Variable reconstruction and high-order schemes

Introduction

As already pointed out, the Godunov method presented earlier (§.4.2) corresponds to a simple piecewise constant distribution of the data employed during the reconstruction step (Eq. 4.41, Fig. 4.2) which formally achieves a first order spatial accuracy.

However, Godunov versatile formulation allows the extension of the method to higher-order recon-structions, and thus greater spatial accuracy through the use of reconstruction methods. The underlying principle of these methods is the approximation of the initial data at the cell edges by using fitting functions. Two main class of methods are available in the literature, the Monotone Upstream-centred Scheme for Conservation Laws (MUSCL) [153,178,278,284] and the Essentially Non-Oscillatory (ENO) and Weighted Non-Oscillatory (WENO) [15]. Only the MUSCL class of schemes is considered in this work.

Achieving high-order spatial accuracy for hyperbolic conservation laws remains a challenge as discon-tinuities are allowed. As pointed out earlier, data need to be interpolated in both upwind and downwind directions. First order methods would not cause oscillations or create new extrema, but any sharp fea-tures of the solution would be smeared by numerical dissipation. Modern reconstruction methods use high-order interpolation in smooth regions of the flow and revert to low-order interpolation when sharp gradients are encountered.

Choice of the reconstructed variables

It is well known that the variables set (primitive, conservative, characteristics, etc.) to interpolate has a significant impact on the solution.

Interpolating conservative variables is sometimes noisy and can lead to pressure and temperature oscillations near material or contact surfaces, which is exactly what this work aims to avoid in multi-component flows. For this reason, this approach will not be considered here, even though it is the solution associated with the smallest computational cost.

Interpolating primitive variables is efficient computationally (although less so that reconstructing the conservative ones), but is prone to produce small oscillations near shock waves, especially in the case of interacting shock waves such as the colliding shocks problem presented in §.5.1.2. In this case, primitive variables are extracted from conservative variables, then limited. Eventually, the limited conserved vari-ables used in the Riemann solver are recovered using the limited primitive varivari-ables.

The last solution, interpolating characteristics variables, is interesting in the vicinity of shocks as it reduces the oscillations amplitude observed with the primitive variable interpolation. However, the computational cost associated with this approach is a lot larger than limiting primitives variables. An additional issue arising when using this limiting technique is the appearance of small temperature spikes near contact waves, which is not acceptable when reacting problems are considered. This could be avoided by using a hybrid scheme switching between primitive variables reconstruction everywhere in the flow, and characteristics near shock waves, thus taking advantage of both methods.

Despite the advantages of the hybrid method, primitive variables will be limited in this work as it was found that this interpolation worked well in most encountered situations. The primitive variables used for the different schemes are as follows,

◦ QCVF : V = [u, v, w, P, z^k, ρk]^T

◦ FCVF : V = [u, v, w, P, zk, ρk]^T

◦ QCMF : V = [u, v, w, P, T, Y^k]^T

◦ FCMF : V = [u, v, w, P, ρ, Yk]^T

1^st order Upwind

The first order Upwind scheme corresponds simply to a piecewise constant reconstruction used in the standard Godunov method, yielding,

Vⁿ_L=Vⁿ_i−1 (4.78)

Vⁿ_R=Vⁿ_i (4.79)

for the RP Uⁿ_i−1, Uⁿ_i

. This scheme will be referred to as Upwind1 in the remainder of this report.

Second-order MUSCL reconstruction

To increase the order of accuracy obtained through the Upwind1 scheme, a piecewise linear distribution is considered within the cell, which results in a second-order accurate scheme. An example of such distribution is pictured in Fig.4.5 and the data distribution within cell i follows,

V (x) = Vi+x − xⁱ

∆x ∆i, x ∈

xi−1/2; xi+1/2

 (4.80)

where Vi represents the cell averaged value of V. From this distribution, values at the left and right inter-cell boundaries are expressed as,

Vi,L=Vi−1

2∆i (4.81)

Vi,R=Vi+1

2∆i (4.82)

where ∆i is a suitably chosen slope. It can also be remarked that the mean integral value of Eq.4.80is still Vi independently of the slope choice ∆i.

The Riemann problem presented in Eqs.4.42-4.43is thus slightly modified, and becomes,

∂U

∂t +∂F

∂x = 0 (4.83)

V (x, tⁿ) =







Vi+1/2,L = Vⁿ_i +1

2∆i x < xi+1/2

V_i+1/2,Rⁿ = Vⁿ_i+1−1

2∆i+1 x > xi+1/2

(4.84)

The determination of ∆i is at the heart of the MUSCL reconstruction technique and there exists numerous approaches for its estimation, all sharing the common goal of enforcing a monotone character to avoid spurious oscillations near discontinuities. According to Godunov’s theorem [109], there are no linear monotone reconstruction schemes of order of accuracy higher than one. The slope calculation therefore relies on non-linear monotone reconstruction to circumvent this culprit.

The slope calculation retained in the present work is the MinMod slope featuring a second-order accurate reconstruction. Two slopes are defined for each component V_i^j of the primitive variables vector Vi,

∆^j_i−1/2=V_i^j− Vi−1^j (4.85)

∆^j_i+1/2=V_i+1^j − Vi^j (4.86)

Figure 4.5: Piecewise linear MUSCL reconstruction of the cell averaged data in a single cell

Slopes ∆^j_i are then estimated as the minimum value of ∆^j_i+1/2 and ∆^j_i−1/2 provided that they have similar signs, and zero otherwise to avoid an oscillatory behaviour [178].

∆^j_i =







∆^j_i−1/2 ∆^j_i−1/2< ∆^j_i+1/2 & ∆^j_i−1/2× ∆^ji+1/2 > 0

∆^j_i+1/2 ∆^j_i−1/2> ∆^j_i+1/2 & ∆^j_i−1/2× ∆^j_i+1/2 > 0 0 ∆^j_i−1/2× ∆^ji+1/2< 0

(4.87)

The computed slope is obviously different for each component of the vector and the reconstructed bound-ary states VL and VRare described by each of their components,

V_L^j=V_i^j−1

2∆^j_i (4.88)

V_R^j=V_i^j+1

2∆^j_i (4.89)

A simplified formulation used in the present work features a slope limiter function Φ defined by,

Φ = max 0, min 1, ∆^j_i+1/2

∆^j_i−1/2

!!

(4.90)

which has the advantage of accounting for the three different situations in a single expression. Recon-structed inter-cell values are thus given by,

V_L^j=V_i^j−1

2Φ∆^j_i−1/2 (4.91)

V_R^j=V_i^j+1

2Φ∆^j_i−1/2 (4.92)

This reconstruction scheme will be referred to as MinMod2 in the remainder of this work.

Fifth-order MUSCL reconstruction

A formulation similar to the MinMod2 is used for a fifth-order accurate MUSCL scheme. The piecewise linear reconstruction presented above is also used, but the slopes computation and their limiting is more complex. It has been presented by Kim and Kim [153], and successfully applied to numerous compressible flows by Thornber et al. [275,276] where its increased resolution compared to the MinMod2 has been highlighted.

Similarly to Eq.4.84, the initial conditions of the local Riemann problem at x_i+1/2 are now computed The monotonicity is enforced by limiting the linear extrapolation through the left and right limiters Φ^lim,swhich are defined by,

Φ^lim,s= max

0, min

2, 2r^lim,s_i , φ^s

(4.95) where the terms φ^s are given by,

φ^L=

and the slope ratios are estimated as follows,

r^lim,L_i =Vi+1− Vⁱ

One of the main issues of the finite volume Godunov method is its large numerical dissipation at low Mach number making them hardly suitable for LES or DNS calculations. Thornber et al. [276] showed that the inviscid fluxes computation using a standard Godunov method produced a leading order ki-netic energy dissipation rate proportional to the speed of sound, which smears all features at low-Mach numbers. A simple and very effective technique for reducing this induced dissipation was derived by Thornber et al. [276] by modifying the velocity jump at the inter-cell boundaries. The velocity is altered by a blending function behaving like an upwind scheme for large Mach numbers and like a fully centred scheme towards low-Mach numbers. The modified dissipation is not a function of the speed of sound any more, but follows the four-fifth law of Kolmogorov [165]. This simple modification has been shown to work very well in compressible mixing problems, and produces the proper decay of turbulent kinetic energy in homogeneous decaying turbulence [86,276].

The corrected velocities are expressed by, u^c_L=uL+ uR

where the blending function z is defined by,

z = min (max (ML, MR) , 1)