Applied Computational Fluid Dynamics Techniques - Wiley Episode 1 Part 9 doc

First derivatives: second form While 10.10 is valid for any finite element shape function, a more desirable form of theRHS for the first-order fluxes is ri = e ij k Fk j+ Fk i , e ij k =

ipoi2 ipoi1

unkno, rhspo

geoed, redge

rhspo

ipoi2 ipoi1

ipoi2 ipoi1

(a)

(b)

(c)

Figure 10.1 Edge-based Laplacian

10.2 First derivatives: first form

We now proceed to first derivatives Typical examples are the Euler fluxes, the advective termsfor pollution transport simulations or the Maxwell equations that govern electromagneticwave propagation The RHS is given by an expression of the form

One may observe that:

- for a change in indices ij versus ji we obtain

- an extra boundary integral leads to a separate loop over boundary edges, adding

(unsymmetrically) only to node j

The flow of information for this first form of the first derivatives is shown in Figure 10.2

j i

j i

(a)

(b)

Figure 10.2 First derivatives: first form

Observe that we take a difference on the edge level, and then add contributions to bothendpoints This implies that the conservation law given for the first derivatives is not reflected

at the edge level, although it is still maintained at the point level This leads us to a secondform, which reflects the conservation property on the edge level

10.3 First derivatives: second form

While (10.10) is valid for any finite element shape function, a more desirable form of theRHS for the first-order fluxes is

ri = e ij

k (Fk j+ Fk

i ), e ij k = −e ji

In what follows, we will derive such an approximation for linear elements As before, we start

by separating the Galerkin integral into shape functions that are not equal to N iand those thatare equal:

One may observe that this form is anti-symmetric in the pairing of indices i, j for d k ij and

symmetric for b ij k Moreover, an additive form for the fluxes on each edge has been achieved

However, it is important to bear in mind that this pairing of fluxes is only possible for linear

elements.

A typical edge-based evaluation of the first derivatives RHS for a scalar variable wouldlook like the following:

redge=cedge(1,iedge)*(fluxp(1,ipoi2)+fluxp(1,ipoi1)

enddo

The conservation law expressed by∇ · F = 0 is reflected on each edge: whatever is added to

a point is subtracted from another The flow of information and the sequence of operationsperformed in the loop are shown in Figure 10.3

j i

j i

(a)

(b)

Figure 10.3 First derivatives: second form

10.4 Edge-based schemes for advection-dominated PDEs

The RHS obtained along edges was of the form

ri = d ij

(Fk i + Fk

The inner product over the dimensions k may be written in compact form as

D ij , D ij=$d k ij d k ij (10.30)and

d k ij=12

Comparing this expression to a 1-D analysis, we see that it corresponds to a central differenceapproximation of the first derivative fluxes This is known to be an unstable discretization, andmust be augmented by stabilizing terms In what follows, the most commonly used optionsare enumerated

10.4.1 EXACT RIEMANN SOLVER (GODUNOV SCHEME)

The Galerkin approximation to the first derivatives resulted in (10.32), i.e twice the average

flux obtained from the unknowns ui ,uj If we assume that the flow variables are constant in

the vicinity of the edge endpoints i, j , a discontinuity will occur at the edge midpoint The

evolution in time of this local flowfield was first obtained analytically by Riemann (1860),and consists of a shock, a contact discontinuity and an expansion wave More importantly,the flux at the discontinuity remains constant in time One can therefore replace the averageflux of the Galerkin approximation by this so-called Riemann flux This stable scheme, whichuses the flux obtained from an exact Riemann solver, was first proposed by Godunov (1959).The flux is given by

This is a first-order scheme A scheme of higher-order accuracy can be achieved by

a better approximation to ur and ul, e.g., via a reconstruction process and monotonelimiting (van Leer (1974), Colella (1990), Harten (1983), Woodward and Colella (1984),

Barth (1991)) The major disadvantage of Godunov’s approach is the extensive computationalwork introduced through the Riemann solver, as well as the limiting procedures In thefollowing, we will summarize the possible simplifications to this most expensive yet most

‘accurate’ of schemes, in order to arrive at schemes that offer better CPU versus accuracyratios

10.4.2 APPROXIMATE RIEMANN SOLVERS

A first simplification can be achieved by replacing the computationally costly exact Riemannsolver by an approximate Riemann solver A variety of possibilities have been considered(Roe (1981), Osher and Solomon (1982)) Among them, by far the most popular one is theone derived by Roe (1981) The first-order flux for this solver is of the form

F ij= fi+ fj − |Aij |(u i− uj ) (10.36)where|Aij| denotes the standard Roe matrix evaluated in the direction dij In order to achieve

a higher-order scheme, the amount of dissipation must be reduced This implies reducing the

magnitude of the difference ui − uj by ‘guessing’ a smaller difference of the unknowns atthe location where the approximate Riemann flux is evaluated (i.e the middle of the edge).The assumption is made that the function behaves smoothly in the vicinity of the edge Thisallows the construction or ‘reconstruction’ of alternate values for the unknowns at the middle

of the edge, denoted by u−

j ,u+

i , leading to a flux function of the form

F ij= f++ f−− |A(u+i ,u−

j ) |(u−j − u+i ), (10.37)where

−

i = ui − ui−1= 2lji· ∇ui − (u j− ui ), (10.40a)

+

j = uj+1− uj= 2lj i· ∇uj − (u j− ui ), (10.40b)

and lj i denotes the edge difference vector lji= xj− xi The parameter k can be chosen to

control the degree of approximation (Hirsch (1991)) The additional information required for

ui+1,uj+1can be obtained in a variety of ways (see Figure 10.4):

- through continuation and interpolation from neighbouring elements (Billey et al.

(1986));

- via extension along the most aligned edge (Weatherill et al (1993b)); or

- by evaluation of gradients (Whitaker et al (1989), Luo et al (1993, 1994a)).

i j

i j j+1

i Ŧ 1

i j

i j

Figure 10.4 Higher-order approximations

The inescapable fact stated in Godunov’s theorem that no linear scheme of order higherthan one is free of oscillations implies that with these higher-order extensions, some form of

limiting will be required The flux limiter modifies the upwind-biased interpolations ui ,uj,replacing them by

where s is the flux limiter For s = 0, 1, the first- and high-order schemes are recovered,

respectively A number of limiters have been proposed in the literature, and this area is still amatter of active research (see Sweby (1984) for a review) We include the van Albada limiterhere, one that is commonly used This limiter acts in a continuously differentiable mannerand is defined by

s i = max 0, 2

−

i (uj − ui ) + (
−

j )2+ (u j − ui )2+

!

where is a very small number to prevent division by zero in smooth regions of the flow.

For systems of PDEs a new question arises: which variables should one use to determine

the limiter function s? Three (and possibly many more) possibilities can be considered:

conservative variables, primitive variables and characteristic variables Using limiters oncharacteristic variables seems to give the best results, but due to the lengthy algebra thisoption is very costly For this reason, primitive variables are more often used for practicalcalculations as they provide a better accuracy versus CPU ratio Before going on, we remarkthat the bulk of the improvement when going from the first-order scheme given by (10.36)

to higher-order schemes stems from the decrease in dissipation when ui− uj is replaced by

u−

j − u+i In most cases, the change from fi+ fj to f++ f−only has a minor effect and may

be omitted

10.4.3 SCALAR LIMITED DISSIPATION

A further possible simplification can be made by replacing the Roe matrix by its spectralradius This leads to a numerical flux function of the form

F ij= fi+ fj − |λ ij |(u j− ui ), (10.43)where

a second-order dissipation operator, leading to a first-order, monotone scheme As before,

a higher-order scheme can be obtained by a better approximation to the ‘right’ and ‘left’

states of the ‘Riemann problem’, which have been set to ur= ui, ul= uj This reduces the

difference between ui ,uj, decreasing in turn the dissipation The resulting flux function isgiven by

F ij= fi+ fj − |λ ij |(u−j − u+i ), (10.45)

with u−

j ,u+

i given by (10.41) and (10.42) Consider the special case of smooth flows where

the limiters are switched off, i.e s i = s j= 1 This results in

u−

j − u+i = (1 − k)[u i− uj+1

2lji · (∇u i+ ∇uj ) ], (10.47)

i.e fourth-order dissipation The important result is that schemes that claim not to require a

fourth-order dissipation inherently do so through limiting The fact that at least some form offourth-order dissipation is required to stabilize hyperbolic systems of PDEs is well known inthe mathematical literature

10.4.4 SCALAR DISSIPATION WITH PRESSURE SENSORS

Given that for smooth problems the second-order dissipation |ui− uj| reverts to a fourthorder, and that limiting requires a considerable number of operations, the next possiblesimplification is to replace the limiting procedure by a pressure sensor function A scheme of

this type may be written as

is one that has proven reliable In a first pass over the mesh, the highest of the edge-based

β-values given by (10.49) is kept for each point In a second pass the highest value of the two

points belonging to an edge is kept as the final β-value Although this discretization of the

Euler fluxes looks like a blend of second- and fourth-order dissipation, it has no adjustableparameters

10.4.5 SCALAR DISSIPATION WITHOUT GRADIENTS

The scalar dissipation operator presented above still requires the evaluation of gradients Thiscan be quite costly for Euler simulations: for a typical multistage scheme, more than 40%

of the CPU time is spent in gradient operations, even if a new dissipation operator is onlyrequired at every other stage The reason for this lies in the very large number of gradientsrequired: 15 for the unknowns in three dimensions, and an additional three for the pressure

An alternative would be to simplify the combination of second- and fourth-order dampingoperators by writing these operators out explicitly:

10.4.6 TAYLOR–GALERKIN SCHEMES

Owing to their importance for transient calculations, it is worth considering possible based Taylor–Galerkin schemes The essential feature of any Taylor–Galerkin scheme is thecombination of time and space discretizations, leading to second-order accuracy in bothtime and space An edge-based two-step Taylor–Galerkin scheme can readily be obtained

edge-by setting the numerical flux to

10.4.7 FLUX-CORRECTED TRANSPORT SCHEMES

The idea behind FCT (Boris and Book (1973, 1976), Book and Boris (1975), Zalesak (1979))

is to combine a order scheme with a low-order scheme in such a way that the order scheme is employed in smooth regions of the flow, whereas the low-order scheme isused near discontinuities in a conservative way, in an attempt to yield a monotonic solution.The implementation of an edge-based FCT scheme is exactly the same as its element-based

high-counterpart (Löhner et al (1987, 1998)) The use of edge-based data structures makes the

implementation more efficient, and this is especially attractive for 3-D problems The based two-step Taylor–Galerkin scheme will lead to a high-order increment of the form

edge-Ml
uh = r + (M l− Mc )
uh (10.58)

Here Mldenotes the diagonal, lumped mass matrix and Mcthe consistent finite element massmatrix The low-order scheme is simply

Ml
ul = r + c d (Mc− Ml )un , (10.59)i.e lumped mass matrix plus a lot of diffusion Subtracting (10.59) from (10.58) yields theantidiffusive edge contributions

(
uh −
u l )= M−1(Ml− Mc )(c dun +
u h ). (10.60)

This avoids any need for physical flux recomputations, leading to a very fast overall scheme.

As with other limiters (see above), for systems of PDEs one faces several possible choicesfor the variables on which to limit

Table 10.3 summarizes the main ingredients of high-resolution schemes for compressibleflows, indirectly comparing the cost of most current flow solvers

Table 10.3 Ingredients of edge-based compressible flow solvers

Solver Riemann Gradient Char Transf Limiting

‘Char Transf.’ denotes transformation to characteristic variable, and ‘Roe’, ‘HLLC’ denote

approximate Riemann solvers.

of literature (various conferences1, Thomasset (1981), Gunzburger and Nicolaides (1993),Hafez (2003)).

The equations describing incompressible, Newtonian flows may be written as

Here p denotes the pressure, v the velocity vector and both the pressure p and the viscosity

µ have been normalized by the (constant) density ρ By taking the divergence of (11.1) and

using (11.2) we can immediately derive the so-called pressure-Poisson equation

What sets incompressible flow solvers apart from compressible flow solvers is the fact that

the pressure is not obtained from an equation of state p = p(ρ, T ), but from the divergence

constraint This implies that the pressure field establishes itself instantaneously (reflecting theinfinite speed of sound assumption of incompressible fluids) and must therefore be integratedimplicitly in time From a numerical point of view, the difficulties in solving (11.1)–(11.3)are the usual ones First-order derivatives are problematic, while second-order derivatives can

be discretized by a straightforward Galerkin approximation We will first treat the advectionoperator and then proceed to the divergence operator

11.1 The advection operator

As with the compressible Euler/Navier–Stokes equations, there are three ways of modifyingthe unstable Galerkin discretization of the advection terms:

1See the following conference series: Finite Elements in Fluids I–IX, John Wiley & Sons; Int Conf Num Meth.

Fluid Dyn I–XII, Springer Lecture Notes in Physics; AIAA CFD Conf I–XII, AIAA CP; Num Meth Laminar and Turbulent Flow, Pineridge Press, and others.

Applied Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods, Second Edition.

(a) integration along characteristics;

(b) Taylor–Galerkin (or streamline diffusion); and

(c) edge-based upwinding

11.1.1 INTEGRATION ALONG CHARACTERISTICS

If we consider the advection of a scalar quantity φ, the Eulerian description

can be recast in the Lagrangian frame

i.e there should be no change in the unknown along the characteristics given by the

instantaneous streamlines This implies that if we desire to know the value of φ n+1

The integration along the streamlines is usually carried out using multistage Runge–Kutta

schemes (Gregoire et al (1985)) Employing linear interpolation for (11.6) results in a

monotonic, first-order scheme Higher-order interpolation has been attempted, but can result

in overshoots and should be used in conjunction with limiters

Observe the following

(a) An additional factor appears in front of the streamline upwind diffusion (Brooks and

Hughes (1982)) or balancing tensor diffusivity (Kelly et al (1980)) Usually, only the

advection-diffusion equation is studied This assumes that the transport velocity field issteady In the present case, the velocity field itself is being convected This introducesthe additional factor