practically all Mach regimes. Sample results are given to show the advantages of this new scheme. Introduction. Proposed for its improved accuracy and ...
Pressure-Based Multidimensional Upwind Residual Distributive Scheme for All-Speed Flow Simulations. Doru Caraeni
and D.C. Hill t
Fluent Inc., Lebanon, NH, 03766, USA
An all-speed pressure-based multidimensional upwind residual distributive algorithm is proposed, for solution of the steady-state Navier-Stokes equations. This is a compact stencil algorithm which provides both high order of accuracy and robust convergence for practically all Mach regimes. Sample results are given to show the advantages of this new scheme.
Introduction Proposed for its improved accuracy and robustness, the niultidimensional upwind residual distribution methods have gained in popularity and beCame an attractive alternative to finite-volume (FVM) and finiteelement (FEM) methods.’ The original idea of these schemes was proposed by professor Roe 23 and has been further clevcloped by professor Dcconinck ‘ Pailk n Edwin van d( I Weide Issiii rn Struijs 8 Sidilkovei Barth and Abgrall 10 11 Caiaeni and Fuchs., 1 Rad 13 and Nishikawa. 14 Unlike for the FV’I approach, the residual (listribution method assumes a continuous flow field, with data stored at the vertices of the cell. Formulated as a pseudo-time dependent problem, the method computes a cell-residual which is the cell volume integral of the PDE (Euler/Navier-Stokes) and then splits this residual among the cell’s vertices according with a specific iiumerical scheme to construct the updates of the nodalvalues and to advance the solution towards the (pseudo-time) stea(ly-state. One of the key ingredients of the multidimensional upwind residual distribution schemes is the residual splitting/distribution mechanism. Over the years only a few very successful nmltidimensional upwind scalar and matrix distribution schemes have been proposed, e.g. the positive Narrow (N), the Low Diffusion A (LDA), and the Positive Streaniwise Invariant (PSI) schemes’ High-order (higher than order) residual distribution schemes have been proposed by Caraeni’ 5 and 6 The original idea proposed in Abgrall) 15 was to use a high-order computation of the cell-residual together with a linearity preserving distribution scheme to obtain a third-order compact scheme suitable for Large Eddy Simulations (LES) and for Computational Aeroacoustics (CAA). Despite all these advances the multidimensional upwind residual (listribution schemes proposed to-date showed two main drawbacks: due to a density-based pressure velocity coupling, these schemes require some sort of (low-Mach) pre conditioning, in order to be able to accurately simulate low-speed (incompressible) regimes, very accurate schemes have only been formulated for simplicials (triangles in 2D and tetrahedra in 3D). The very few attempts made so far to come up with distribution schemes for general finite-elements have only been partially successful.’ 9 Those schemes for the Euler/Navier-Stokes equations on finite-elements other than simplicials could not prove comparable accuracy with the similar schemes devised on simplicials. Here, the authors present a new higll-or(ler multidimensional residual distribution method in an all-speed formulation and applicable to general finite-elements. There are two key elements for this new algorithm: a) use a pressure-based equations coupling, and multidimensional residual distribution schemes for spatial discretization, to allow the solution of flow equations for all regimes, without resorting to a low-Mach preconditioning, .
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*Iea(l Development Engineer, Flucift Inc., 10 CavcidisIi Court, Senior AIAA I\leml)er Lcad Development Engineer, Fluent Inc. , 10 Caveridish Court , AIAA J\Icniber
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b) use an oJ)timized residual distribution schemes for the resulting advection-diffusioii equations of this pressure— based algoritluii. T3oth the classjcal (central—upwind) and a iriore natural Green—functions—based (Greeii functions of the advection-diffusion eqn.) residual distribution have been considered. These scheiiies are linearity-preserving ( LP) schemes, thus can be extended to high order of accuracy (third arni forth order) As a result, the overall pressure-based algorithm has a high order of accuracy, at convergence. Sample results, showing the accuracy in all fviach regimes of the new scheme and future work plans will conclude the paper. .
I.
Flow equations
Consider the Navier-Stokes system of equations in 2D written in conservative form:
+L= 0x
DT
where as:
Q
i9x
(1
= {p, PUi , pE} is the conservative variable vector. The convective/diffusive flux vectors are expressed
Uj UU()
FC
+ PöiO
(2)
puH 0
11=
(3) +
with i = 0, 1 Here p, p, u = (u 1 , ‘u ),E aiid H = E + 2
are the pressure, density, velocity vector, total energy
.
E = T+
and the total enthalpy, respectively. The stress tensor -r is given by:
= 2i(S where
[L
—
5LI:)
(4)
is the dynamic viscosity and Sjj is the deformation rate tensor:
sij
=
1 8•u ;-(-T----
2 dx 1
+
thU
—) 8x
(5)
The equation of state p = p(p, T) , close the system, where T is the static temperature. These equations express a balance between the convective and diffusive terms. In the pure diffusion regime (i.e. when the diffusion term is dominant) it is appropriate to use a central discretization, while in the advection dominated regime one needs an (full or biased) upwind discretization. In fact, for a correct solution of the Navier-Stokes equations one needs a discretization that will naturally adapt its character to the local flow reginie, changing automatically from a central to an upwind discretization as the regime changes from a diffusion dominated to a convection dominated (and for all the regimes in between). Here we present what we call the classical central-upwind residual distribution treatment. We also show how a more natural multidimensional residual distribution scheme can be constructed, based on a set of local Green-functions for the advection-diffusion equation. These two schemes can be used for the solution of the Navier-Stokes equations, in the new pressure-based resi(Iual distribution formulation presented here.
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Advection-difflision equation discretization
The scelar advection-diffusion equation that we use as the building block for our Navier-Stokee discretiza tion is written as,
a
82
(6)
Uiw#V#8,
This is a second order PDE that can be transformed into an equivalent system of two first order PDE’s,
8 8 uiE#_v;;ti
=
“—a;
=
8,
(7)
0
(8)
or equivalent, when using it in the context of a pseudo-time marching algorithm, one can write itas,
L#+(&#_vkwi_8,)
(
#) 4 tJ_
=
0
(9)
=
0
(10)
Consider that the computational domain V has been decomposed in a set of non-overlapping elements, {fQ=i,..,fl SUCh that V = First we present here the so ceDed clanicxd residual distribution scheme for the advection-difusion equation, as proposed by Nishikawa 17 In this approach the system 7 is solved by iterating in pseudo-time the following two distinct steps: - firstly, advance the scalar equation in in pseudo-time using central-upwind distribution coefficients, constructed based on the upwind distribution coefficients and an estimate of the (cell) Reynolds number
01
=
(ftUP
+
3)’(1 +
—)
(11)
Here the upwind distribution coefficients fl are constructed based only on the local transport velocity it and the opposite-face inward normal 4, ag. fl” = flflit, iifl, while k is a scheme parazneter The (scalar) cell residual or fluctuation is computed using the cell volume integral: 8 8 If 4,a = I 12’r4+i’ri+8•1 (12) 40 Xj u;&j Ja\ / while the upwind (scalar) distribution coefficients are computed (here we give as an example the LDA scheme) as: maxI—fr U .llj, mn(irai;, 0) LIED •—
jQUP
1
The scaler
13
is then updated using:
J#c.Ea,a (14) ajEa - secondly, update the subsequent equations for t using a central residual distribution (Galerkin dis cretization). At convergence, the solution should satisfy in a weak-form the original advection-cliffusion equation 6. This residual distribution approach for the solution of scales advection-dufusion equation is linearity preserving, e.g. it will provide us a high-order multidimensional residual distribution discretization. lb date, we’ve tested only this approach in the new Euler/Navier-Stokes pressure-based residual distribution discretization. Next we would like to introduce a different residual distribution scheme for the solution of the advection-difusion equation 6. 3 of 20
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III.
Adjoint-based residual distribution scheme for advection-diffusion equation
In this section ai approach to creating a residual distribution scheme is presented that differs significantly ill flavor from previous (levelopments. The scheme is developed as a quasi-Newton method in which the coiivolution of the residual field with the adjoiiit Green function for the partial (hiferential system guides
the field updates. The faniiliar features of a residual distribution sclienw that enierge during the analysis include the partition of unity for the distribution coefficients, and the role of the choice of the time step size. F’urtliermore, there is no fundameiital restriction on cell topology that must be imposed when using this approach. The convection-diffusion equation for a scalar is used as an example in which this approach is applied, and a scheme is created that transitions naturally from a strongly-convective to a strongly-diffusive regime. The scheme is demonstrated to be second order accurate by performing a gri(l refinement study. An example is presented for which an analytic solution exists. The viscosity is varied by three orders of magnitude and it is denionstrated that the scheme adjusts automatically throughout the range without any nlo(lificatioll.
A.
Continuous form for Residual Distribution
A time-evolving nonlinear system is considered, where q(r, t) is the vector of independent variables at any z:, at time t within the spatial doinaiii V on which the probleni is (lefined. The systeni is denoted by
point,
Oq
+(q)=2
(15)
where (q) is a nonlinear operator. A set of initial conditions, and boundary conditions are considered as being inlpose(l on q such that the problem is well-posed. \Vitliiii this system, a sul)—domain V C V with boundary F is selected. This sub—domain will correspond ultimately to the cells surrounding a particular node of interest. Note that the boundary F may or may not coinci(le in part with the boundary of the full domain. Interest is restricted further to a time interval [—At, 0], corresponding to a single time step. Consider a scenario in which a field q(r, t) has been chosen on ‘I-’, with the consequence that
+N(q) =7(r,t),r where :g(L, t) now (lenotes a resi(lual, with solution (j. It is further noted that
V,
t
[—At,Oj
> 0, that appears because of a
1k:, —st)
=
Vr
,
(16)
poor initial choice of the
V
(17)
by construction. The goal is now to consider how the imperfect solution q can be updated so that the residual, 7 approaches zero on V for t E [—At, 0]. The strategy that is adopted here is a continuous form of Newton’s method. A correction öq is introduced on V, for t e [—At, 0], that satisfies the linearized system
(q)
+(öq)
=
-7(r,t)
(18)
with Sq(z:, —At) = Q. The addition of öq to the original q constitutes a Newton update of the solution. The next step is to express the update in terms of the observed residual. This is accomplished by constructuig a generalized or exten(led Green theorem for the honiogeneous linearized oJ)erator. An adjoint field (r, t) is iiitroduced that is used to weight the homogeneous form of the linearized system (18). A spatial integral over V and a time integration over the interval [—At, c], with c > 0, of the weighted linearized system are performed sinmitaneously. This yields
fdfdt{((öq)+q(5q))d
[I
Jv
dr(q.)] ——
—
+ t—-Lt
4
I
J—it
dt
I
Jr
j(öq, ; q).dA
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Aiiicrican Institute of Aeronautics arid Astroiiautics
(19)
where 1 ‘ the adjoint to the original linearized operator, and .7(öq,4;q) is the bilinear concomitant. It is bilinear in the fields Sq and e and may depend on the underlying (as yet imperfect) solution field q. Notethatatthispointnospecificchoiceofflasyetbeenmade. Itisnowchosentobesnadjoint Green vector resulting from the introduction of a point source on the right hand side of the adjoint system. A point source at some position rf E V at time 9 E [—At, 01 is introduced on the ith component of the acijoint system, resulting in a field The temptation is avoided to introduce a tensor here for simplicity of notation. The governing system is (I)
_si__ +
k’)
=
6(t
—
t’)6(
—
r.’)Q
(20)
where is a vector with 1 in the ith position, and zero elsewhere. Since we are dealing with a time-dependent system that represents physical processes, a causal Green function is chosen. That condition can be expressed ss p,t)=fl, Vt>!?.
(21)
Itisimportanttonotethattheadjointsystemismarchedbackwsrdsintima It is further noted that the Green function has no far-field boundary condition that is prescribed explicitly. A Green function of this kind is sometimes refered to as a free-space Green function or fundamental solution. Naturally, any solution of the homogeneous form of the adjoint system (20) can be sdded to the particular Green function and still qualify as a Green function for the system (as long as it satisfies the causality condition). In practice the causality condition is usually sufficient to identify an appropriate Green function for time-dependent systems. It follows that
6!ft, t’).ç =
:Ldr
[at
R(r t).’(r, t;p’,
ill
Volumo contribution
(22)
Boundary contribution
This rather general result expresses, on the bsais of Newton’s method, what update should be made to the ith component of the candidate solution at position pf at time V as a consequence of the observed residual field throughout the region of interest and at its boundary. The space-time structure of the adjoint Green function has a significant bearing on the update that is to be made, and already a similarity with residual distribution schemes is apparent. The connection will be made even more precise in the next section. In order to construct a discrete form, a single node is considered, at position j, which is surrounded by Nc 2ll5. The volume V of interest is the union of all the celia, and the boundary 1’ is the external shell formed by the faces not shared between cells. As suggested already, the time interval [—At, 0] corresponds to a single time-step of the discrete problem. The observed cell residual, , as may be computed from a discretization of the original differential equation, corresponds to the integral over the cell of E (y, 0). For the present purposes the residual during the interval t [—At, 0] is approximated as being spatially constant over each cell, and increasing linearly from zero to the observed final value during this interval. Namely x(Lt)(1+)
(23)
Here V, is the cell volume. The boundary contribution is ignored for the moment. The influence of this term will be considered later. With the shove approximations the general scheme gives rise to a nodal update q(r,0).ç
-
cccells
where i)
2
# c
Xc€
J d 0 rJ.Idt(i + )%,t;r.,o) 5 of 20
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(24)
(25)
The space and time-dependeiice in the Greeii function has now been formally integrated out to give a single vector for each cell. On the assumption that all cells in the domain are included, then
:
(i)
(26)
cEcells
The volume integral of the Green function, over all space, is unity at all times. This is a consequence of the first—order time—derivative in the gOvenhing equation. This partition of unity behavior is reassuring since it echoes the conditions enforced commonly on dis— trihution coefficients. However, note here that the normalization is for distribution to a single node from the surrounding cells, whereas conventionally the distribution of a cell residual to the surrounding nodes is considered.
B.
Convection-Diffusion equation
The convection—diffusion equation for a scalar b is considered, and expressions for (listribution coefficients derived from the adjoint Green function. The transporting field is considered to he uniform locally, of magnitude U , aIl(l aligned with the x—axis (without loss of generality) , and with constant diffusion coefficient
v:
+U_lJ = 2 o dt dx dx
(27)
The adjoint Green function then satisfies
dO
d —-:-—
at
—
U-;----
dx
—
u— 5 Ox;
5(r)S(t)
=
(28)
where, once again without loss of generality, the Green function is centered on the space—time origin. The causal solution to these equations in 2D is C72D(r,t)
= =
22 ____et )/4vt ) ,
4rvt
()
(29)
As has been described above, these functions describe how inlperfecta)ns in the governing system influence the solution as the system evolves. The Green functions indicate a region of influence extending upstream of the origin, with the region of influence growing larger the further back in history we look. The rate of broadening of the influence off the streaniline through the origin depends on the strength or weakness of the (liffusive I)rocess at work. The Green function clearly provides a very natural upwinding by virtue of this structure. It is also re-assuring to note that in the limit of zero convection the Green function is isotropic in space. In this way it is observed that the Green function encapsulates all of the characteristic behavior that will be encountered. That this behavior can provide specific guidance on residual distribution is a primary result of this paper. The goal of this section is to compute the distribution coefficient for an arbitrary cell, given the above Green functions. From above, the expression for the distribution coefficient is 2D
frf°dt(1 +
)O2D(rt)
(30)
As a first step, the convective part of the exponential in the Green function is re-expressed in terms of Modified Bessel functions of the first kind &2D(r, —
t)
=
=
COSO
4-irzit _e72/4vt2t/4U{Io()
2i
47rl’t
+2Ik() cosk6} 2v k=
6 of 20 Anicricari Institute of Aeronaut ics and Astronautics
(31)
where the substitutions r = •\/x2 + y 2 , and x = —r cos 8 have been made. The angle 0 is measured relative to the reverse flow direction. The contribution from the cell is given formally by the sum of the volume and boundary contribntions 22). Since the contributions from the boundary integrals for faces shared by contributing cells are equal ( and opposite they can be ignored safely. This leaves the boundary integral at the outer perimeter of the cell group. For the present development, not only is that boundary integral ignored, but it is treated as though the boundary has been pushed out to oc. Such an approximation is considered legitimate as long as the time-step, t, is maintained sufficiently small that the Green function itself is small at the original outer boundary. The choice of a time step that preserves this condition is developed as follows. Let L 0 define the charac teristic size of the cell. Define also a convective length ut, and a diffusive length The convective length indicates how far the maximum of the adjoint Green function travels in t, while the diffusive length defines the characteristic radius of the area covered by the Green function. The maximum allowed time step is determined by the constraint that the sum of these lengths equals L . From which 0
(Lou + 2v) + (Lou + 2v) 2
_
(32)
2 Lu
With the above time-step constraint met, the cell integral is approximated by fl2D
d8f
rdrf dt(1 +
(33)
D(r,t) 2 )
where o and 6i are the angles of the edges of the cell, relative to the reverse flow direction. The radial inegration of the Green function is
L
00
00
dr
rG2D(r,
t)
{i + 2
=
Ak
2 (‘i) cos k6}
(34)
where Ak(x)
/xk/2 e
(1 + k/2, 1 + k,x), F 1
=
k > 0,
(35)
2’FQ4i)
and i Fi (n, rn, x) is the Kummer confluent hypergeometric function. The Ak have the property that Ak (x) 1 as x —+ Do, and Ak(O) = 0. Performing the time integration gives fl2D
d6{i + 2ZBk( ) coskéI} 4
:
—
(36)
where Bk(x)
fXdz(1_x/z)Ak(z) =
=
rk/2
/2) [F(1
:
-
:
2
((i + k/2, k/2), (2 + k/2, 1 + k),
F ((2 + k/2, k/2), (3 + k/2, 1 + k), _x)] 2
-) (37)
where 212 ((k, £), (ri, rn), x) denotes a generalized hypergeometric function. The functions Bk(x) retain the 1 as x same general behavior as Ak(x), namely Bk(x) cc, and Bk(0) = 0 t/4v 2 In the limit of a strongly convective flow, u oc, and —
—
—
2D
01
,
L0
dO
(—
01
100
cos i)
+
If the angular range contains the upwind direction 9 upwind scheme.
=
0, then
=
L
/2D
d6 ö(8) otherwise /3
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(38) 0. This is a pure
In the limit of high viscosity, ‘u t/4v 2
0, and
—*
/ L 3 )
I
,
2ir
’ 0 f
d6
O() =
2ir
(39)
The size of the weight in this limit is deternuned simply to he the fraction of the total angular range. This is a 1)ue isotropic scheme.
C.
Coiivection Diffusion Examples
A siniple twO—(linlensiOnal iinplenientation of a solver for the convection and diffusion of a scalar has been im— plemented. A tri-iiiesh is considered, the advecting field is prescribed, and Neumann and Dirichiet boundary COfl(litiOns are considered. The cell residual on each cell is the sum of the face fluxes of the conserved quantity using nodal values of the scalar field and its gradients. Gradients at nodes are constructed by computing cell gradients in the cells surrounding each node using the Green—Gauss forniula, and performing an inverse—distance weighted average over the J)articipatmg cells. The distribution coefficients are computed by tabulating the Furier coefficients Ck(a)
=
Bk(x),
k
=
0,
...,
K
(40)
for a sequence of values of the argunient x. The iiiimber of coefficients retaine(1 is such that the last coefficient is smaller than 10 F’r a value of x = 1 this requires 7 coefficients, while for x = 5000 a total of 224 coefficients are needed. These coefficients are those of the sine series resulting froni the angular integral of the integrand in (36). A grid convergence study was l)erforllle(l to deterniine the convergence rate. A unit magnitude advecting field, aligned with the +x (lirection was chosen Ofi the unit circle, with the analytic solution .
Io (Ur/211) (Jx/2ii
(41) Io(U/2v) prescribed :n the boundary, aiid U = 1, ii = 0.5. The R’1S deviation at the iiodes from the analytic result was taken as the nieasure of the error. The solution was coniputed on uniform triangular meshs with 20, 40, 80, 160, and 320 segments on the circunference. The cell diineiision apj)roxiluately halves with each refinenent. The results are plotted in Figure 1. The results show the second order accuracy of the scheme. The a(laptability of the scheme to different levels of conpetition between convection an(l diffusion is explored in a wake diffusion example. The analytic solution for the wake of a point source in a uniform unit flow is employed to (lefine an uJ)streanl boundary condition for a 2 x 2 box with the x-axis aligned with the flow. The mesh is a uniform triangular mesh with 50 segments per e(lge. The analytic solution is
(y)
=
A1)()eU421’
(42)
where the constant A is chosen so that the peak aniplitude of the inlet profile is unity, and r is the distance to the source. The viscosity is varied by 3 orders of magnitude from 0.0001 to 0.1 in multiples of 10, and the source location shifted from -100 to -0.1 respectively in niiiltiples of 10 relative to the left side of the box. The velocity magnitude U = 1 The cases were run with the maximum choice of time step, as given by (32), .
and without any niodification
to
the operation
the scheme between cases. shown in figures 2 to 5. The transition froni the diffusive as the viscosity decreases. The exit J)rofiles for the scalar are shown of
The contour plots of the transported scalar are to highly-convective regimes is evident
in Figure 6. A direct comparis n is made with the analytic result (42) and excellent agreement is observed.
Iv.
High order residual computation
To obtain a high-order residual distribution scheme we need to combine a linearity-preserving scheme
( central—upwind) and the high—order cell-residuals. To compute the cell residual with high or(ler of accuracy 8
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