Applied Computational Fluid Dynamics Techniques - Wiley Episode 1 Part 8 docx

8.35The salient features of this type of artificial viscosity are as follows: - it is invariant under coordinate rotation; - ∇2may be approximated by M l− Mc, yielding a fast scheme; - b

This is the same as central differencing! A stability analysis of the RHS,

r i = −1

2h (u i+1− ui−1), (8.15)indicates that only every second node is coupled, allowing zero-energy or chequerboardmodes in the solution Therefore, stabilizing terms have to be added to re-couple neighbouringnodes

The two most common stabilizing terms added are as follows

1 Terms of second order: h2∂2/∂x2, which result in an additional RHS term of

which equals 4 for the ( −1, 1, −1) chequerboard pattern on a uniform mesh shown in

Figure 8.1 Most TVD schemes use this type of stabilization term

1

-10u

i - 1 i + 1 i + 2

Figure 8.1 Chequerboard mode on a uniform 1-D grid

2 Terms of fourth order: h4∂4/∂x4, which result in an additional RHS term of

u i+2− 4ui+1+ 6ui − 4ui−1+ ui−2, (8.17)

which equals 16 for the ( −1, 1, −1) chequerboard pattern on a uniform mesh shown

in Figure 8.1 Observe that this type of stabilization term has a much more pronouncedeffect than second-order terms Therefore, one may use much smaller constants whenadding them The fourth-order operator can be obtained in several ways One obviouschoice is to perform two ∇2-passes over the mesh (Jameson et al (1986), Mavriplis and Jameson (1990)) Another option is to first obtain the gradients of u at points,

and then to approximate the third derivatives by taking a difference between firstderivatives obtained from the gradients and the first derivatives obtained directly from

the unknowns (Peraire et al (1992a)) The implications of choosing either of these

approaches will be discussed in more depth in Chapter 10

Figure 8.2 Equivalency of linear GFEM and FVM

The same equivalency between the GFEM with linear elements and FVMs can also beshown for 3-D tetrahedra

8.2 Lax–Wendroff (Taylor–Galerkin)

All Lax–Wendroff type schemes are derived by performing a Taylor-series expansion in time

(Lax and Wendroff (1960), Donea (1984), Löhner et al (1984))

u =
tu,t+
t2

2 u,tt



Then the original equation,

8.2.1 EXPEDITING THE RHS EVALUATION

The appearance of the JacobiansA makes the evaluation of the RHS tedious Substantial

savings may be realized by developing two-step schemes that approximate the RHS withoutrecourse toA (Burstein (1967), Lapidus (1967), MacCormack (1969)) The simplest two-step

scheme that accomplishes this goal is given by:

Several possibilities exist for the spatial discretization If one chooses linear shape functions

for both half-steps (N i , N i ), a five-point stencil is obtained in one dimension The schemeobtained is identical to a two-step Runge–Kutta GFEM Thus, for the steady state no damping

is present Another possibility is to choose constant shape functions for the half-step solution

(P e , N i ) This choice recovers the original three-point stencil in one dimension, and for thelinear advection equation yields the same scheme as that given by (8.27) Observe that thisscheme has a second-order damping operator for the steady state Thus, this second schemerequires no additional stabilization operator for ‘mild’ problems (e.g subsonic flows withoutshocks) The difference between these schemes is shown conceptually in Figure 8.3

Figure 8.3 Two-step schemes

8.2.2 LINEAR ELEMENTS (TRIANGLES, TETRAHEDRA)

Because of the widespread use of the latter scheme, a more detailed description of it will begiven The spatial discretization is given by

where Nn denotes the number of nodes of the element.

(b) Second step The second step involvesSCATTER-ADDoperations:

a) b)

Figure 8.4 Two-step Taylor–Galerkin scheme

8.3 Solving for the consistent mass matrix

Due to its very favourable conditioning, (8.30) is best solved iteratively as

of the shock The shock width is typically orders of magnitude smaller than the smallestmesh size used in current simulations Therefore, this viscosity effect has to be scaled to thecurrent mesh size At the same time, the effects of these artificial viscosity terms should only

be confined to shock or discontinuity regions Thus, a sensing function must be provided tolocate these regions in space These considerations lead to artificial viscosity operators of theform

d=
t

t l ∇(h2f (u)) ∇u. (8.32)

Here f (u) denotes the sensing function and h the element size The ratio of timestep taken

(
t) to allowable timestep (
t l) is necessary to avoid a change in solution when shiftingfrom local to global timesteps Some popular artificial viscosities are listed below

(a) Lapidus Defined by (see Figure 8.5)

Shear Layer v

l

Shock

v

v v

l

Figure 8.5 Lapidus artificial viscosity

- it is invariant under coordinate rotation;

- it is essentially 1-D and thus fast;

- it produces the desired effects at shocks;

- k llvanishes at shear and boundary layers;

- the identification of shocks (k ll < 0) or expansions (k ll >0) is simple

The Lapidus artificial viscosity (Lapidus (1967), Löhner et al (1985a)) is most often employed for transient simulations in conjunction with more sophisticated FCT (Löhner et al.

(1987)) or TVD schemes (Woodward and Colella (1984)) in order to add some backgrounddamping

(b) Pressure-based Given by

d=
t

t l ∇(h2f (p)) ∇u. (8.35)The salient features of this type of artificial viscosity are as follows:

- it is invariant under coordinate rotation;

- ∇2may be approximated by (M l− Mc), yielding a fast scheme;

- because the pressure p is near constant in shear and boundary layers, f (p) should

vanish there;

- usually, the enthalpy is taken for the energy equation in u.

Many variations have been proposed for the sensing function f (p) The three more popular

ones are as follows:

8.5 Boundary conditions

No numerical scheme is complete without proper ways of imposing boundary conditions TheEuler equations represent a hyperbolic system of PDEs Linearization around any arbitrarystate yields a system of advection equations for so-called characteristic variables (Usaband Murman (1983), Thomas and Salas (1985)) The proper handling of these boundaryconditions will be discussed below The following notation will be employed:

ρ: the density;

v n: the normal velocity component at the boundary (pointing inwards);

v t: the tangential velocity component at the boundary;

p: the pressure; and

c: the velocity of sound

Furthermore, use will be made of the Bernoulli equation, given by the total pressure relation:

Suppose that an explicit evaluation of the fluxes (i.e the RHS) has been performed andthe unknowns have been updated These predicted unknowns, that have to be corrected

at boundaries, will be denoted by ‘∗’, e.g the predicted density as ρ∗ A linearizedcharacteristics analysis can be used to correct the predicted (∗) values; performing a 1-D

analysis at the boundary, in the direction normal to it, we have, with the inward normal n (see

Figure 8.6), the following set of eigenvalues and eigenvectors:

Wing Wall

Engine Wall

n n

Figure 8.6 Boundary conditions for wing

One is now in a position to correct the predicted variables depending on the Mach number

of the flow The possible states are: supersonic inflow, subsonic inflow, subsonic outflow andsupersonic outflow Let us consider them in turn

(a) Supersonic inflow In this case, there should be no change in the variables, as no

information can propagate upstream Therefore, the variables are reset to the values at thebeginning of the timestep Thus, the predicted values are discarded in this case

(b) Subsonic inflow The incoming characteristics are λ1−3, while the outgoing characteristic

is λ4 The values for the variables W1, W2, W3, corresponding to the incoming characteristics,

are taken from the ‘state at infinity’, and W4is taken from the predicted state This results in

v t = vt∞; vn = vn∗

→ p = 0.5(p∞+ p∗) ; → ρ = ρ∞+ (p − p∞)/c2. (8.48)

(c) Subsonic outflow The incoming characteristics are λ3, while the outgoing characteristics

are λ 1,2,4 The value for the variable W3, corresponding to the incoming characteristic, is

taken from the ‘state at infinity’, and W1, W2, W4 are taken from the predicted state Theresulting equations are

(c4) Prescribed mass flux (ρvn )∞:

v t = vt∗; p = p∗+ c∗((ρv n )∞− ρ∗v n∗)

→ ρ = ρ∗+ (p − p∗)/c2; → vn = (ρvn )∞/ρ. (8.53)

(d) Supersonic outflow In this case, no information can propagate into the computational

domain from the exterior Therefore, the predicted variables are left unchanged

(e) Solid walls For walls, the only boundary condition one can impose is the no-penetration

condition, taking into consideration the wall velocity w The predicted momentum at the

surface
ρv∗is decomposed as

ρv∗=
[ρ(w + αt + βn)], (8.54)

where t and n are the tangential and normal vectors, respectively The desired momentum at

the new timestep should, however, have no normal velocity component (β = 0) and it has the

(a1) Evaluation of viscous fluxes at points, e.g.,

Here nj denotes the component of the surface normal vector in the j th direction Inserting

the appropriate expressions for the stresses results in

i, j components of the tensor of second derivatives would more than double the memoryrequirements For this reason, most edge-based solvers use the first form when evaluating theviscous terms

A third way to evaluate the viscous fluxes is found by separating the terms formingLaplacian operators from the rest of the viscous fluxes For the momentum equations thisresults in

so there is almost no extra cost involved

9 FLUX-CORRECTED TRANSPORT

SCHEMES

Although flux-corrected transport (FCT) schemes have not enjoyed the popularity of totalvariation diminishing (TVD) schemes in the steady-state CFD community, they are includedhere for the following reasons

(a) FCT schemes were the first to introduce the concept of limiters (Boris and Book(1973, 1976), Book and Boris (1975)), and particularly for didactic purposes it neverhurts to go back to the source

(b) FCT schemes are used almost exclusively in the unsteady CFD and plasma physicscommunities

(c) Unlike TVD schemes, FCT limiting is based directly on the unknowns chosen Thismakes FCT ideally suited to introduce the concept of nonlinear schemes and limiting.FCT schemes were developed from the observation that linear schemes, however good,perform poorly for transient problems with steep gradients For these linear schemes, thechoice is between high-order, oscillatory, ‘noisy’ solutions, or low-order, overdiffusive,

‘smooth’ solutions Godunov’s theorem (Godunov (1959)) states this same observation as:

Theorem (Godunov) No linear scheme of order greater than 1 will yield monotonic

(wiggle-free, ripple-free) solutions.

The way to circumvent this stringent theorem is to develop nonlinear schemes This is

most easily accomplished by mixing a high-order and a low-order scheme The process of

combining these two schemes in a rational way is called limiting.

Any high-order scheme used to advance the solution either in time or between iterationstowards the steady state may be written as

u n+1= u n +
u h = u n +
u l + (
u h −
u l ) = u l + (
u h −
u l ). (9.1)

Here
u h and
u l denote the increments obtained by the high- and low-order scheme,

respectively, and u l is the monotone, ripple-free solution at time t = t n+1 of the low-orderscheme The idea behind FCT is to limit the second term on the RHS of (9.1),

u n+1= u l + lim(
u h −
u l ), (9.2)

in such a way that no new over/undershoots are created It is at this point that a furtherconstraint, given by the original conservation law itself, must be taken into account: strictconservation on the discrete level should be maintained This means that the limiting processwill have to be carried out wherever RHS contributions are computed: for element-basedschemes at the element level, for edge-based schemes at the edges

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

9.1 Algorithmic implementation

The concept of FCT can now be translated into a numerical scheme The description is givenfor element-based schemes For edge-based schemes, replace ‘element’ by ‘edge’ in whatfollows For finite volume schemes, one may replace ‘element’ by ‘cell’ FCT consists of thefollowing six algorithmic steps:

1 Compute LEC: the element contributions from some low-order scheme guaranteed to

give monotonic results for the problem at hand;

2 Compute HEC: the element contributions given by some high-order scheme;

3 Define the anti-diffusive element contributions:

4 Compute the updated low-order solution u l:

AEC c = Cel · AEC, 0 ≤ Cel≤ 1; (9.5)

6 Apply the limited AEC:

9.1.1 THE LIMITING PROCEDURE

Obviously, the whole approach depends critically on step 5 above If we consider an isolatedpoint surrounded by elements, the task of the limiting procedure is to insure that the

increments or decrements due to the anti-diffusive element contributions AEC do not exceed

a prescribed tolerance (see Figure 9.1)

In the most general case, the contributions to a point will be a mix of positive and

negative contributions Given that the AECs will be limited, i.e multiplied by a number

0≤ Cel≤ 1, it may happen that after limiting all positive or negative contributions vanish.The largest increment (decrement) will occur when only the positive (negative) contributionsare considered For this reason, we must consider what happens if only positive or onlynegative contributions are added to a point The comparison of the allowable increments anddecrements with these all-positive and all-negative contributions then yields the maximum

allowable percentage of the AECs that may be added or subtracted to a point On the other

hand, an element may contribute to a number of nodes and, in order to maintain strictconservation, the limiting must be performed for all the element node contributions in thesame way Therefore, a comparison for all the nodes of an element is performed, and thesmallest of the evaluated percentages that applies is retained

Define the following quantities:

allowable range for u n+1 i

Figure 9.1 Limiting procedure

i is taken between each point and its nearest-neighbours For based schemes, it may be obtained in three steps as follows:

element-(a) maximum (minimum) nodal unknowns of u n and u l:

u∗

i = maxmin

!

(u l i , u n i ); (9.11)(b) maximum (minimum) nodal value of element:

(c) maximum (minimum) unknowns of all elements surrounding node i:

u

max min

i For example, the so-called ‘clipping limiter’ is

obtained by setting u∗

i ="max min

#

u l iin (a), i.e by not looking back to the solution at the previoustimestep or iteration, but simply comparing nearest-neighbour values at the new timestep oriteration for the low-order scheme As remarked before, the limiting is based solely on the

unknowns u, not on a ratio of differences as in most TVD schemes.

9.2 Steepening

The anti-diffusive step was designed to steepen the low-order solution obtained at the newtimestep or iteration In some cases, particularly for high-order schemes of order greater thantwo, the anti-diffusive step can flatten the profile of the solution even further, or lead to anincrease of wiggles and noise This is the case even though the solution remains within itsallowed limits A situation where this is the case can be seen from Figure 9.2

x

u

x AEC

(D (D

=-1)

= 0)

Figure 9.2 Steepener for FCT

This type of behaviour can be avoided if the anti-diffusive flux is either set to zero orreversed A simple way to decide when to reverse the anti-diffusive fluxes is to compute thescalar product of the low-order solution at the new timestep or iteration and the anti-diffusiveelement contributions:

∇u l · AEC < 0 ⇒ AEC = −α · AEC, (9.14)

with 0 < α < 1 Values of α greater than unity lead to steepening This can be beneficial

in some cases, but is highly dangerous, as it can lead to unphysical solutions (e.g spuriouscontact discontinuities) in complex applications