Compact upwind algorithms for viscous compressible flow

Sadhan& Vol. 24, Parts 1 & 2, February & April 1999, pp. 147-174. © Printed in India.

Compact upwind algorithms for viscous compressible flow calculations

K S RAVICHANDRAN

Computational & Theoretical Fluid Dynamics Division, National Aerospace Laboratories, Bangalore, India e-mail: ksr @ ctfd.cmmacs.ernet.in

Abstract. A family of high-order accurate compact upwind difference operators have been used together with the split fluxes of the KFVS (Kinetic Flux Vector Splitting) scheme to obtain high order semi-discretizations of nonlinear convective terms in 2-D Navier-Stokes equations governing viscous compressible flows. A TVD multi-stage Runge-Kutta time stepping scheme is used to compute steady states for selected transonic/supersonic inviscid and viscous flow problems which indicate the higher accuracy and low diffusion realizable in such schemes.

Keywords. Euler equations; compressible flow; compact schemes.

1. Introduction

It is generally believed that accurate simulation of fluid flow with multiple and wide range of spatial scales and structures is a difficult task except through spectral approximations. However, at present the use of spectral approximations is limited by their applicability to simple geometries with generally periodic boundary conditions. While it is possible in principle to devise very high order accurate finite-difference approximations to the spatial derivatives occurring in the Euler or Navier-Stokes equations, recent studies (Lele 1992) indicate that considerations based on mere formal accuracy (low truncation error) are not sufficient to ensure uniform resolution of the wide range of scales that may be present in the solution that one wants to approximate. In particular, the smaller length scales are often poorly represented in the conventional high order difference approximations. While uniform resolution of the whole range of scales (ignoring aliasing errors) is exclusively the property of spectral methods, Lele (1992) has shown that it is possible to devise, on a given stenci l, difference schemes that have much better resolution properties than conventional difference schemes of comparable order of accuracy. The price

Abbreviations: CUD - compact upwind difference; FDS - flux difference splitting; MUSCL - monotone upstream schemes for conservation laws; MUSCL3 - third-order MUSCL; TVD - total variation diminishing; TVDSYM - TVD symmetne.

147

148 K S Ravichandran

paid is that one is required in general to invert a tri-diagonal system of linear equations to compute the derivatives. This class of difference approximations, which are called compact difference schemes, may be of the centred or upwind type and are classically known as rational function (as opposed to polynomial function) or Pade' approximants to the derivatives. The high cost as well as the limited geometry adaptability of the spectral methods has in recent years given the impetus to investigate the effectiveness of compact difference approximations for a wide range of inviscid and viscous flow problems.

In this work, we study a class of compact upwind difference approximations to the Euler and Navier-Stokes equations governing compressible fluid flow. The motivation is to investigate the suitability of these schemes for the computation of viscous transonic internal/external flows, wherein isolated and/or interacting shock waves may be present, giving rise to complex shock wave/boundary layer interaction zones. Since the viscous terms may be approximated to a high degree of accuracy by centred compact formulae, a pre-requisite for accurate viscous computations is the sharp and monotone resolution of shock discontinuities in such flows. This work is an attempt to study the suitability of compact schemes with this purpose in mind.

The compact upwind algorithms proposed in this paper preserve high order accuracy in the smooth regions, but reduce to first-order accuracy in the vicinity of shocks and other solution extrema because of the use of minmod flux limiters. Such a compromise is obvi- ously unavoidable if one wants to prevent spurious, high frequency numerical oscillations from polluting the accuracy of the computed solution in such narrow transition regions. The centred compact approximations proposed in (Lele 1992) are unsuitable for computing flows with discontinuities because of the odd--even decoupling which gives rise to high frequency oscillations even in smooth regions. Killing such oscillations requires the introduction of commensurately high order background dissipation terms, which implies that the compactness of the stencil is lost. On the other hand, as found by Cockburn & Shu (1992), introduction of minmod limiters in centred schemes, affects the solution accuracy even in smooth regions if there are oscillations to be suppressed. The use of compact upwind operators together with the KFVS split fluxes (Mandal & Deshpande 1994) is mo- tivated by these considerations. Another reason for the choice of KFVS splitting is that, as compared to the van Leer or Steger-Warming splittings which have vanishing eigenvalues, the former is expected to maintain its robustness in high-order computations of mixed flow fields with sonic points present.

From a different point of view, the present work may be viewed as an approach to generating high-order KFVS schemes on structured meshes. Since the schemes are formulated in a finite-volume framework with the equations cast in strong conservation form, the Lax-Wendroff theorem guarantees that limit solutions are indeed weak solutions to the conservation laws, the Euler equations in the present case. Additionally, use of the entropy-consistent KFVS splitting (Mandal & Deshpande 1994) results in a family which has been used to compute transonic flow solutions that are free of sonic glitches or expansion shocks. Hence there is reason to believe, and numerical experience confirms, that the present higher order extensions of the KFVS formalism using compact upwind operators do remain entropy-consistent.

It is pertinent to mention here that upwinding in compact schemes has earlier been incorporated by Tolstykh (1973, 1986) through a Murman-type switch. This gives rise to

Compact upwind algorithms for viscous compressible flow calculations 149

compact schemes that are Riemann solver-based and hence of the flux-difference splitting type. A generalization of this approach to other approximate Riemann solvers was proposed by Tolstykh & Ravichandran (1991) and its extension to a finite-volume framework in general co-ordinates may be found (Ravichandran 1994). Transonic flow computations using Roe's approximate Riemann solver (Ravichandran 1997a) showed that while crisp shock capture is possible, computation times are uneconomical, especially for viscous flow calculation, mainly because the algorithm involves the inversion of block tri-diagonal linear systems whose sub-blocks (4 x 4 in the case of 2-D Euler equations) are dense. As opposed to this drawback of the FDS-based compact schemes, the present approach based on flux-vector splitting gives rise to sub-blocks which are diagonal matrices with constant entries. In some cases, the tri-diagonal linear system only involves a Toeplitz matrix and therefore may be inverted by special faster algorithms. In any case the effort involved in the inversion and the computation of the high-order accurate derivatives is now only proportional to m and not to its power (m being the number of dependent variables).

Steady state compressible flow calculations using fourth-order centred compact schemes with or without artificial dissipation have been carded out by Abarbanel & Kumar (1988). Oblique shock reflection computations show that the computed solution is oscillatory, ba- sically because flux limiters were not introduced near discontinuities. Applying a fourth order compact operator to the conventional MacCormack scheme, Carpenter (1990) has concluded that the resulting upwind C-Mac scheme gives considerable improvement in accuracy for the supersonic mixing layer problem. More recently, Cockburn & Shu (1992) have formulated a family of nonlinearly stable compact schemes for nonlinear scalar conservation laws in one and two space dimensions, introducing upwinding through flux-vector splitting. In spite of the BV-stability property, they found that their centred approximations fared badly in terms of accuracy in actual computations because of the odd-even decoupling problem mentioned above. This motivates the emphasis on the use of upwind compact operators in the present work. Thus the present approach is essentially similar to theirs though modifications have been necessitated from the point of view of computational economy. Also while the restriction by Cockburn & Shu (1992) to the class of symmetric compact operators and the introduction of a solution mean enables them to prove results concerning formal accuracy in the scalar case, we use a wider class of compact upwind operators which do not possess such symmetry. Formal accuracy of the flux-limited scheme therefore remains unverified for the present.'However nonlinear TVD stability of the flux-limited semi-discrete scheme is easily established following Tadmor (1988).

Compressible viscous flow computations for the supersonic mixing layer using third and fifth order compact schemes have also been reported by Ma & Fu (1993). They have used the Steger-Warming flux-splitting for upwinding. Although no comparisons have been made, computed results show the capture of sharp shock transitions and smooth rollup structures. In what follows, the numerical scheme for the Euler equations in one space dimension is described and then extended in a finite-volume framework to two dimensions in general curvilinear coordinates. The algorithm is applied to compute solutions to inviscid transonic/supersonic flows. Validation is shown by computational comparisons with results obtained from well-known second-order upwind TVD schemes (Yee 1989).


2. Compact differencing: An introduction

2.1 Compact difference approximation to the first derivative - a simple example

Consider a smooth function f ( x ) in the interval (0, 1). Let the grid points be given by xj = jh, h being the uniform mesh width and j = 0, 1, 2 . . . . . N. A conventional second- order difference approximation to the first derivative is

Oxfj = - (1 /2h) ( f j+ l - J~-l) + O(h2)- (1)

By Taylor expansion

3~+1 = J~ 4- hOxfj 4- (h2/2)O2fj 4- O(h3), (2)

f j -1 = f j - hOxfj 4- (h2/2)02fj 4- O(h3). (3)

Subtracting (3) from (2) gives, after cancellation of the O (h 2) term, formula ( 1 ) for the first derivative. To obtain formally fourth-order truncation accuracy, we need the grid points J j -2 and fj+2 and using the same Taylor series procedure we obtain

Oxfj = - (1 /12h ) (~+2 - 8fj+l 4- 8 f j - 1 - f j -2) 4- O(h4).

Fourth-order accuracy is obtained at the cost of widening the stencil by including 2 additional grid points.

Let us now consider (2) and (3) but retain up to order O(h 5) terms in the Taylor expansion,

h2 ~ h4 fj+l = fj 4- hOxfj 4- --O2 fj 4- 03f j4- 04 . 2 ~ x J) + O(h5), (4)

h2 ~ h4 04 . fj--1 = f j - - hOxfj 4- -~02 fj - 03xfj 4- ~-~ x fJ 4- O(hS) • (5)

(1/2h){(4)-(5)} gives,

(1/2h)(f j+l - J~-l) = Oxfj 4- (h2/6)O3fj 4- O(h4), (6)

so that we may write

(1 /2h)A0f j = [I + (h2/6)O2x]Oxfj + O(h4). (7)

If we now have an O(h 2) approximation to the operator 0 2, such as

(1/hZ)A2)~ = (1/h2)(3~+1 - 2j) + f j -1) + O(h2),

then

(1 /2h)A0f j = {I + (h2/6)((1/h2)A 2 + O(h2))}Oxfj + O(h 4)

= (I 4- (1/6)A2)Oxfj + O(h4).

A fourth-order accurate approximation to Ox f j is now obtained as

Oxfj = (I 4- (1 /6)A2)- l (1 /2h)Aof j 4- O(h4).

(8)

(9)


Denoting the fourth-order approximation to Ox fj by Fj we may write

(I + (1/6)A2)Fj ----- (1/2h)A0fj

or in expanded notation,

Fj + (1/6)(Fj_l -- 2Fj + Vj+l) = (1/2h)A0J), (10)

(1/6)Fj_I - (4/6)Fj + (1/6)Fj+I = (1/2h)Aoj~, j = 0, 1, 2 . . . . N. (11)

Thus we must solve a tri-diagonal linear system to obtain the fourth-order approximations Fj to the first derivative Ox fj . Note however that information on f j from the entire domain affects the computed value of Fj at the grid point j . This gives the compact difference approximation a proximity to the spectral approximation of the derivative, where also global information is used to compute local derivatives. A spectral analysis of the resolution properties of centred compact formulae of orders up to 10 is given by Lele (1992).

2.2 Approximation to the second derivative

In Navier-Stokes computations we need approximations to the second derivative as well. Denote the fourth-order approximation to the second derivative of f ( x ) at xj by Sj. Pro- ceeding in a manner similar to that above, it can be shown

1 S . lOs . 1 S h- ~ 1 12 j-1 - - 12 J + ]-2 j + l = A2fj = ~--~(fj-1 -- 2j~ + f j+l ) . (12)

Once again we must solve a tri-diagonal linear system to obtain the fourth-order approximations Sj to the second derivative 82j~).

2.3 Compact upwind approximations to the first derivative

In numerical solutions of hyperbolic problems, we use upwind differencing to account for the direction of signal propagation and ensure a proper domain of dependence. The corresponding compact upwind difference operators are required for higher order approximations. These compact upwind difference (CUD) operators were introduced by Tolstykh way back in 1973 (Tolstykh 1973). We now proceed to derive a third-order compact upwind difference operator for the first derivative of a function. Consider the backward difference approximation

Oxfj = ( 1 / h ) A _ ~ + O(h). (13)

Expanding in a Taylor series as before,

__0 2 -~A_fj = ! - - ~Ox + 6 x Oxfj + O(h3).

We replace the differential operators 8x and Ox 2 by their second-order difference approximations,

Ox = (1/2h)Ao + O(h2), 02 = (1/h2)A 2 + O(h2).


Then

-~A_fj = I - ~Ao + /~2 oxfj + O(h 3)

5 8 1 = -~Fj-1 + --~Fj - -i-~Fj+l + O(h3). (14)

A third-order accurate upwind approximation to the first derivative is thus obtained by solving the tri-diagonal system

5 8 1 1 -~ Fj_] + -~ Fj - -~ Fj+I = -~ A _ f j , j = O, 1, 2 . . . . N. (15)

If we use forward difference instead of backward difference, we obtain a corresponding third-order downwind approximation by solving

1 8 5 1 12Fj-1 + T~Fj + -~Fj+I = ~A+j~ , j = 0, 1,2 . . . . U. (16)

We now use the above ideas to construct a third-order accurate compact upwind approximation to the scalar conservation law.

3. Third order compact upwind approximations (scalar conservation law)

Consider

ut + fx(U) = 0, (17)

where f ( u ) is a convex flux function so that f ' ( u ) > O. The flux function f ( u ) may be split into positive and negative parts so that

f ( u ) = f+ (u ) + f - ( u ) , d f + / d u > O, d f - / d u < O, Vu. (18)

For example, the Lax-Friedrich splitting gives

f+(u ) = f ( u ) + otu, y - ( u ) = f ( u ) - otu, ot = max If ' (u)l . (19) u Then rewriting (17),

ut + f+(u ) + f x (U) = O, (20)

one may approximate by backward differencing for the positive part and by forward differencing for the negative part of the flux derivatives. We may obtain third-order accuracy by approximating

1 +

1 12FT__l -lt- ~ F f - ~--~FAI = 1A+ fj - .

j = 0 , 1 , 2 . . . . N

Denoting the tri-diagonal operators by A+ and A_ we may write

A+t~+ = ( 1 / h ) A - f j +,

a_v j - = (1/ h)zx+ f 7 ,

(21)

(22)


where A+ = (5, 8, - 1 ) / 1 2 and A_ = ( -1 , 8, 5)/12. By way of notation we may define a three-point grid function operator by a triplet of numbers (al, a2, a3) so that

(al, a2, a3)fj = al f j -1 + a2j~ + a3j~+l.

For example, in this notation, the backward and forward difference operators are respectively given by ( -1 , 1, 0) and (0, - 1 , 1) and the centred difference operator is ( -1 , 0, 1).

We may now put the above difference scheme in the semi-discrete conservative form

Ouj (1 fj+)A~_I(1 f)_) 0--7- + A+I ~A_ + ~ A + = 0, (23)

or

where,

Ouj 1 -1 + + A-I a--~ + ~{A+ (fj f j_ l ) + ( f j+ l f j - )} = 0 , (24a)

Ouj 1 ^ 0---7- + ~{fj+l/2 - )~-1/2} = 0, (24b)

)~+1/2 = A+l f 7 + A-l f~+t , )~-112 = A+ 1 fj+-I A-1 f j-- (25)

We emphasize again that (24)-(25) is a third-order accurate compact upwind semi- discretization of (17). Since it is in conservative form, it can be easily extended in a finite-volume framework to multi-dimensional problems.

Note that if the numerical flux is expressed as a sum of positive and negative contributions at the cell interface j + 1/2,

D+,/2 = f ,i2 +

then

j~+ = A+I fj+, k+,/2 A - ' f j+ 1 • j+l/2 (26)

In a conventional first-order upwind scheme we have

= D +, -- D+, . (27)

One may therefore regard the third-order compact upwind fluxes as having been generated by post-processing of the first-order split fluxes at the cell interface.

3.1 Extension to Euler equations in 1-space dimension

The Euler equations in one-space dimension are given by

OU/Ot + OF/Ox = 0, (28)


where U = [p, pu, e] and F = [pu, p u 2 q- p, (e + p)u], p, u, e, p being respectively the density, gas speed, total energy and pressure.

The flux function F may be split as

F = F + + F - . (29)

For example the KFVS splitting for the Euler flux is (Mandal & Deshpande 1994)

F + = [pua ± -4- pb, (p + pu2)a + ,4- pub, (e + p)ua + ,4- (p/2 + e)b] t

= [f±, u f ± + pa ±, Hf+-4-pb/2] t with f ± = pua±+pb, (30)

where,

a + = (1 ±erf(s (u)))/2, b = e x p ( - (s (u))2)/2(yrfl) 1/2,

S(X) = X1~ 1/2 and fl = p/2p,

and eft() is the error function. H denotes the total enthalpy, H = (e + p)/p. We know that the Jacob±an matrices OF+/OU and OF-/OU respectively have non-

negative and non-positive eigenvalues. A conservative first-order accurate upwind semi-discretization to (28) is given by

OUj/Ot = - ( 1 / h ) ( F j + l / 2 - F j - 1 / 2 ) . (31 )

A first-order conventional numerical flux is given by

/~j+l/2 = F+(Uj) + F-(Uj+I). (32)

To obtain third-order spatial accuracy, using (26) and (27) we write

/~ j+l /2 ~- A+I(F+(Uj)) + A~-I(F-(Uj+I)) • (33)

3.2 Higher order compact upwind approximations

Equation (33) suggests that a more general higher order numerical flux corresponding to any compact difference operator may be decomposed into positive and negative parts given by

/l~-k - 1 + . ^ = A - 1 j + I / 2 - - A + (Fj+I/2), Fj+I/2 _ (Fj7_~_l/2),

Fj+,/2+ = b l F / _ 1 + b2F/ + b3F~l + b4F~2 , (34)

F-j+I/2 = b4Fj-_ 1 + b 3 F j +b2FT+ 1 + bl Fj7+2 •

The coefficients b~ satisfy the consistency condition ~ bk = 1. A similar condition holds for the coefficients of the operator A+.

The following is a summary of some possible schemes that may be derived.

(1) Third-order compact upwind scheme with one-point upwind numerical flux (Tolstykh 1973) bl = b3 = b4 = 0, b2 = 1; al = 5/12, a2 = 8/12, a3 = -1 /12 .

(2) Third-order compact upwind scheme with two-point fully upwind numerical flux (Cockbum & Shu 1992): bl = - 1 / 2 , b3 = b4 = 0, b2 = 3/2; al = - 1 / 3 , a2 = 5/3, a3 = - 1 / 3 .


(3) Third-order compact upwind scheme with two-point upwind weighted numerical flux (Ma & Fu 1993): bl = 0, b3 = i /6 , b4 = 0, b2 = 5/6; al = 1/3, a2 = 2/3, a3 = 0.

(4) Fifth-order compact upwind scheme with four-point upwind weighted numerical flux (Ma & Fu 1993): bl = 3/60, b3 = 11/60, b4 = - 1 / 6 0 , b2 = 47/60; al = 2/5, a2 = 3 / 5 , a3 = 0.

3.3 Flux limiters for shock computations

The schemes as detailed above are not suitable for computing flows with shocks. Non- physical oscillations generated in the vicinity of shocks by these high-order accurate schemes have to be eliminated by the introduction of flux limiters. The resulting schemes are therefore first-order accurate in the neighbourhood of shocks and other flow discontinuities and extrema. Limiters are defined through flux differences of the higher order numerical flux with the first order values as follows.

_ ^ + _ F + ; ^ = d~'~l/2--F)+l/2 dFj+l/2 -Fj+I/2 + F~+ 1, (m)dF~l/2 = minmod(d/~j~l/2, A+F +, A+F+_I ),

(m)dp~+,/2=minmod(dP7+,/2, ex+F~, ex+Fj+,), (m) ~ l / e = F~ -q- (m) d ~ i / 2 ,

'm) F~l/2= X~l - (m)dfi~j+l/2.

(35)

As usual, the minmod limiter function vanishes if any of the arguments differ in sign and otherwise returns the value of that argument which has the least absolute value. The minmod limited values as defined above, replace the numerical fluxes appearing in (31) and (34).

The residual may now be defined as

Rj = -(1/ h)((m) ~'j+l/2 -(m)Fj_l/2). (36)

Third-order accuracy in the spatial discretization requires that the time integration be carried out by a multi-stage TVD Runge-Kutta scheme (Cockburn & Shu 1992).

The spatial discretization as described above is in some important respects different from the one developed for scalar equations by Cockburn & Shu (1992). They carry out a pre-multiplication by the compact difference operators A+ which, in their case, coincide with one another and are symmetric, making such an operation feasible. Consequently their algorithm involves the definition of a mean value for the unknown state vector U and the flux differences are defined in terms of values defined in terms of the mean. This implies that the Euler fluxes have to be evaluated for both U and its mean. The scheme is therefore computationally much more expensive, especially in multi-dimensions. The present scheme is simpler and economical and at the same time retains the property that, like the scheme of Cockburn & Shu (1992), a simple modification in the limiters, as below, renders it nonlinearly stable for a scalar conservation law. Introducing the following condition,

(m)dP~+l/2=O i f (m)d~'~+l/2=O, (35a)


and using the Lax-Friedrich splitting (Cockburn & Shu 1992), the net numerical flux in the j th cell is given by

/ ~ j+ l / 2 - - Fj-1/2 = (rn) Fjw1/2 --(m)Fj-1/2

= F+ + + V ; + , -

--Fj+i l - (m)dfi '7_ll 2 - F j - -P (m)dFj-_..ll 2.

Consider now an extremum, say a maximum of Uj (being scalar) such that Uj-1, Uj+l < Uj. Then

(m),;E+,,, j+l/2 = minm°d(d/3j~-l/2, A+ F+ ' A + F + I )

= minmod(d/~j++l/2, ~A+Uj,/3A+Uj_I),

where c~,/3 > 0 in view of the Lax-Friedrich splitting. Now if Uj is a maximum, and A+Uj_I > O, A+Uj > 0 and the minmod function vanishes. A similar argument holds

for m d ~--1/2" With the additional condition (35a) introduced above, the net flux is given by

Fj+l l 2 i 6 1 i 1 /2 = ~ + Fj~ 1 -- Fj+i- Fi, (35b)

which is simply the first-order upwind contribution to the j th cell and is hence monotone. The same argument is valid for a minimum of Uj. This implies that the Scheme is TVD at solution extrema. Noting that the numerical flux at solution extrema has two-point support, if successive solution extrema are separated by at least two points, as happens in many practical cases, by the local TVD criteria of Tadmor (Lemma 2.1 of Tadmor 1988) the scheme is TVD. Computations with and without the modification (35a) show only minor differences in the results but the scheme with (35a) has a tendency to go into a limit cycle kind of behaviour, after the residual drops by two orders of magnitude. For this reason all computations have been carried out without the modification (35a). A proof of the TVD property of the scheme without (35a) is given by Ravichandran (1997b).

3.4 Boundary conditions and solution of the tri-diagonal system

The tri-diagonal linear system indicated in (34) involves matrices with constant entries in- dependent of the grid point. Secondly, in the case of a system of conservation laws, the sub- matrices in the block tri-diagonal system are diagonal and hence easily inverted by a scalar algorithm. In expanded form the linear system in the first row of (34) may be written as

Diag(aj)F+_l/2 + Diag(bj)F~+l/2 + Diag(cj)FD3/2 -- F + j + l / 2 ' j = 0, 1, 2 . . . . N. (37)

An inspection of (37) indicates that the value of F?_ 1/2 is required at j = 0 and is unknown. A similar situation obtains at the right boundary with respect to the grid point j = N. At both boundaries, we replace the high-order flux by their first-order values, thus to some extent compromising on the accuracy realized at_the boundaries. In all computations reported here, two dummy points have been introduced at a boundary in such a manner that the prescribed boundary condition is satisfied at the boundary whether it is a solid wall or


an inflow or an outflow. Solid wall boundary conditions are obtained by reflection while inflow/outflow boundary conditions are obtained by the use of 1-D Riemann invariants. It is always assumed that the number and type of boundary conditions prescribed are consistent with the theory of 1-D wave propagation for a system of hyperbolic equations.

4. E x t e n s i o n to E u l e r e q u a t i o n s in t w o d i m e n s i o n s in g e n e r a l c o o r d i n a t e s

The Euler equations in two dimensions in general coordinates may be written as

a0 - - - - = 0, (38) at + - - ~ + On

where 0 = U/J, F = (~xF + ~yG)/J, G = (rlxV + ~TyG)/J and J = (~xrly - ~yOx), is the Jacobian of the transformation. It is assumed that J # 0. The state vector U = [p, pu, pv, e],andthefluxes F = [pu, puZ + p, puv, (e + p)u ) ] and G = [pv, p u v , pv2 + p, (e+p)v], where p, u, v, p and e are the density, x-component of velocity, y-component of velocity, the pressure and the total energy per unit volume respectively. For a perfect gas, e and p are connected through e = P/(Y - 1) + p(u 2 + v2)/2. Entropy is denoted by e = p/pr . As in the 1-D case the fluxes are split and written as a sum of positive and negative parts. Denoting these by F+ and (~± respectively, (38) is written as

00 ap+ 0P- a0 + at + a0

~ O~ (39)

It is natural to use upwind weighted differencing for the positive and downwind weighted differencing for the negative flux derivatives. Higher order accuracy is subsequently achieved by using the compact difference relations individually in the ~ and 0 directions. Typically

^+ -4- A±Fi+I /2 = El+l~ 2,

+ bl/~L1 b2P~ + b3/W/~_l Fi+l/2 = + + b4F~2,

F,-+l/2=b.Pi-a + b3;i- + b3PT+. + bl;i+2. which can be written in expanded form as

a~i Fi-1/2 + b~/7i+1/2 + c~/~i+3/2 =/~i+1/2, 0 < i < i max. (40)

In (40), the ~- or q: signs and the subscript j have been dropped for convenience. Note that the block tri-diagonal system (40) has to be solved for every value of j , 0 <

j < j max. Fortunately as in the 1-D case the block matrices which are 4 x 4, are diagonal with constant diagonal entries. Their inversion therefore entails no more effort than that required for a scalar tri-diagonal system. An equation simil~ to (40) may be written for the 0 direction and these will have to be solved for every i, 0 < i < i max. Having solved

-̂4- ^4_ for Fi+l/2, j and Gi,j+l/2, the total residual may be defined by,

^ ÷ ^

Gi, j+l /2 = Gi , j+l /2 + G-£,j+I/2,

Rij = (1/A~)(Pi+I/Z,j - [gi-1/2,:) + ( 1 / A l T ) ( a i , j + l / 2 -- Gi,j-1/2,). (41)


The higher order fluxes may be replaced by their flux-limited values using (35) for each of the directions ~ and 17. Time integration may again be done using a TVD multi-stage Runge-Kutta scheme (Tolshykh & Ravichandran 1991).

As in the one-dimensional case, it is assumed that suitable boundary values are specified at the dummy points i = - 2 , - 1 and i = i max +1 and i max +2 for all j and similarly for j for all i. The block tri-diagonal linear system may therefore be closed by suitable first-order approximations for the numerical fluxes at the boundaries. Numerical experiments suggest that, while this does not unduly affect the accuracy of the local solution whenever the latter is close to uniform conditions such as farfield boundaries, near solid wall boundaries, the accuracy of the computed solution appears to be satisfactory. The present first-order scheme at the boundary at least has the merit that the boundary scheme is reasonably robust and stable.

It now remains to specify the KFVS split fluxes in the curvilinear coordinates. Define the unit normals and the contravariant velocities as follows:

k = (~2 + ~2)1/2, (~x, ~y)= (~x, ~y)/k,?~ = ~xU + ~yV, (42)

l = (r/x2 -t- r/y)212/, (~x, t l y ) = (r/x, r/y)/l , ~) = ~xU n t- ~yV.

The KFVS splitting is given by (Mandal & Deshpande 1994),

/~+ = [f-4-, f+u + ~xpa +, f+v + ~ypa ±, f ± H m pb/2] t, ~4- = [~a:, ~4- u + Flxpa +, ~4- v + ~lypa ±, ~4- H 7: pb/2] t,

f ± = pfia ± 4- pb, ~± = pf)c ± 4- pd, (43)

a ± = (1 4- erf(s(~)))/2, c ± = (1 4- erf(s(O)))/2,

b = e x p ( - (s (~))2)/2(7r/3) 1/2, c = e x p ( - (s 03)) 2)/2(zr/3) 1/2.

The definitions of s(x), ~ etc. are the same as in the 1-D case. For viscous computations, the N-S terms have to be added to the RHS of the Euler

system (38). Centred compact formulae as given by (12) may be used for discretization of the second-order velocity derivatives appearing in the viscous terms. This ensures fourth- order truncation accuracy in viscous term discretization consistent with the third- or fifth- order accuracy in the representation of the convective fluxes. In the computations reported here we have only used the conventional three-point second-order discretization. Even so, the improvement in the resolution of the flowfield as compared to computations with the MUSCL or the Roe scheme is apparent.

5. Numerical results

The algorithm described above has been coded in FORTRAN and run on the ALPHA 3000 machine at the VAX computing facility at the National Aerospace Laboratories (NAL). We show some sample results to demonstrate the improved accuracy. The following cases are considered:

(i) External flow past a 10% semi-thickness parabolic profile at transonic Mach numbers of 0.85;

"(~8"0 = IAI) oigoad ~![oq13xed lsgd ~ao U lgUaOlx~I "[ ~n~[.~

/uepunoq uJo:goq uo uo!;nq!J),s!p JequJnu qoe~

o't e't O'L ere e'~-

I i

O'G S'~ OZ S'L O't S'O 0"0-

O't S*~ 0"~ S'L O't S'O 0"0"

9E'Q'~'~ J-'" L=Xlq'~ lq~.SOAJ. O:~'ON SHOY'~;OS!

S'O

0'|

6gI suol.mlnHva 6tolf alqt.ssaaduaoo snoos!a aof sua~lZl.aoglv pu?~adn goDduto D

1.25

(ii) External flow past a 10% semi-thickness parabolic profile at transonic Mach numbers of 1.3;

(iii) Internal flow past a 4% parabolic hump at a Mach number of 1.4;

(iv) M = 3 flow in converging diverging type channel;

(v) Mach 3 supersonic flow past a curved concave wall;

(vi) Mach 3 supersonic flow in 2-D tunnel with forward facing step;

(vii) Oblique shock wave/boundary layer interaction in supersonic flow (Hakkinen's data).

A feature of the present computations is that comparisons have been made with conventional second-order symmetric or upwind TVD scheme (Harten-Yee) using the same

(a) ISOMACHS NC,=20 TVDSYM M-MX-1 .98 M-MN=0.50

; / 1.00

0.7S

0.50

0.25

1.25

1.00

0.75

0.50

0.25


-0,0 1.0 2.0

Figure 2. Continued.


M

1.75

1.50

1.26 (

1.00

0.75

0.50

1

I I i I " " ' ' " ~ l I l I I -1.0 0.0 1.0

T V D S Y M o C U D - T O L

~;E~LL{[ ( ( [ - [ ( [ [

i l l l l , , . , l 2.0 3.0

E 0.775

0.750

0.725

- - [ ( i t ( { ( ( I

-1.0 0.0 1.0

T V D S Y M O C U D - T O L

t • . • • !

2.0 3.0

Mach number and entropy distribution on bottom boundary

Figure 2. External flow past 10% parabolic profile (M = 1.3).

computer code. Therefore flow initialization, grid and associated metric quantities and boundary condition implementation are identical for all the computations reported here.

Case 1: Figure 1 shows results of computations for a subsonic freestream Mach number of 0.85. The profile has 10% semi-thickness. The 120 x 60 computational grid extends about 3 chords upstream and downstream of the profile and about 4 chords in the transverse direction. The results are presented in the form of Mach contours affd surface distributions of the Mach number. Figure 1 shows a comparison of the results of second order accurate TVDSYM computations (Yee 1989) with that of CUD-TOL (scheme 1) which is third-order accurate. The Mach contours show that the shock is almost normal and well- aligned locally withthe grid. The shock transition on the surface is fairly sharp and free of over- or under-shoots.


1,2S

1.00

0.76

0 ,50

0 2 S A .4.0 I .o

ISOMACHS NC=20 TVDSYM M-MX=l.68 M-MN=0.93

L.

=,0 ,I.o

t.,?S

0.1'5

O.SO

o.=m

ISOMACHS NC=20 CUD-TOL M-MX=l.74 M-MN=0.83

-Ct 0 1.0 1.0 =.0 4.0

,,00 t.O t ,O 3,0

TVDSYM O CUD-TOL

m f * 4~P

Mach number distribution on bottom wall

Figure 3. Flow past 10% parabolic hump in channel ( M = 1.4) .

Case 2: This concerns Mach 1.3 free air flow past the same profile. The grid in this case extends about one chord upstream and three chords downstream. The transverse extent of the grid is the same as in the previous case. For the specified Mach number and geometry, a detached shock forms upstream of the profile. An oblique shock forms at the trailing edge so that in this problem we can see the effects of shock misalignment with the grid. Figure 2a shows the Mach contours from the second-order TVDSYM scheme (Yee 1989) compared with CUD-TOL, scheme 1 calculations. The detached shock ahead of the leading edge and the oblique shock at the trailing edge are captured well. Note that the respective computed maxima and minima of the Mach number field are in close agreement. Figure 2b shows the comparisons for surface Mach number and entropy distributions. While the Mach number

'ponu.rluoD "t' ~D'Ht~!,~I

9P'~=NIN'I#I ~g'¢==Xlhl'~ "IOSfII~I'SA.-DI 0~=ON SHOYI~OSI

IFe"

WO

SrO

WO

rv ere .re erl e~-

iwe .

ul

ii" LS Z NI/I IN 9~; S XN Ve "IO.L ano 0Z=ON SI-IOVI~OSI

1|¢0" ~'o I~1"0

t St' ~ NIN ~ 6g ~=XB iN :I:IA N3J.NVH 0~='ON SI-IOViNOSI

E9I suoi.lDlnal.a ~totf alq~.ssaadmoa snoasl.a aof sm~ilt.aOglV pul.~dn laDdmo D


3.5

3.0

M

2.5

2.0

Harten-Yee

[ ] KFVS-MUSCL

(~ Scheme 1

A Scheme 4 • 0

QQ r'l

I I I I I l I I I I I . . • • | i • I • I

-O.O 1.0 ,~ 2.0 3.0 4.0 / k

Mach number distribution on bottom boundary

Figure 4. Flow in a convergent-divergent channel (M = 3.0).

distribution shows that the CUD-TOL shock transitions are slightly sharper as expected, the surface entropy distribution is non-monotone in the vicinity of the shocks. Such behaviour in the entropy distribution has also been observed in third-order ENO (Yang et al 1990) as well as to a minor extent in second order TVD computations (Spekreijse 1987). In figure 2b, the Mach number distribution obtained from a first-order KFVS computation is also included. It is apparent that higher dissipation in the vicinity of the shock affects the accuracy even at points away from the shock. We remark that in both problems 1 and 2, the resolution achieved by the present scheme is not markedly different from the second-order results.

Case 3: In this case we compute M = 1.4 flow through a channel that has a 4% parabolic hump on the bottom wall. This problem has been used as a test case for many other algorithms. The main features of the flow are the reflection of the leading edge oblique shock from the upper wall and its interaction with the oblique shock emanating from the trailing edge.

Figure 3 shows the computed Mach contours on a 120 x 60 grid, using the second order TVDSYM. The reflected shock from the top wall interacts with the oblique shock from the trailing edge, giving rise to two triangular regions of uniform flow separated by a shock which hits the bottom wall. This reflected shock from the bottom wall subsequently merges with the trailing edge shock which reflects back and forth till it becomes con- siderably weakened at the exit. The bottom surface Mach number distribution shows the captured shocks clearly. These are sharp and free of wiggles. Note that the shock transitions computed by CUD-TOL are sharper, especially downstream of the hump, showing the higher resolution achievable by the present schemes. Also the Mach number distribution ahead of the trailing edge shock is slightly more peaky for the present schemes, perhaps due to the smaller dissipation as compared to the second order TVDSYM.


O,?S

O.SO

0.2S

0.00

.02S

KFVS-MUSCL GRID=120X60 DELT=0.0044 ITER=10000

,.0,0 t .o =.0 lt.O 4.0

t ,00

0.71

O.r*O

02S

O,00

-02S

SCHEME 3 GRID:90X40 DELT=0.008 ITER=4500

-0.0 ~ .0 * | i

~Z.O :l.O 4.0

Figure 5. Flow in C-D channel, pressure contours (M = 3). Comparison of scheme 3 with KFVS-MUSCL, NC = 20, DP = 0.1.

Case 4: Figure 4a shows the isomach contours for M = 3 flow through a converging- diverging channel using the following schemes, (i) second-order upwind Harten-Yee (flux-difference splitting), (ii) second-order KFVS-MUSCL post-processing, (iii) CUD- TOL (third order), and (iv) CUD-MAFU (fifth order). Note that while the MUSCL post- processing approach gives inferior shock resolution compared to that of the Harten-Yee upwind scheme, the compact scheme shock transitions are much better resolved, especially in the interaction regions downstream of the channel. The surface Mach number and Cp distributions shown in figure 4b also confirm these observations.

Figure 5 shows a comparison of the pressure contours obtained using the KFVS-MUSCL scheme and the present scheme on a 120 x 60 and 90 × 40 grid respectively. The KFVS- MUSCL scheme took about 10,000 time steps for the residual to drop by three orders of magnitude while, with the present scheme 1, a residual drop of four decades was achieved in about 4500 time steps. Numerical experiments reveal that on a given grid, CPU time required by the present scheme per time step is about three times that needed by the KFVS-MUSCL scheme. This directly corresponds to the 3 stage Runge-Kutta scheme used for time marching in the compact algorithms. This also implies that the present post- processing approach for generating the higher order fluxes, is only slightly more expensive than the conventional MUSCL approach. This indicates that the compact algorithms may be made more attractive computationally, if one can fully exploit the scope of the multi- stage schemes by increasing the stability range using techniques such as implicit residual smoothing.

!i o.

s tJ

t4

L

o

:e.s

s.

e si

s

(b)

t

' .

..

..

..

.

| ~

"

" °

' .

..

..

"

" "

| .

..

.

sJ

e~

lo

t~

t~

e

L

Le

, (c) ".

..

..

; ..

....

...

~ ..

....

....

Bs

10

iS

2

~

ts

LO

U

i 1.

0 1

J ,t

.O

=.l

)~

U

e.s

ta

tJ

ia

t4

U

LS

o,s

i~e

iJ

• o

:t.s

:.

¢

s.|

Fig

ure

6.

Flo

w p

ast

a cu

rved

wal

l (M

=

3.0)

. (a

) S

chem

e 1;

(b)

KF

VS

- M

US

CL

; (e

) H

arte

n-Y

ee.

(Fig

ures

on

the

left

are

pre

ssur

e co

ntou

rs a

nd th

ose

on

the

righ

t ar

e en

trop

y co

ntou

rs.

NC

=

20,

DM

=

0.17

5,

Den

t =

0.02

.)

o.s

(a)

1.0

0.5

0.5

1.0

1.5

2.0

2.5

(b)

1.0

0.5

0.5

1.0

1.5

2.0

2.5

g~

g~

2.

..,g r~

Fig

ure

7.

Flow

thro

ugh

tunn

el w

ith

20%

ste

p, G

: 15

0 x

40 (

M =

3).

(a)

KFV

S, C

UD

3-T

olst

ykh.

(b)

Den

sity

DM

N =

O, D

MX

= 5

, KFV

S, M

USC

L3.

0.40

0,30

0.20

0.10

Pressure contours, streamlines, separation bubble

f j

0 . . i ", " J " "" "" -'"

0 . ~ ~ 0.~ 0,10 0.20

!i!

pressure, NC=30, INC=0.01

1

-0.5 O0 ~ Oi l 1.0

mach NC=40, INC=0.01


-0.5 0.0 0.5 1•0

density, NC=40, INC= 0.02

0,40

0.30

0.20

0.10

-0•5 0.0 0.5 t .0

Figure 8. (a) S W / L B L interaction in supersonic flow. M = 2, u = 32•6, R e = 2 .96 x 105, Harten-Yee, G : 1 2 0 x 100.

Case 5: In figure 6 we present the results for M = 3 supersonic flow past a curved channel. Zeroth order extrapolation is used on the upper computational boundary for this calculation• This is a simple test case for generating a contact discontinuity in the interior of a flow field.


0,100

0.060

0.000 -0.10


"-., ,,,/.. / . , / " J - - I / '

\ \ F ' / / j / -

-0.00 0.10 0.20


0.10

• .o.$ o.0 o.5 1.o

Mach NC=40, INC=0.01

.o.~ o'.o o'.~ ,.o


0.40

0,30

0.20

0.$0

-O.S 0.0 0.5 1.0

Figure 8. (b) SW/LBL interaction in supersonic flow. M ---- 2, ~ = 32.6, Re ---- 2.96 x 105, KFVS-MUSC2, G:120 x 100.

The objective in computing this case is to see if the present compact schemes can resolve the contact discontinuity any better than the second-order upwind schemes. It is apparent from figure 6 however that the present scheme in this respect performs only marginally better than the second-order schemes.


Pressure contours, streamlines, separat=on bubble

\.\\ i i / ~~ "x \ , \ t ( / ~ -

f / / \ ~ - S A ) / 01" L ( ' ' I , ~ ~ . . . . ¢ I ' ' ¢ ,

-0.10 -0.00 0,10 0.20

0 .40

0.30

0.20

0.10


=

-O.S 0.0 0.5 1.0


0.30

0.20

0.10

0.40

0.30

0.20

0~10

-O.S 0.0 0.5 1.0

F i g u r e 8. (e) SW/LBL interaction in supersonic flow. M = 2, ~ = 32.6, Re = 2.96 x 105, KFVS-MUSC3, G:120 x 100.

Case 6: This test case was first introduced by van Leer and subsequently studied by Woodward & Colella (1984) in their classic paper comparing several difference schemes with regard to their resolution properties, while computing inviscid high-speed flow. Since

0.160



171

0.100

0.050

0.000 -0.10 0.00 0.10 0,20


0.40

0,30

0.20

0.10

-O.S 0.0 O.S t.O


0.40 I ~

• i

-O.S 0.0 O.S 1.0


0,40

0,30

0.20

0.10

r -0,5 0.0 0.5 1.0

Figure 8. (d) S W / L B L interaction in supersonic flow. M = 2, c~ = 32.6, Re = 2.96 x 105, C V D - K F V S , G:120 x 100.

at steady state the flow features are somewhat uninteresting, the unsteady evolving flow at non-dimensional time t = 4.0 is generally depicted in the literature. Figure 7 shows the density contours computed using CUD-TOL and KFVS-MUSCL3. The improved


4 . 0 B o

0

3.0 i a O

2.0

~ o

o 1.0

0.0

-1 .0

zO

[ ]

~7

C, ©

KFVS-CUD3

Harten-Yee

KFVS-MUSCL2-PRP

KFVS-MUSCL3-PRP

KFVS-MUSCL2-PSP

KFVS-MUSCL3-PSP

XSHK=0.1 XLE=-0.9

Figure 8.

-0.5 0.0 0.5 1.0 X

(e) SW/LBL interaction in supersonic flow. Comparison of all results.

resolution of the flow by the former scheme is evident in the capture of the contact at the triple points both near the upper and the lower wall. It must be mentioned here that no special numerical treatment of the boundary condition at the comer is used in the present computations which accounts for a numerical boundary layer aft of the comer and the consequent appearance of the Mach stem.

Case 7: By way of an example of a viscous computation, we present a comparison of the results obtained for the problem of an oblique shock/laminar boundary layer interaction in supersonic flow. The oblique shock is generated by an incoming M = 2 flow. The shock is inclined at an angle of 32.6 ° to the horizontal axis and impinges on a flat plate at a unit distance from the leading edge of the plate. The Reynolds number is 2.96 x 105. Separation is known to occur and many schemes have been used in the past to compute the interaction flow field. Figures 8(a--d) show a comparison of the pressure contours computed using the present scheme with those obtained using the conventional MUSCL as well as the Roe schemes. The improved resolution of the various flow regions is apparent. Figures 8e and f show the skin friction and pressure distributions on the bottom wall. The low dissipation realised in the calculations using the present compact approximations is evident in figure 8e.

LL z o.

1.50

1.25

1.00

0.75


KFVS-CUD3 [] Harten-Yee /~ KFVS-MUSCL2-PRP

KFVS-MUSCL3-PRP '~ KFVS-MUSCL2-PSP C) K F V S - M U ~

XSHK=0.1 XLE=-0.9

Figure 8.

, . I . . . . I . . . . I . . .

-0.5 0.0 0.5 X

(t3 SW/LBL interaction in supersonic flow. Comparison of results.

, I

1.0

173

6. Conclusions

High-order compact upwind schemes for the compressible Euler and Navier-Stokes equations in general geometries have been formulated. Upwinding is incorporated through flux- vector splitting. With the introduction of minmod flux limiters, these schemes are shown to be TVD for the case of a scalar conservation law in one-space dimension. These spatial discretizations have been combined with three-stage TVD Runge-Kutta time marching to obtain an explicit family of schemes that may be used for steady state computations. In the present work we have used the KFVS split fluxes in actual numerical computa -- tions. Computed results show that higher accuracy and resolution is indeed achievable via the post-processing approach using compact upwind difference" operators and split fluxes. In particular, for viscous flow calculations, use of compact upwinding in higher order algorithms may help overcome the known deficiencies of flux-vector splitting in accu- rately resolving shear layers without imposing a high penalty in terms of computational cost.

This work was done as part of the Indo-Russian Long-Term integrated pro~amme and was supported by the Department of Science & Technology, Government of India. The computing facilities extended by the Experimental Aerodynamics Division and the cooperation of the staff at the VAX computing centre is gratefully acknowledged. The author thanks the anonymous reviewers for comments that improved the quality of this paper.


References

Abarbanel S, Kumar A 1988 Compact high order schemes for the Euler equations. NASA CR 181625

Carpenter M H 1990 Fourth order compact schemes for shock calculations. 12th ICNMFD Lecture Notes in Physics 371:254-258

Cockburn B, Shu C W 1992 Nonlinearly stable compact schemes for shock calculations. NASA CR 189663, ICASE Report No. 92-21

Lele S 1992 Compact finite difference schemes with spectral like resolution. J. Comput. Phys. 103:16--42

Ma Yanwen, Fu Dexun 1993 Numerical simulation of complex compressible viscous flow. Proc. 5th ISCFD, vol. 7, pp 158-163

Mandal J C, Deshpande S M 1994 Kinetic flux vector splitting for Euler equations. Comput. Fluids 23:447-478

Ravichandran K S 1994 CUD-3 schemes with Runge-Kutta time stepping for Euler equations. NAL PD CD 9401

Ravichandran K S 1997a Explicit third order compact upwind difference schemes for compressible flow calculations. Int. J. Comput. Fluid Dyn. 8:311-316

Ravichandran K S 1997b Higher order KFVS algorithms using compact upwind difference operators. Int. J. Comput. Phys. 130:161-173

Spekreijse S P 1987 Multi grid solution of the steady Euler equations. Proefschrift, Centrum voor Wiskunde Informatica, Amsterdam

Tadmor E 1988 Convenient TVD criteria for difference schemes for hyperbolic conservation laws. SIAM J. Numer. Anal. 25 (5)

Tolstykh A I 1973 On a method for the numerical solution of Navier-Stokes equations of a compressible gas over a wide range of Reynold's numbers. Dokl. Akad. Nauk SSSR 210:1-48

Tolstykh A 1 1986 High accuracy methods for solving fluid dynamic problems. Current problems in computationalfluid dynamics (eds) O M Belotserkovskii, V P Shidlovsky (Moscow: Mir)

Tolstykh A I, Ravichandran K S 1991 Compact upwind difference schemes for scalar conservation laws. NAL TM CF9101

Woodward P R, Colella P 1984 High resolution difference schemes for compressible gas dynamics. J. Comput. Phys. 54:115-130

Yang J Yet al, 1990 Third order ENO schemes for compressible flow calculatio~ns. 12-th ICNMFD Lecture Notes Phys. 371: 248-255

Yee H 1989 A class of implicit and explicit high resolution shock capturing methods. NASA TM 101088

Documents

Compact upwind algorithms for viscous compressible flow