scheme. The solver used was FLO76.

Figure 7. Surface Cp distribution for the linearized Riemann solver and CUSP schemes for a NationalAdvisory Committee for Aeronautics 0012 airfoil with M1 D 0.8 and D 1.25. The solver used was

FLO76.

6.3. Three-dimensional results

The LRS scheme was also tested for a couple of three-dimensional configurations on FLO3XX

[16].

1. An ONERA M6 wing with M1 D 0.84 and D 3.06. The computation was done on 316, 000node tetrahedral mesh. The pressure contours on the surface wing are shown in Figure 8. The

surface Cp plots for the LRS and CUSP schemes are compared at the cross-sections x=c D 0.2and x=c D 0.4 in Figures 9 and 10.

2. A complete aircraft configuration : The Falcon business jet on a 356, 000 tetrahedral mesh.Density contours are plotted in Figure 11, and a comparison of the multigrid convergence

history with CUSP is given in Figure 12.

7. QUALITATIVE DIFFERENCES WITH THE ROE SCHEME

As can be seen from the results in the preceding section, LRS is a competitive approximate Riemann

solver, producing qualitatively similar results to Roes scheme on a number of numerical test


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Figure 8. Pressure contours with the linearized Riemann solver for the ONERA M6 wing at M1 D 0.84and D 3.06.

1 2 3 4 5 6 7 8 9 10 111

0.5

0

0.5

1

1.5

X

Cp

Cp

distribution at 20% chord

LRSCUSP

Figure 9. Surface Cp distribution at x=c D 0.2 for the ONERA M6 wing at M1 D 0.84 and D 3.06.Comparison of the linearized Riemann solver and CUSP schemes.

3 4 5 6 7 8 9 10 11 121

0.5

0

0.5

1

1.5

X

Cp

Cp

distribution at 40% chord

LRSCUSP

Figure 10. Surface Cp distribution at x=c D 0.4 for the ONERA M6 wing at M1 D 0.84 and D 3.06.Comparison of the linearized Riemann solver and CUSP schemes.


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Figure 11. Falcon business jet. Nondimensional density contours from 0.6 to 1.1.

0 50 100 150 200 250 300 350 400 450 50010

10

109

108

107

106

105

104

103

102

101

100

Work

Log(Error)

Convergence history for the Falcon business jet

LRSCUSP

Figure 12. Convergence history of the density residual for the Falcon business jet. Comparison of thelinearized Riemann solver and CUSP schemes.

cases. Further, both Roe and LRS behave identically on stationary contacts, stationary shocks,

and both produce expansion shocks in the absence of additional dissipation. This, combined with

Equation (51), would lead one to expect that the LRS scheme would at best offer a cheaper alterna-

tive to Roes scheme, because the two are identical up to the first order dissipation term. A simple test

case reveals that the higher order terms can cause surprising qualitative differences between the two.

7.1. A shock stability test

The test case that reveals this difference is that of a planar stationary shock subject to two dimen-

sional perturbations. Roes scheme is known to be unstable for this case beyond a certain Mach

number. In fact, an analysis in [6] shows that for schemes of the form

F .WL, WR/D1

2.FLCFR/

1

2AD.WL, WR/.WL WR/, (86)

strict stability on this problem is incompatible with exact resolution of contact discontinuities. Roes

scheme, which is unstable to small perturbations, is of the aforementioned form, whereas LRS is


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Figure 13. Roes scheme for shock subject to a two-dimensional perturbations with Ms D 5.0 , t D 3.3.The situation shown here is on its way to instability.

not. Because LRS and Roe are effectively equivalent in the first-order terms, one would expect thatLRS would also be unstable. We explore here if this is actually true in practice.

The problem being considered in the following is a simpler version of a problem proposed by

Quirk [5] as a diagnostic tool for the hypersonic carbuncle phenomenon. Dumbser et al. [17] have

done a linearized matrix stability analysis for this problem for Roes scheme. They used a uniform

25 25 grid with a normal steady shock as the initial unperturbed flow. The initial condition wasperturbed by a uniform random perturbation of relative order 106 102 in every cell.

Roes scheme exhibits three different ranges of behavior when subjected to perturbations

1. It was found to be stable for Mach numbers lower than 1.5.

2. It was found to be quasi stable for intermediate Mach numbers.

3. For high Mach numbers, the shock was unstable to mean flow perturbations, and for suffi-

ciently large Mach numbers, even round-off errors were found to be sufficient to set off the

instability. Figure 13 shows the solution for Ms D 5 on its way to instability.

The LRS scheme on the other hand, was found to be stable even for an upstream Mach number

of100 even with perturbations of relative order as high as 102 of mean flow (higher perturbationswere not tested). Figures 14 and 15 are representative solutions. The LRS scheme seems to be stable

for this problem.

7.1.1. Hypersonic blunt body. The aforementioned shock stability case is used as a diagnostic tool

for determining if a given scheme would exhibit the carbuncle phenomenon [7] for the hypersonic

blunt body test case. The carbuncle phenomenon is an incorrect but entropy satisfying solution to

a hypersonic flow past a blunt body. This occurs usually when the solution has nearly grid-aligned

shocks, computed with schemes which have low dissipation. The origins for this are unknown and

it is still a topic of active research (see [17] for a recent review ).

Roes scheme exhibits this phenomenon. The simplest fix for this phenomenon for Roes schemeis to add an empirically determined large dissipation to the linear vorticity mode via Hartens fix

[18] applied to the eigenvalue D u corresponding to the linear vorticity mode. Apart from beingphysically and mathematically unjustified, [5], this is a somewhat undesirable fix in that it might

affect computations of viscous shear layers adversely. Further, because of its empirical nature, the

amount of dissipation that needs to be added for a physically valid solution cannot be determined a

priori. This makes the fix unreliable for new test cases.

This high Mach number is used here only to indicate the robustness of the underlying Euler solver even though the Eulerand Navier-Stokes equations are no longer valid for this Mach number.


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Figure 14. Linearized Riemann solver scheme for shock subject to two-dimensional perturbations withMs D 2.0 , t D 10. The perturbations show no growth.

Figure 15. Linearized Riemann solver scheme for shock subject to two-dimensional perturbations withMs D 100.0 , t D 100. The shock is stable to perturbations.

Because the LRS scheme was stable for the planar shock, it was tested to see if it exhibited the car-

buncle. The test case chosen was one which was known to produce the carbuncle for Roes scheme.

A blunt body with the free stream Mach number M1 D 6 on a 160 64 grid was chosen.Figures 16 and 17 show the carbuncle phenomenon for this configuration with the use of Roes

scheme without Hartens fix on the D u mode (the fix was still applied to the other modes for con-vergence). As can be seen, the solution is not a detached shock as desired. It should be emphasized

here that this is a completely convergent solution, albeit the physically incorrect one.

The converged LRS results for the same problem (without any fix on any mode) are shown in

Figures 18 and 19. The solutions are indeed carbuncle free through a post shock overshoot that

is seen. This is probably because of the LRS not being able to support an ideal one-point shock

structure. It has been argued in the literature (see [8] for example) that numerical shock structure is

intimately tied to the carbuncle phenomenon.

Ismail et al. [9] suggest a one-dimensional carbuncle problem which tests a one-dimensional shock

with an artificially introduced intermediate state. They find that almost no scheme leads to an ulti-

mately stable shock structure for all possible intermediate states, but the better scheme performs on

this test, the better it performs on the carbuncle. Figure 20 shows LRS performance on this test case

for a MD 6 shock. The remarkable similarity of LRS behavior on this problem to its performanceon the hypersonic problem reinforces the belief that shock structure might be the key to finding the

reason behind the carbuncle phenomenon. An appropriate way to fix the overshoot, while keeping


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X

Y

-2 -1 0 1 2

-2

0

2

4

Figure 16. Roe scheme for a hypersonic blunt body with M1 D 6. Convergent scheme exhibiting thecarbuncle solution. Isothermal contours are plotted.

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 10.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

X

Cp

Figure 17. Cp along the centerline for the Roe scheme. Hypersonic blunt body with M1 D 6.

accuracy intact could not yet to be found. We did find, however, that using a Roe average instead of

an arithmetic average for the intermediate state does alleviate (but not eliminate) the problem.

Further insight into this overshoot can be gained by looking at the corresponding entropy plots.

Figures 22 and 24 show that the situation is seemingly reversed with entropy viz. The entropy in

CUSP shows an overshoot, whereas LRS shows a (nearly) smooth entropy profile. Figures 21 and

23 show entropy contours over the blunt body for LRS and CUSP respectively. The local entropy

overshoot, however, maybe interpreted as an approximation to the entropy profile for a viscous shock

profile; as is known [19], the viscous shock entropy profile is not monotonic and shows a maximum

within the shock. This then, might give a non ad hoc method for correcting the pressure overshoot

in LRS via adding an appropriate viscous term to Equation 35. We are currently investigating

this approach.

On the pessimistic end, it is entirely possible, the fact that LRS does not show that the carbuncle

might be intimately tied to the fact that it exhibits an overshoot. However, its excellent performance

on the other one-dimensional tests in Section 6.1 leads us to believe that this is probably not the case.


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Figure 18. Linearized Riemann solver scheme for a hypersonic blunt body with M1 D 6. Convergentscheme exhibiting no carbuncle. Isothermal contours are plotted.

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 12.5

2

1.5

1

0.5

0

X

Cp

Figure 19. Cp across stagnation line using linearized Riemann solver scheme for a hypersonic blunt bodywith M1 D 6. Slight overshoot is observed across the shock.

0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.51.4

1.2

1

0.8

0.6

0.4

0.2

0

0.2

x

p

1D "carbuncle" using LRS (M0= 6, = 0.7)

Figure 20. 1-D carbuncle using linearized Riemann solver at MD 6. Note the similarity with the behaviorfor hypersonic flow over a blunt body.


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Figure 21. Linearized Riemann solver scheme for a hypersonic blunt body with M1 D 6. Contours of

constant entropy are plotted.

X

S

-2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -10

0.5

1

1.5

2

2.5

3

3.5

Figure 22. Entropy profile along the centerline for the linearized Riemann solver scheme (M1 D 6).

Figures 25 and 26 show that the CUSP scheme does not exhibit the carbuncle phenomenon either,

even when no additional dissipation is added to any mode.

Further, it gives an extremely clean and sharp shock and does not exhibit the overshoot that the

LRS scheme exhibits. The sharp shock resolution here is much better than the shock profiles typ-

ically obtained by the use of other schemes not exhibiting the carbuncle such as the HLL: Harten

Lax Van-Leer.

The LRS and CUSP schemes were also tested for freestream Mach numbers as high as M1 D 50and were found to be carbuncle free. The LRS, however, had problems in convergence for high

Mach numbers while the CUSP was very well behaved. This very robust behavior of CUSP for the


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Figure 23. CUSP scheme for a hypersonic blunt body with M1 D 6. Contours of constant entropyare plotted.

X

S

-2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -10

0.5

1

1.5

2

2.5

3

3.5

Figure 24. Entropy profile along the centerline for the CUSP scheme. (M1 D 6).

hypersonic case seems to have been unreported in the literature. We therefore think that, because of

its excellent performance on the hypersonic problem, CUSP offers one of the best practical schemes

for such flows. However, as with all Riemann solvers, CUSP does have its lacunae CUSP, unlike

LRS, does not resolve steady contacts exactly.

Finally, we note that LRS does have a pressure difference term in the mass flux, as can be easily

from Equations (87) and (88) and yet it does not, strictly speaking, exhibit the carbuncle phe-

nomenon. This therefore seems to show that Lious conjecture [20] that it is this pressure difference

in the mass flux that causes the carbuncle is probably wrong.


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Figure 25. CUSP scheme for a hypersonic blunt body with M1 D 6. Convergent scheme exhibiting nocarbuncle. Isothermal contours are plotted.

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 12

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

X

Cp

Figure 26. Cp along the centerline for the CUSP scheme. A clean one point shock is captured. Hypersonicblunt body with M1 D 6.

8. CONCLUSIONS

Using a simple linearized model for the propagation of information along the characteristics, an

upwinded-state scheme, LRS, was derived to compute interface fluxes for the Euler and Navier

Stokes equations. This scheme is significantly cheaper compared with the popular Roe scheme andperforms similarly to Roe on a number of one-dimensional and multidimensional problems. Most

interestingly, it seems to be closely related mathematically to Roe in that they are identical to first-

order terms; yet, it is stable on a shock stability problem where the Roe scheme exhibits instability.

The scheme also seems to violate Lious conjecture on the carbuncle problem despite an unphysical

overshoot while not exhibiting the carbuncle.

We foresee the following applications for LRS

1. As an approximate Riemann solver wherever Roe scheme is used. LRS offers a cheaper,

less complicated alternative. This includes, for instance, incompressible flows using pseudo-

compressibility.


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2. As a solver for the hypersonic blunt body problem where Roe cannot be applied in an

unmodified form.

3. As a starting point for analysis of the reasons for the carbuncle and shock stability phe-

nomenon. The fact that the LRS and Roe schemes exhibit vastly different behaviors on the

same problem when they are closely related mathematically might offer a new way to analyze

this difficult problem.

Finally, as an ancillary result, the CUSP scheme was found to be a very efficient and robust solver

for the hypersonic blunt body problem.

ACKNOWLEDGEMENTS

This work was partially funded by a grant from the Aeronautics Research and Development Board(Aerodynamics), DRDO, Government of India. Balaji Srinivasan would like to thank Mr. Vivek Hariharanfor his help in generating some of the results.

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