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Abstract

As part of this work, high-speed viscous flow is modelled using the computational fluid dynamics toolset OpenFOAM®. The existing density based solver, rhoCentralFoam, is extended to evaluate possible improvements in stability and accuracy through the implementation of alternative discretisation schemes. This extension includes evaluation of different spatial interpolation techniques to compute the face fluxes as well as a comparison of multistage explicit and matrix-free implicit methods when discretising the system of equations in time. To evaluate the different numerical schemes a number test cases for high-speed viscous flow given in literature are considered.
9th OpenFOAM Workshop
23-26 June 2014 in Zagreb, Croatia
Modelling high-speed flow using a matrix-free
coupled solver
Johan A Heyns1, Oliver F Oxtoby1and Adriaan Steenkamp2
1Aeronautic Systems, Council for Scientific and Industrial Research (CSIR), Meiring Naude
Road, Pretoria, South Africa, jheyns@csir.co.za and ooxtoby@csir.co.za
2Flamengro a division of Armscor Research and Development SOC Ltd., Armscor Building,
Nossob Street, Pretoria, South Africa, adriaans@armscor.co.za
Key words: Density-based, Compressible flow, Coupled solver, Matrix-free
This work considers the development and implementation of a matrix-free, implicit solver for the
analysis of high speed compressible flow using the density based approach. The work can broadly
divided into two parts: First, the HLLC and AUSM+up flux interpolation schemes are imple-
mented and compared with the Central-Upwind scheme currently employed in rhoCentralFoam.
The second part of the work considers the temporal discretisation of the governing equations.
With the aim of improving efficiency a matrix-free implicit solver is implemented and evaluated
against segregated and multi-stage temporal schemes.
1 Introduction
This study considers the development of the compressible density based solver for im-
proved accuracy and efficiency when modelling high-speed viscous flow. When considering
high-speed external aerodynamics, density based solvers are typically preferred as they
allow for the efficient modelling of sharp numerical discontinuities as encountered with
propagating shock waves. These solvers usually employ central schemes with artificial
dissipation or flux splitting schemes to resolve the shock waves [2].
As part of this study different spatial discretisation or flux interpolation schemes are
implemented in the OpenFOAMrfinite volume tool set and evaluated. With the aim of
improving the stability and efficiency of the solver, explicit multi-stage schemes as well
as matrix-free implicit methods are investigated to discretise the system of equations in
time.
Modelling high-speed flow using a matrix-free coupled solver
2 Numerical implementation
The conservative continuity, momentum and energy equations governing unsteady com-
pressible viscous flow read [14]
(ρ)
∂t +(ρuj)
∂xj
= 0 (1)
(ρui)
∂t +(ρuiuj)
∂xj
+∂p
∂xi
=σij (2)
(ρE)
∂t +(ρujE)
∂xj
+(ujp)
∂xj
=ulσlj +k∂T
∂xj
(3)
where ρis the density; uithe velocity in the Cartesian direction i;pis the pressure;
E=e+|u|2/2 denotes total energy and ethe specific energy. The set of equations are
closed using the equation of state for a perfect gas
p= (γ1)ρe, T =e/Cv(4)
where Cvdenotes the specific heat at constant volume and γis the specific heat ratio,
Cp/Cv. The components of the viscous stress tensor are given by
σij =µ∂ui
∂xj
+∂uj
∂xi+λuk
∂xk
δij (5)
with µand kbeing the viscosity coefficient and thermal conductivity. If Stokes hypothesis
can be assumed the following relation can be employed, λ=2µ/3.
The governing equations provided above can be described using a single unified equa-
tion. Applying the finite volume approach the equation can be cast into weak form by
integrating over the control volume, V, and from Gauss’ divergence theorem it follows
∂t ZV
WdV+ZS
Fj
cnjdS=ZS
Fj
vnjdS(6)
where Wis the conservative variable vector and contains the components
W= [ρ;ρui;ρE] (7)
and the vector of convective fluxes reads
Fc= [ρuj;ρujui+δijp;ρHuj] (8)
with the total enthalpy is defined as H=E+p/ρ.
If the effect of viscosity and thermal diffusivity are neglected, Fv= 0, the Euler equations
are obtained. For the initial evaluation of the numerical discretisation the Euler equations
will be used as it will allow for a more direct evaluation of the interpolation schemes,
without the introduction of additional numerical diffusion stemming from the viscous and
diffusive terms.
2
Modelling high-speed flow using a matrix-free coupled solver
3 Spatial discretisation
Additional to the Central-Upwind scheme of Kurganov and Tadmor [5] used in rhoCen-
tralFoam [4], AUSM+up [6] and HLLC [1] are implemented to compute the face fluxes. As
a detailed account of these schemes have been presented in literature only a brief overview
of them are given here. In the presentation of the various formulations it is attempted to
relate it as closely as possible to the numerical implementation of the respective solvers.
3.1 AUSM+up
Liou [6] extended the AUSM scheme for improved solutions on a wider range of speed
regimes. It employs a flux splitting method where the inviscid flux is split into convective
and pressure parts
F=ρujψ+P(9)
where
ψ= [1, ui, H]; P= [0, p δij,0]
The convective flux, ψis simply interpolated using a non-linear higher-order scheme
ψf=(ψlif Mf>0
ψrif Mf0(10)
where land ris is the left and right hand states which are computed using the non-linear
slope limiting scheme MUSCL [13]. Above, Mdenotes the Mach number.
Liou [6] notes that the interpolated mass flux can be computed as follows
(ρuj)f=cfMf(ρlif Mf>0
ρrif Mf0(11)
For the AUSM+up scheme, Liou [6] recommends the following definition for the acoustic
velocity face value
cf= min(˜cl,˜cr) (12)
where
˜cl/r =c
l/rc
l/r
max(c
l/r,|u|)(13)
3
Modelling high-speed flow using a matrix-free coupled solver
and cis the critical speed of sound and in the case of a perfect gas reads
c=s2(γ1)
γ+ 1 H(14)
The Mach number flux can be computed based on the left and right hand states
Mf=M+
(m)(Ml) + M
(m)(Mr) + Mp(15)
where Ml/r =ul/r/cfand the split Mach numbers are polynomial functions of degree
m= 1,2,4
M±
(1)(M) = 1
2(M± |M|),(16)
M±
(2)(M) = ±1
4(M±1)2(17)
and
M±
(4)(M) = (M±
(1) if M > 1
M±
(2)(1 16βM
(2)) if M0(18)
To improve the solution for low Mach number flows a pressure diffusion term is introduced
Mp=Kpmax(1 σM, 0)prpl
ρ
fc2
f
, σ 1, ρ
f= 1/2(ρl+ρr) (19)
The general pressure flux formulation reads
pf=P+
(n)(Ml)pl+P
(m)(Mr)pr+pu(20)
where Liou [6] recommends a fifth degree polynomial which is also expressed in terms of
the split Mach number functions
P±
(5)(M) = (1
MM±
(1) if M > 1
M±
(2)[(±2M)16αM M
(2)] if M0(21)
and the following velocity difference or diffusion term is suggested
pu=kuP+
(5)(Ml)P
(5)(Mr)(ρl+ρr)cf(urul) (22)
with the coefficient 0 Ku1.
4
Modelling high-speed flow using a matrix-free coupled solver
3.2 HLLC
Toro et al. [12] proposed an extension of the HLL scheme [9] to restore the contact
surface. Batten et al. [1] evaluated an implicit version of the HLLC scheme which is
positivity-preserving. It is suggested that the computational cost of the scheme is similar
to the Roe scheme and that it resolves both shock and contact angles. With HLLC the
convective flux is defined as [1]
Fj
c=
Fj
lif Sl>0
Fj(U?
l) if Sl0< Sm
Fj(U?
r) if Sm0Sr
Fj
rif Sr<0
(23)
where
Fj(U?
l/r) =
Smρ?
l/r
Sm(ρui)?
l/r +p?δij
Sm((ρE)?
l/r +p?)
(24)
U?
l/r =
ρ?
l/r
(ρui)?
l/r
(ρE)?
l/r
= Ωl/r
(Sl/r ul/r)ρl/r
(Sl/r ul/r)(ρui)l/r + (p?pl/r)δij
(Sl/r ul/r)(ρE)l/r ul/r pl/r +Smp?
(25)
l/r = (Sl/r Sm)1(26)
p?
l/r =ρl/r(ul/r Sl/r)(ul/r Sm) + pl/r (27)
In these equations the velocity magnitude is computed using the relation ul/r = (ui)l/r ni
where niis the outward facing unit vector normal to the cell face. Batten et al. [1] suggest
the following relation for signal velocity
Sm=ρrur(Srur)ρlul(Slul) + plpr
ρr(Srur)ρl(Slul)(28)
and Toro et al. [?] recommends
Sl= min[ulcl,˜u˜c] (29)
Sr= max[urcr,˜u+ ˜c] (30)
where ˜uand ˜care the Roe averages for the flow velocity and acoustic velocity.
5
Modelling high-speed flow using a matrix-free coupled solver
The Roe-averaged variables are computed based on the density ratio
˜
R=rρr
ρl
(31)
from which the Roe-averaged velocity, specific enthalpy and acoustic velocity then follows
˜uj=˜
Ruj
r+uj
l
˜
R+ 1 (32)
˜
H=˜
RHr+Hl
˜
R+ 1 (33)
˜c=q(γ1)( ˜
H1/2|˜u2
i|(34)
4 Temporal discretisation
The general form of the governing equations can be expressed as a system of coupled
differential equations, of which the semi-discrete form reads
Wn+1 Wn
tV=βRn+1 + (1 β)Rn(35)
were Rdenotes the right-hand side of the equations. First order implicit and explicit
formulations are obtained for βvalues of respectively unity and zero. The well known
second-order Crank-Nicolson scheme is recovered for β= 1/2.
4.1 Stability
Implicit time stepping methods have no stability constraints and the time-step size is
only governed by accuracy. These are, however, subjected to large computational and
memory overheads. Explicit methods, despite being computationally efficient and easy
to implement, are restricted by stability constraints. For a stable explicit solution the
time-step size is restricted by the CFL number, for which the follow expression can be
used
t=CF Lχ
λ(36)
where the CFL number must be smaller than unity for β= 0. λrefers to the eigenvalues
associated with the system and the length scale is computed using the relation ∆χ=
xixi.
6
Modelling high-speed flow using a matrix-free coupled solver
4.2 Multi-stage Runge-Kutta
To evaluate the possible improvement in both stability and accuracy a multistage Runge-
Kutta scheme is implemented. With this multistage time-stepping scheme the residual is
evaluated at intermediate stages and weighted as the solution is advanced. The coefficients
used to weight the residual can be optimised based on the required stability and accuracy.
For the purpose of describing the Runge-Kutta temporal discretisation the semi-discrete
system of governing equations may be written as
W
t=1
VZS
Fj
cnjdS=R(W) (37)
The generalised form of the multistage Runge-Kutta then reads [7]
Wn+1 =Wn+ ∆tbiri(38)
ri=R(tn+cit, Wn+ ∆taij rj), i = 1, s;j= 0, s 1 (39)
with sbeing the stage of the scheme and the coefficients aij, 1 j < i s,bi,i= 1, s
and ci,i= 2, s. These coefficients are typically derived using the Butcher table [7].
4.3 Matrix-free implicit solver
If an implicit approach is employed to discretise the flow equations, a sparse non-symmetric
system is recovered. As direct solvers are not practical for large systems, iterative solu-
tion methods and approximate factorization methods are typically employed to solve
such a system. To reduce the memory overhead typically associated with implicit solvers
a matrix-free approach [8] is implemented. The semi-discrete form of the Euler implicit
scheme reads
Wn+1 Wn
tV=Rn+1 (40)
which can be linearised in time as
Wn
p
tVp=Rn
p+Rn
WWp(41)
where W=Wn+1 Wn. The equation can be written for all nodes as a function of the
Jacobian matrix
V
tIRn
WW=Rn(42)
It is noted that the left-hand side coefficient matrix, A, contains a diagonal contribution
from the temporal discretisation, V/t, as well as the Jacobian matrix, Rn/∂W.
7
Modelling high-speed flow using a matrix-free coupled solver
The Jacobian matrix involves the linearisation of both the convective and viscous flux
vectors. As an explicit formulation of the flux Jacobian is not always feasible, a simplified
first-order approximation of fluxes is linearised. This implies that the quadratic conver-
gence of Newton’s method would not be achieved, but it reduces storage requirements
and improves computational efficiency. It then follows that the number of non-zeros in
each matrix row is related to the number edges associated with each node. To simplify
the computation of the left-hand side matrix the Jacobian fluxes are approximated using
Lax-Friedrich
R(Wp,Wn,nf) = 1
2(F(Wp,nf) + F(Wn,nf)) 1
2|λf|(WnWp) (43)
where ndenotes the neighbouring nodes of node p. The largest wave speed is computed
as
|λf|=|uj
fnj
f|+cf(44)
with cfbeing the speed of sound and nbeing the unit vector normal to the cell face f.
The left-hand side coefficient matrix, A, can be stored as strictly upper, lower and diagonal
matrices so that for a node pthe following holds
LW|p=X
n:n<p
Rp
Wn
Wn=X
n:n<p
1
2(J(Wn)− |λf|I) ∆Wn
UW|p=X
n:n>p
Rp
Wn
Wn=X
n:n>p
1
2(J(Wn)− |λf|I) ∆Wn
D="Vp
tX
n
Rp
Wp#="Vp
tX
n
1
2(J(Wp)− |λf|I)#
where J=F/Wrepresents the Jacobian of the inviscid flux vector.
Luo et al. [8] suggested a matrix-free implicit method to solve the system of equations
given above. This method consisted of solving a system of approximated linear equa-
tions through the use of a Generalised Minimum Residual (GMRES) method [11] with
Lower-Upper Symmetric Gauss-Seidel (LU-SGS) preconditioning [16]. It is suggested
that the LU-SGS preconditioning provides good stability properties and compared to ex-
plicit methods is competitive in term of computational cost, while the iterative GMRES
solver improves the convergence of the solution. The GMRES method is a generalisa-
tion of the conjugate gradient method and allows for the solution of linear systems with
a non-symmetric and/or positive definite coefficient matrix, as commonly found when
considering the flow equations.
5 Results
To evaluate the implemented flux splitting schemes and the matrix-free implicit solver a
number of benchmark test cases are considered. These include the forward facing step
8
Modelling high-speed flow using a matrix-free coupled solver
(a) Cockburn and Shu (b) Central-Upwind
(c) AUSM+up (d) HLLC
Figure 1: Forward facing step
[15], the NACA0012 airfoil at 2 degrees angle-of-attack and the RAE plane test problem
[10]. Based on availability these test cases the numerical predicted results are compared
to experimental and simulation results.
The first test case considers the forward facing step as discussed in the work of Woodward
and Collella [15]. Uniform flow at Mach 3 enters a rectangular channel and is obstructed
by a forward facing step. The flow structure at t = 4 s are evaluated and the ability of the
discretisation schemes to capture the shock formations are investigated. The numerical
predicted results for the Central-Upwind, AUSM+up and HLLC schemes are shown and
compared to the results of Cockburn and Shu [3] in Figure 1. It is found that there
is good agreement between the Central-Upwind scheme and AUSM+up, but that the
accuracy deteriorates if HLLC is used. On structure meshes AUSM+up is able to pick
up higher-order flow features like the Kelvin–Helmholtz instability forming at the shear
layer.
To evaluate the analysis of flow over a 3D geometry the RAE plane as discussed in the work
of Romanelli [10] is considered. In Figure 2 a comparison of the numerically predicted
pressure coefficients and experimental measurements are shown. These include results
along the fuselage as well as the wing chord at different span positions. It is noted that
the implemented fourth-order Runge-Kutta solver with high-resolution with AUSM+up
scheme provides a reasonably accurate approximation of the experimental measurements
and agrees with numerical results obtained from the commercial code Fluent [10].
Lastly, to evaluate the implemented multi-stage explicit and matrix-free implicit solvers
the NACA0012 airfoil at a 2 degree angle-of-attack is simulated. In Figure 3 the simulation
9
Modelling high-speed flow using a matrix-free coupled solver
(a) Pressure contour (b) Fuselage, φ= 15·
(c) Wing, Y/B = 0.6 (d) Wing, Y/B = 0.85
Figure 2: RAE plane
time for the respective temporal discretisation schemes are compared. It is shown that the
matrix-free implicit approach provides a notable improvement in stability and efficiency
over its explicit and multi-stage counterparts. For this case it is found that the matrix-free
LU-SGS solver and GMRES solver with LU-SGS preconditioning provides respectively a
15 and 95 time speed-up when compared to the segregated explicit solver.
6 Conclusion
In conclusion, it is found that the OpenFOAMrtoolset easily allows for the development
of customised CFD solutions. Furthermore, it is shown that it can be used to efficiently
obtain accurate solutions for high-speed compressible flows.
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10
Modelling high-speed flow using a matrix-free coupled solver
(a) Pressure contour (b) Cut-cell Cartesian Mesh
(c) CPU time (Explicit and Implicit) (d) CPU time (Implicit)
Figure 3: NACA 0012: Comparison of CPU times for the explicit, multi-stage and implicit
coupled solvers
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12
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An accurate, fast, matrix-free implicit method has been developed to solve the three-dimensional compressible unsteady flows on unstructured grids. A nonlinear system of equations as a result of a fully implicit temporal discretization is solved at each time step using a pseudo-time marching approach. A newly developed fast, matrix-free implicit method is then used to obtain the steady-state solution to the pseudo-time system. The developed method is applied to compute a variety of unsteady flow problems involving moving boundaries. The numerical results obtained indicate that the use of the present implicit method leads to a significant increase in performance over its explicit counterpart, while maintaining a similar memory requirement.
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Central schemes may serve as universal finite-difference methods for solving nonlinear convection–diffusion equations in the sense that they are not tied to the specific eigenstructure of the problem, and hence can be implemented in a straightforward manner as black-box solvers for general conservation laws and related equations governing the spontaneous evolution of large gradient phenomena. The first-order Lax–Friedrichs scheme (P. D. Lax, 1954) is the forerunner for such central schemes. The central Nessyahu–Tadmor (NT) scheme (H. Nessyahu and E. Tadmor, 1990) offers higher resolution while retaining the simplicity of the Riemann-solver-free approach. The numerical viscosity present in these central schemes is of order ((Δx)2r/Δt). In the convective regime where Δt∼Δx, the improved resolution of the NT scheme and its generalizations is achieved by lowering the amount of numerical viscosity with increasing r. At the same time, this family of central schemes suffers from excessive numerical viscosity when a sufficiently small time step is enforced, e.g., due to the presence of degenerate diffusion terms.