Modified multi-dimensional limiting process with enhanced shock stability on unstructured grids Fan Zhanga,b,∗, Jun Liua,b , Biaosong Chenb b State
a State Key Laboratory of Aerodynamics, Mianyang, Sichuan, China Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, China
Abstract The basic concept of multi-dimensional limiting process (MLP) on unstructured grids is inherited and modified in this work for improving shock stability and reducing numerical dissipation in smooth flows. A relaxed version of MLP condition, simply named as weak-MLP, is proposed for reducing dissipation. Moreover, a stricter condition, that is the strict-MLP condition, is proposed to enhance the numerical stability. The maximum and minimum principles are satisfied by both the strict- and weak-MLP conditions. A differentiable pressure weight function is applied to combine two novel conditions, and thus the modified limiter is named as MLP-pw (pressure-weighted) limiter. A series of numerical test cases show that MLP-pw limiter has improved stability and convergence, especially in hypersonic simulations. Moreover, the limiter also shows low numerical dissipation in simulating flow fields without shock-waves. Keywords: multi-dimensional limiting process; unstructured grid; shock stability; numerical dissipation; strict/weak-MLP
1. Introduction Unstructured grids are commonly used for the spatial discretization of current industrial computational fluid dynamics codes that simulate aerodynamics or gas dynamics phenomena. The advantages of using unstructured grid include the conveniences ∗ Corresponding
author. Email address:
[email protected] (Fan Zhang)
Preprint submitted to Elsevier
December 2, 2017
5
in automatic grid generation [1, 2, 3], grid adaptation [4, 5, 6] and moving mesh techniques [7, 8], for complex geometries and flow phenomena. However, the accuracy and stability of unstructured schemes are usually challenged by the irregularity of grid connectivity and deterioration of grid quality [9, 10], which are inevitable in the automatic discretization for complicated geometries. Especially, simulating transonic and super-
10
sonic flows requires accurate approximation of nonlinear multi-dimensional physical phenomena, such as shock wave, shock waves interaction, and shock-vortex interaction, and thus excellent accuracy and stability are indispensable. As a key factor that affects spatial accuracy and stability, slope limiters, or for short, limiters, have been investigated for decades. As well known, second-order or higher
15
than second-order schemes suffer from numerical oscillations across discontinuities, a typical one of which is shock wave [11, 12]. Therefore, limiters are used to suppress these oscillations while keeping second-order reconstruction in the smooth region of flow field. On structured grids, the finite difference method (FDM) and finite volume method (FVM) have been applied along with mature limiting method based on sol-
20
id theories. The typical strategy is MUSCL (Monotonic Upstream-Centered Scheme for Conservation Laws) scheme [13] with limiters that subject to TVD (Total variation diminishing) condition [14, 15, 16]. However, these structured schemes can not be extended onto unstructured grids directly due to various reasons. Firstly, the schemes for structured grids are usually developed based on one-dimensional analysis and extended
25
to multi-dimensional structured grids by dimensional-splitting, which is infeasible for unstructured grids. Secondly, one-dimensional principles, for instance, the TVD condition, are not necessary feasible on multi-dimensional unstructured grids. An example of Jameson had shown that flow field on which the total variation is smaller could be more oscillatory than flow field on which the total variation is larger [17]. Furthermore,
30
a scheme applying TVD condition will cause accuracy deterioration at extrema even in smooth regions, and thus the TVB [18] and ENO [19] schemes were developed. By extending Spekreijse’s monotone condition [20], Barth and Jespersen designed a limiter on unstructured grids [21], which modifies the piecewise linear distribution at each control volume. Barth-Jespersen limiter removes local extrema and insures stabil-
35
ity. However, this limiter shows similar effects as that of TVD condition, which reduces 2
accuracy at smooth extrema. Furthermore, the limiting function of Barth and Jespersen is non-differentiable, and thus the convergence is less satisfactory. Therefore, an improvement was introduced by Venkatakrishnan [22], who used a differentiable function similar to that of van Albada limiter [23] which is designed for structured grids. 40
Venkatakrishnan limiter archives better convergence compared with Barth-Jespersen limiter. Whereas, Venkatakrishnan limiter is not strictly monotone, and thus it might produce oscillations across shock wave. Generally speaking, Barth-Jespersen limiter and Venkatakrishnan limiter have been successfully applied on unstructured grids since their inventions.
45
Many researches have been focusing on the improvement of limiters. In order to reduce the dissipations of two aforementioned unstructured limiters, a strategy was introduced, which is turning off limiter in subsonic region. Nejat and Ollivier-Gooch introduced hyperbolic tangent function in their application of Venkatakrishnan limiter, by which the limiter only activates in limited region [24]. Michalak and Ollivier-Gooch
50
further improved this method [25]. Thereafter, Kitamura and Shima introduced the concept of second limiter, which also uses a hyperbolic tangent function to turn off limiter in stagnation or subsonic zone, but removes predefined parameters [26]. It was proved by numerical results that second limiters can reduce dissipations effectively. A relatively new method on unstructured grids is MLP (Multi-dimensional Limit-
55
ing Process) limiter, which was first introduced on structured grids [27, 28]. By using the MLP condition which satisfies maximum/minimum principles, MLP limiter properly introduces multidimensional information. Therefore, the method has been showing better accuracy, robustness and convergence in various circumstances. Park, et al. designed unstructured MLP limter [29]. Thereby, Park and Kim [30] had constructed
60
three-dimensional unstructured MLP limiter and proved that the limiter obeys LED (Local Extremum Diminishing) condition [17]. Gerlinger designed a low dissipation MLP limiter, MLPld , on structured grids, and simulated combustion problem [31]. Do, et al. defined a low dissipation MLP limiter for central-upwind schemes [32]. Kang, et al. [33] reduced dissipation by only turning on the MLP limiter in the vicinity of shock
65
waves/nonlinear discontinuities. MLP limiter had also been developed for higher order unstructured numerical schemes, including Discontinuous Galerkin method [34] and 3
Flux-Reconstruction or Correction Procedure via Reconstruction [35, 36, 37]. Li, et al. developed a multi-dimensional limiter, WBAP, which modifies the gradients by a component by component approach [38, 39]. This method is not rotationally invariant 70
but shows good accuracy, robustness and convergence in numerical tests. In spite of the successful applications, there is still room for MLP limiter to improve the stability and convergence, especially for hypersonic flow simulations. Therefore, the presented research is focusing on this topic. This paper is organized as follows. The finite volume method and spatial reconstruction are briefly described in Section
75
2. Then, the Barth-Jespersen limiter, Venkatakrishnan limiter and MLP limiter are briefly introduced in section 3, where the differences are emphasised. In Section 4, the presented modifications on MLP limiter are formulated. A series of numerical test cases along with corresponding discussions are given in section 5. Finally, section 6 concludes the whole work.
80
2. Finite volume method and second-order reconstruction The discretization for the compressible Navier-Stokes equations is introduced as follows. The integral form of the equations is ∫ Ω
∂Q dΩ + ∂t
∫
[Fc (Q) − Fv (Q)] · ndS = 0,
(1)
∂Ω
where Q are the conservative variables in the flow field, Fc (Q) is convective flux, and Fv (Q) is viscous flux, which could be solved by using a central scheme for unstructured grids [40]. In this paper, solutions of the convective flux are emphatically investigated. Therefore, in the following discussions the Fv (Q) term is neglected, and thus the 85
equations are simplified as Euler equations. Q and Fc (Q) are given as
ρ
ρ Vn
ρ uVn + pnx ρu , Q = , Fc (Q) = ρ vVn + pny ρv ρE ρ HVn
4
(2)
where Vn = V · n = (unx + vny ). E is the total energy, H is the enthalpy, given as E=
1 p 1 2 + (u + v2 ), γ −1 ρ 2
H =E+
p , ρ
(3)
(4)
where γ is the ratio of specific heat. For air at moderate pressures and temperatures one may use γ = 1.4. The governing equations are discretized by using cell-centered finite volume formulation which is applied to a polygon computational cell i sharing a interface k with a neighbouring cell j. Therefore, the spatial discretization at cell i for the Euler equations can be expressed as N
f ∂ (QΩ)i = −( ∑ Fc,k · nk Sk )i , ∂t k=1
(5)
where Sk = |∂ Ωk | is the interface area, nk is the unit norm vector outward from the interface, N f is the interface number of cell i. Although the exact convective flux function Fc,k is nonlinear, it is usually solved by a linearized numerical flux instead of the exact formula [41]. Furthermore, the numerical flux function could be simplified as an one-dimensional scheme that calculates in the direction of vector nk . In fact, upwind schemes, such as FDS (Flux Difference Splitting) scheme or FVS (Flux Vector Splitting) scheme, are mostly designed based on one-dimensional hypothesis. FDS schemes or FVS schemes could be defined as a function of conservative variables Q, and thus the flux is given as − Fc,k = FFDS/FVS (Q+ k , Qk , nk ),
(6)
where the superscript (·)± denote the left and right values of interface k respectively. In the following paragraphs, the subscripts c and k are neglected for simplicity. The cell interface values are extrapolated from the cell centre values by using gradient ∇q: q+ k = qi + ϕi ∇qi · ∆rik , q− k = q j + ϕ j ∇q j · ∆r jk ,
5
(7)
where ∆(·)ik = (·)k − (·)i and q could be any of the conservative variables. ∇q is calculated by nodal averaging procedure [42] and Gauss-Green scheme [21], and the slope limiter value ϕ is employed to suppress oscillations at captured discontinuities. In the following sections, the calculation of ϕ will be investigated. Reconstruction becomes conservative if the integration of q over a cell equals to the cell-averaged value, i.e. q= 90
1 |Ω|
∫
qdΩ.
(8)
Ω
The time derivative in Eq.5 could be solved by explicit and implicit schemes. Due to the limited topic of the presented article, temporal solutions will not be further discussed.
3. Limiters In order to give the background information of designing the improved limiter, the 95
success classical limiters are briefly introduced in this section, with emphasising important features. For the detailed information readers may examine the original articles introducing the limiters. 3.1. Barth-Jespersen limiter and Venkatakrishnan Limiter As aforementioned, Barth-Jespersen limiter [21] and Venkatakrishnan limiter [22]
100
are two typical and common used limiters on unstructured grids. They are efficient and simple to apply on various element types, including trilateral, quadrilateral, tetrahedron, hexahedron, and so on. More importantly, these two limiters are following a similar formulation, or framework, and thus they are introduced in this subsection. The primary idea of these two limiters is maintaining monotonicity during variable
105
reconstruction. For calculating the scalar variable ϕi , in Eq.7, which eventually serves the purpose of monotonicity, the Barth-Jespersen limiter is given as ( max ) qi − qi , if qt − qi > 0 fBJ qt − qi ( min ) ϕBJ = min fBJ qi − qi , if qt − qi < 0 qt − qi 1, if qt − qi = 0 6
(9)
where the subscript t indicates a test value, which could be different based on definitions, and the function fBJ is defined as ( ) ( ) ∆+ ∆+ fBJ = min 1, . ∆− ∆−
(10)
where the ∆+ = qmax,min − q and ∆− = qt − q. This minimum function will limit the gradient, i.e. ϕi < 1, if the test value is larger than the maximum, qmax , or smaller than i the minimum, qmin i . Therefore, the new local extrema introduced by gradient recon110
struction will be removed by the limiter, and then the monotonicity can be attained. It has been commonly reported that the non-differentiable function fBJ is a major drawback that could cause accuracy lost in smooth regions and convergence deterioration in steady state computations. Therefore, an improvement, the limiter of Venkatakrishnan, is given as ) ( max qi − qi , if qt − qi > 0 fV qt − qi ( min ) ϕV = min fV qi − qi , if qt − qi < 0 qt − qi 1, if qt − qi = 0 where the function fV is ( ) ] [ ∆+ 1 (∆+ 2 + ε 2 )∆− + 2∆− 2 ∆+ , fV = ∆− ∆− ∆+ 2 + 2∆− 2 + ∆+ ∆− + ε 2
(11)
(12)
which is differentiable. The small parameter ε 2 in the function is defined as
ε 2 = (K∆h)3 ,
(13)
by which the limitation effect of limiter is tunable. Parameter ∆h is the cell scale, and 115
K usually is defined by users to adjust numerical dissipation and was evaluated in [22]. In general, if K = 0, the limiter will be very dissipative because it is activated even in smooth regions. Conversely, the limiter will be actually turned off if K ≫ 1. The difference between Barth-Jespersen limiter and Venkatakrishnan limiter is produced by introducing the function fV , by which the convergence is improved, and the
120
dissipation is reduced, but the monotonicity will not be strictly guaranteed [22]. Despite the minor shortcoming, Venkatakrishnan limiter has been successful applied for 7
simulating various flow conditions involving shock waves, and the parameter K also gives flexibility in tuning dissipation. For Barth-Jespersen limiter or Venkatakrishnan limiter, the maximum and minimum values among the direct neighbouring cells are given as qmax = max(qi , max q j ), i
qmin = min(qi , min q j ), i j∈V (i)
j∈V (i)
(14)
where the subscript V (i) indicates a set of cells connected with cell i by a common 125
interface, and the V means Volume. In order to avoid any new extrema within each cell, the following condition is expected qmin ≤ qt ≤ qmax , i i
(15)
where the qt could be any reconstructed variables within the cell. Two different choices could be used for the definition of qt , in a discretized manner. One way is to restrict the variable distribution in the whole cell by applying Eq.15 at each vertex, and thus the following formula could be given (v)
qt = qi + ∇qi · ∆ril ,
l ∈ v(i),
(16)
where the lower-case v indicates vertex, and thus the superscript (v) means the value is defined at a vertex, and v(i) indicates the set of grid vertexes of cell i. If we only expect that the values at interface centres satisfy the condition of Eq.15, the following equation will be given (f)
qt
= qi + ∇qi · ∆rik ,
∂ Ωk ⊂ ∂ Ωi ,
(17)
where the lower-case superscript ( f ) means the value is defined at an interface centre. It is obvious that Eq.16 will be stricter and more dissipative because a linear reconstruction always shows maximum and minimum at the vertexes which are the most distant points from cell centre. In the numerical cases of this article, the definition in 130
Eq.16 will be applied for Venkatakrishnan limiter, and thus the stability is expected to be enhanced.
8
Figure 1: Stencils involved in MLP condition.
3.2. Unstructured MLP Limiter MLP limiter was first designed on structured grids, and the unstructured MLP limiter is even simpler than its structured version. It should be noted that the MLP limiter shows different standpoint compared with that of Barth-Jespersen’s or Venkatakrishan’s method. A simple but important condition is max qVmin (l) ≤ ql ≤ qV (l) ,
l ∈ v(i),
(18)
where the subscript l indicates a vertex belonging to the vertexes set v(i) of cell i, and the V (l) indicates the set of all the cells connected to vertex l. Moreover, qVmin (l) = min (q j ).
qVmax (l) = max (q j ),
j∈V (l)
j∈V (l)
(19)
Eq.18 is the so call MLP condition. For each vertex of a cell, all the cell-averaged values sharing this vertex are utilised for detecting flow phenomena, including discon135
tinuities, as shown in Fig.1 which is borrowed from [29]. By applying the MLP condition to Eq.7, the permissible range of ϕ is calculated by qVmin (l) − qi ∇q · ∆ril
≤ ϕl ≤
qVmax (l) − qi ∇q · ∆ril
.
(20)
The minimum ϕi = min (ϕl ) of a cell will be applied for linear reconstruction. The l∈v(i)
following maximum/minimum principle ≤ qn+1 ≤ qmax,n qmin,n i V (i) V (i) 9
(21)
is satisfied by the linear reconstruction, where max qmax V (i) = max (qV (l) ),
min qmin V (i) = min (qV (l) ). l∈v(i)
l∈v(i)
(22)
Therefore, MLP limiter will be utilising a wider range of flow information, that is the set of common vertex neighbouring cells j ∈ V (i), compared with Barth-Jespersen and Venkatakrishnan limiters, which use only the direct neighbouring cells, j ∈ V (i). Because more information will be utilised, the limiter can be more accurate and less 140
sensitive to grid perturbation. For detailed discussions one may refer to articles [27] and [29]. Eventually, the formula of MLP limiter is ( max ) qV (l) − qi fMLP , if ql − qi > 0 ql − qi ( min ) qV (l) − qi ϕMLP = min fMLP , if ql − qi < 0 l∈v(i) ql − qi 1, if ql − qi = 0
(23)
where the ql is a vertex value calculated by unlimited linear reconstruction from the centre of cell i. The function fMLP could be the function of Barth-Jespersen limiter, fBJ , or that of Venkatakrishnan Limiter, fV . MLP limiter with using fV , named as MLP-u2 145
limiter, is showing excellent overall performance, including accuracy and convergence [29, 30]. Since the function fV has a free parameter ε 2 to be defined by users, an modified version, MLP-u2(new) limiter, has been designed [30] with an unified definition
ε2 =
K1 ∆q¯2 , 1 + θ V (l)
(24)
∆q¯V (l) . K2 ∆x1.5
(25)
min where ∆q¯V (l) = qVmax (l) − qV (l) , and
θ=
The parameter K1 and K2 are both set as a constant, 5, based on the suggestion of Park 150
and Kim [30], for the non-dimensionalized governing equations. This new definition gives an adaptive strategy, which is involving the scale of cells and the local variation 10
of flow field, to invoke limitation during computations. This strategy also removes the free parameter and thus is more convenient in practical implementations.
4. A modification: MLP-pw limiter 155
A improved limiter following the idea of MLP condition is introduced in this section. The stencils in Fig.1 are also implemented in the presented limiter, which is considered as a development of MLP. The differences are produced because of the purposes of improving shocks stability and reducing dissipation on continuous region or contact discontinuity.
160
4.1. A relaxed version of MLP condition Several researches had managed to reduce the dissipation of slope limiters, on both structured and unstructured grids. A typical and effective example on unstructured grids is the second limiter of Kitamura and Shima [26], which turns off the limiter in subsonic regions. However, on unstructured grids, the slope limiter is securing the
165
numerical schemes from unphysical spatial reconstruction, which is not only the oscillations across discontinuity, but also could be geometrical monotonicity violation [43]. Therefore, an alternative strategy will be developed for reducing dissipation. The strategy used here is to perform limitation at each interface, and thus the following formula is presented / qmax = k
∑
qVmax (l)
/
nv(k) ,
l∈v(k)
qmin k =
∑
qVmin (l)
nv(k) ,
(26)
l∈v(k)
where the qVmax,min are defined in Eq.19, and nv(k) is the number of elements in the set (l) v(k), which includes all the vertexes of interface k. For the two-dimensional cases, nv(k) ≡ 2 because the interface (line) only connects two vertexes. Then the condition is given as ± max qmin k ≤ qk ≤ qk ,
(27)
where q± k is reconstructed values at each side of interface k. This condition is named as weak-MLP condition because it is less restrictive compared with the original MLP 170
condition. 11
Figure 2: Function plane constructed by qVmax,min . (l)
It is necessary to prove the numerical dissipation of weak-MLP condition is indeed reduced. Therefore, the following lemma for linear reconstruction is given based on two-dimensional assumption. Lemma. A linear reconstruction that satisfies the weak-MLP condition in Eq.27 is not 175
more diffusive, and could be less diffusive, compared with the reconstruction satisfying MLP condition. Proof. Assume the reconstruction of cell i is performed at a triangular cell, and v(i) = min {1, 2, 3}. At each vertex l ∈ v(i), there are a maximum, qVmax (l) , and a minimum, qV (l) ,
which are already given. Therefore, an unique linear distribution could be defined for all the maximums or all the minimums, q(max) (r) = qmax + ∇qmax (r − ri ), i
q(min) (r) = qmin + ∇qmin (r − ri ), i
(28)
where the qmax,min are the averages of maximums and minimums respectively. q(max,min) (r) i are linear functions, which means that their distributions are planes in function space, as shown in Fig.2. in Eq.19, the cell centre value qi will be satisfyBased on the definition of qVmax,min (l) ing max max (qVmin (l) ) ≤ qi ≤ min (qV (l) ). l∈v(i)
l∈v(i) 180
max If max (qVmax (l) ) = min (qV (l) ), two possibilities could be given: l∈v(i)
l∈v(i)
p1. qi is the maximum among all the common vertex neighbouring cells, V (i), 12
(29)
Figure 3: The projections on x − y space of the intersection lines in function space.
p2. cell centre values in neighbouring cells are all larger than qi and equal to each other. Therefore, one of the following equation will be valid , qi = q(max) (ri ) = qmax i
or qi = q(min) (ri ) = qmin i .
(30)
min If max (qVmin (l) ) = min (qV (l) ), similar result will be given. In such circumstances, MLP l∈v(i)
l∈v(i)
condition and weak-MLP condition are showing not difference. Otherwise, qi will be satisfying the following inequality q(min) (ri ) < qi < q(max) (ri ). 185
(31)
Considering the maximum constraint q(max) (r), the following deduction could be given. A limit value ϕMLP is given for a linear reconstruction ϕ ∇q, which will be a plane in function space. Three circumstances that satisfy MLP condition are listed follow: c1. If the reconstruction plane is not cutting the plane of q(max) (r) within the cell, it
190
will be not limitation taking place. c2. If the reconstruction plane is cutting the plane of q(max) (r), and the projection of intersection line is across and only across the cell at one of the vertexes of the cell (projectionc2 in Fig.3), the MLP condition will not be actived except at this vertex. Therefore, except at the vertex, following inequality is valid qMLP (r) = qi + ϕMLP ∇q · (r − ri ) < q(max) (r). The weak-MLP condition will be satisfied as well. 13
(32)
c3. If the reconstruction plane is cutting the plane of q(max) (r), and the projection of intersection line coincides with one of the interface of the cell (projectionc3 in Fig.3), the MLP condition will be actived at the two vertexes of the corresponding interface. 195
Except at the intersection line/interface, Eq.32 is satisfied, and thus the weak-MLP condition is satisfied as well. Then, a linear reconstruction that satisfies weak-MLP is given as qw-MLP (r) = qi + ϕw-MLP ∇q · (r − ri ).
(33)
The reconstructed plane in function space is cutting the plane of q(max) (r) in a line of which the projection is across the centres of interface f1 and f2 (projectionw-MLP in Fig.3), and then qw-MLP (r f2 ) = q(max) (r f2 ),
qw-MLP (r f1 ) = q(max) (r f1 ),
(34)
where the r fk is the centre coordinate of the interface. Due to the property of the linear reconstruction, the value at the vertex that is opposite of f3 will be larger than the qVmax (l) at the same vertex, which means the MLP condition is violated. 200
Similar deduction for the minimum constraint qmin (r) could be given, and thus the lemma is proved. The original MLP condition was proved to be satisfying maximum and minimum principles, which guarantee the stability of MLP limiter [29]. By simply following similar procedure, the maximum and minimum principles could be proved for weak-
205
MLP condition. Theorem 1. For a following hyperbolic conservation law in two-dimensional space,
∂ q ∂ f (q) ∂ g(q) + + = 0, ∂t ∂x ∂y
(35)
finite volume method could be used for fully spatial-temporal discretization. While a monotone Lipschitz continue flux function is used for the calculation of numerical fluxes, a linear reconstruction that satisfies Eq.27 under an appropriate CFL condiction will satisfy the maximum/minimum condition. Proof. Assume the reconstruction is performed at a triangular cell. The conservation
14
law of Eq.35 could be discretized into a semi-discrete form N
|Ωi |
f ∂ qi − + ∑ F(q+ k , qk )|∂ Ωk | = 0. ∂ t k=1
(36)
The linear reconstruction gives ∇qi and ∇q j , reconstructing the interface values in the following form q− k = q j + ∇q j · ∆r jk ,
q+ k = qi + ∇qi · ∆rik ,
(37)
satisfying the weak-MLP condition ± max qmin k ≤ qk ≤ qk .
(38)
Then, by applying Eq.26, following inequality is given min ± max qmin ≤ qmax V (i) ≤ qk ≤ qk ≤ qk V (i) .
(39)
Therefore, the proof of the Theorem in subsection 3.2 of [30] could be followed, and then the final formula of maximum/minimum principle qmin,n ≤ qn+1 ≤ qmax,n , i V (i) V (i) will be valid in CFL condition of ∆t
210
Li |Ωi |
1 ∂F sup (q1 , q2 ) ≤ , [ ] ∂q 3 2 min max
(40)
(41)
q1 ,q2 ∈ qV (i) ,qV (i)
where Li is the perimeter of Ωi . The principle in three-dimensional circumstance could be further investigated in the framework of [29]. Therefore, by applying the weak-MLP condition, the numerical schemes will be less dissipative in computations, without violating the maximum and minimum principles.
215
4.2. The strict-MLP condition The strict-MLP condition is defined for improving shock stability. By a direct observation, one may found that the original MLP condition has a minor deficiency in
15
4.5
4
4 3.5
3
3
q
q
2.5
2
2 1.5
1
1 0.5 0
0
1
2
x
3
0
4
(a) Original condition
0
1
2
x
3
4
(b) Strict condition
Figure 4: Limiting conditions based on different principles.
certain situations, for which an example is shown in Fig.4. In Fig.4(a), linear reconstructions are performed at each cell, and then limitations are made based on MLP 220
condition. Therefore, new extrema (compared with cell centre values) will not be produced. In fact, in one-dimensional cases, Barth-Jespersen and Venkatakrishnan limiters present the same result. In Fig.4(a), at the interface/point (x = 1) between left two cells, the reconstructed value in the left side (of the interface/point) is larger than that in the right side. Where-
225
as, the original values in cell centres are showing the contrary. Such a circumstance commonly exists in the conservative piecewise-linear reconstruction of FVM which usually shows discontinuity at interfaces. Usually, this situation does not cause negative effects in computations. However, it could cause instable results near shock waves because this unphysical distribution could affect the pre/post-shock states. As well
230
known, shock wave is a highly non-linear phenomenon. A small change of pre/postshock states could change the strength or position of shock wave, which could lead to remarkable flow variations. Therefore, due to the nonlinearity, the potential effects should be aware of. A strict monotonicity is hence presented in Fig.4(b). An average value is defined at each interface between two cells, and the average value will be used to define a bound
16
for left or right reconstructed values. For each cell, the limitation could be written as max qmin v(i) ≤ q ≤ qv(i)
(42)
where the qmax,min are the maximum/minimum of the averaged vertex values of cell i v(i) qmax v(i) = max (ql ),
qmin v(i) = min (ql ).
(43)
l∈v(i)
l∈v(i)
The overlines indicate that the variables are calculated by the average procedure, which could be a simple weighted average procedure as ql =
∑i∈V (l) ωli qi , ∑i∈V (l) ωli
∀i, ωli ≥ 0,
(44)
where ωli ≥ 0 is the weight of qi . A simple and monotone weight is the inverse distance weight in [42], that is
ωli = 1/|∆rr li |.
(45)
This average method will be used in the following paragraphs. 235
Therefore, in any cases, the strict-MLP condition guarantees that the reconstructed values are strictly monotone. The maximum/minimum principle is obviously satisfied by strict-MLP condition, which could be proved in a similar way as for weak-MLP condition. The brief proof is given as follow. Theorem 2. For a following hyperbolic conservation law in two-dimensional space,
∂ q ∂ f (q) ∂ g(q) + + = 0, ∂t ∂x ∂y
(46)
finite volume method could be used for fully spatial-temporal discretization. While 240
a monotone Lipschitz continue flux function is used for the calculation of numerical fluxes, a linear reconstruction that satisfies Eq.42 under an appropriate CFL condiction will satisfy the maximum/minimum condition. Proof. Assume the reconstruction is made on a triangular cell. The semi-discrete conservation law in Eq.36 is also valid. The linear reconstruction satisfying the strict-MLP condition, ∇qi and ∇q j , which reconstruct the interface values in the following form q+ k = qi + ∇qi · ∆rik ,
q− k = q j + ∇q j · ∆r jk , 17
(47)
give the following inequalities − max qmin v( j) ≤ qk ≤ qv( j) .
+ max qmin v(i) ≤ qk ≤ qv(i) ,
(48)
Then, by applying Eq.26, following inequality is given min ± max max qmin V (i) ≤ qv(i) ≤ qk ≤ qv(i) ≤ qV (i) , 245
(49)
which is the same as that in Eq.39. Therefore, the subsequent conclusion could be made as that of Theorem 1. Remark 1. The strict-MLP condition is more restrictive, and thus more dissipative. In multi-dimensional circumstances, it is unnecessary to restrict the complete linear distribution within a whole cell to satisfy strict-MLP condition. In fact, only the values
250
at several given point, for example the centres of interfaces, should be checked for strict-MLP condition in Eq.42, and thus the scheme will be less dissipative. Remark 2. Three conditions, original MLP, weak-MLP, and strict-MLP, which satisfy maximum/minimum principle, are already given. By satisfying these three conditions, linear reconstructions could be expected to be free from spurious oscillations. Howev-
255
er, as have already been proved, these conditions are different in diffusivity, which will show different numerical performance. 4.3. Pressure weight function Two new conditions are presented in the last two subsection, which will be used to reduce and enhance dissipation respectively. Therefore, how to combine these two conditions in an unified framework is needed to be answered. Obviously, the strictMLP condition is defined for improving shock stabilities, and thus this condition should be used in the vicinity of shock waves. Flow pressure will be increasing drastically across shock waves, and thus the pressure increment had been used for indicating these strong discontinuities. A effective pressure weight function is of the following form ( min )3 pv(i) ωp = , (50) pmax v(i) are maximum and minimum vertex pressure calculated by Eq.43 where the pmax,min v(i) within a cell. 18
Similar polynomial type pressure weight function had been successfully applied for indicating shock waves in upwind schemes [44, 45, 10]. This function will be a small value if the pressure variation within the cell is significant, and thus the strict-MLP condition could be applied. On the contrary, weak-MLP condition will be used while the function giving a relatively large value (but never larger than one). Therefore, the following limiter is introduced ) ( ω p qmax + (1 − ω p )qmax k v(i) − qi f , if qk − qi > 0, qk − qi ( ) min ω p qmin ϕMLP-pw = min (51) k + (1 − ω p )qv(i) − qi , if qk − qi < 0, {k|∂ Ωk ⊂∂ Ωi } f qk − qi 1, if qk − qi = 0, 260
where the qk is an interface centre value calculated by unlimited linear reconstruction from the centre of cell i. To be specific, strict-MLP limiter can be attained by giving
ω p ≡ 0, and weak-MLP limiter can be attained by giving ω p ≡ 1. Remark 3. Here, pressure difference is utilised to indicate the existence of shock waves. It should be noted that density or entropy, or velocity (vector) could be used for 265
indicating discontinuities as well. For instance, in [33], density jump was used to distinguish linear discontinuities from continuous region and nonlinear discontinuity. As well known, flows across shock waves is showing entropy increment. Velocity vector is changed across shock waves, and thus it could be used to indicate shock wave, as in the rotated upwind scheme [46, 47]. Therefore, how to choose the shock indication
270
variable is an open question. Here, pressure is used because of its effectiveness. Furthermore, the computations of contact discontinuity or slip line are expected to be less dissipative, and thus the weak-MLP will be uniformly applied in continuous region and near linear discontinuities. Remark 4. A polynomial function in Eq.50 is used to calculate the weight of strict/weak-
275
MLP condition. The hyperbolic tangent function [26] and exponential function [48] could serve similar purpose. However, the performance of these functions is not discussed in this article. The presented pressure weight function in Eq.50 will be showing satisfactory performance in the numerical cases.
19
Remark 5. The accuracy of piecewise linear approximation is essentially second280
order. Limiters will not elevate the accuracy but reduce it. By using the pressure weight function, the property of limiter will be varying depending on local flow field state. Obviously, the strict-MLP condition will produced more significant numerical dissipation, but the condition will only be activated within limited regions by the pressure weight function.
285
5. Numerical results Numerical test cases are introduced in this section to verify the performance of the presented MLP-pw limiter. Other limiters are also tested for comparison. Since MLP limiter(s) have been outperformed the Barth-Jespersen limiter and Venkatakrishnan limiter in various researches, it is not necessary to compare these two classical
290
limiters specifically, and thus only Venkatakrishnan limiter is used to show the basic behavior of the limiting function fV . For simplicity, MLP-u2 limiter will be simplified as MLP limiter without further explaination. 5.1. Grid refinement test: isentropic vortex advection The well known test case of isentropic vortex advection problem is applied in this
295
subsection to examine the accuracy of the presented limiter in smooth flow with using a series of refined meshes. The meshes are generated by using Delaunay triangulation, and thus the number of cells is only approximately increasing at 4 times during each refinement, as shown in Table 1. The flow field set up and computation duration are the same as in [49]. A mean flow is of (ρ , u, v, p) = (1, 1, 1, 1) and a superimposed isentropic vortex is given as
β 0.5(1−r2 ) e (−y, x), 2π (γ − 1)β 2 1−r2 e , δT = − 8γπ 2 (δ u, δ v) =
r2 = (x − x0 )2 + (y − y0 )2 ,
δ S = 0,
20
(52)
10
8
8
6
6
y
y
10
4
4
2
2
0
0
2
4
6
8
0
10
0
2
4
6
x
x
(a)
(b)
8
10
Figure 5: Density contour and mesh of the isentropic vortex advection problem: 3230 cells.
where β = 5, T is temperature, S is entropy, and the initial location of the vortex 300
centre is (x0 , y0 ) = (5, 5), in the computational domain of [0, 10] × [0, 10], as shown in Fig.5(a). Four steps Runge-Kutta scheme [50, 42] with CFL = 0.6 is used for temporal solution. The results at non-dimensional time t = 0.2 are compared, and the location of the vortex will be (x, y)t=0.2 = (7, 7), as shown in Fig.5(b). HLLC scheme [51] is used for the computations of numerical flux at cell interfaces.
305
The numerical errors and the order of accuracy are shown in Table 1. Since the parameter K affects the performance of the Venkatakrishnan limiting function, two values are given, i.e. K = 1 and K = 10. It is obvious that the L1 error convergence of Venkatakrishnan limiter (K = 1) is approximately reduced to first-order, and the L∞ error convergence is even lower than first-order. MLP limiter (K = 1) shows better
310
behavior, of which the L∞ convergence is nearly second-order. When using K = 10, Venkatakrishnan limiter and MLP limiter show significant improvement in both the absolute accuracy and the order of accuracy. On the other hand, the presented MLP-pw limiter is not very sensitive to the parameter K, although increasing K still improves the accuracy. As a result, MLP-pw limiter
315
shows advantage in accuracy when a smaller K is applied. Moreover, in general, the absolute accuracy of MLP-pw limiter is better than other competitors in this test case,
21
Figure 6: Grid for the shock tube problems.
and its convergence order of L1 error is nearly second-order. MLP-u2(new) limiter is showing better accuracy compared with Venkatakrishnan limiter and MLP limiter (K = 1), but MLP limiter can be showing lower error if the limitation is relaxed, with 320
using K = 10. Therefore, the MLP-pw limiter using weak-MLP condition is in deed reducing the numerical dissipation, as has been proved in subsection 4.1. MLP-u2(new) limiter introduces an automatic mechanism to tune the dissipation and shows advantage compared with MLP limter (K = 1), but the MLP-pw limiter using weak-MLP is still more
325
accurate in this case of which the flow field is relatively smooth. Since in this case the strict-MLP limitation may only be showing minor contribution, following test cases will further investigate the performance in discontinuity-capturing simulations. 5.2. Simple discontinuity-capturing: shock tube problems Two shock tube problems are used to test the performance of limiters in simulating
330
some basic flow phenomena on unstructured gird. The grid is shown in Fig.6, and the computational domain is [0.0, 1.0] × [0.0, 0.1]. There are 101 boundary grid points in the horizontal direction and 11 boundary grid points in the vertical direction, and 2292 triangular cells are created by Delaunay triangulation. A vertical grid line is formed in the middle of the domain, by which the initial discontinuity could be defined
335
accurately. It should be noted that the grid is unsymmetrical, and thus the results will be unsymmetrical even if a symmetry initial condition has applied. Four steps RungeKutta scheme with CFL = 0.2 is used for temporal solutions. Venkatakrishnan limiter, MLP limiter and MLP-pw limiter all use the Venkatakrishnan function fV , of which the parameter K is set as 1. HLLC scheme is used for the computations of numerical
340
fluxes. The Sod shock tube problem [52] is used for testing the performance of simulat-
22
Table 1: Grid refinement test results for isentropic vortex advection. Venkatakrishnan(K = 1)
Venkatakrishnan(K = 10)
MLP(K = 1)
MLP(K = 10)
MLP-pw(K = 1)
MLP-pw(K = 10)
MLP-u2(new)
Boundary cells
Volume cells
L∞
8×4
218
2.2805E-01
order
L1
order
16 × 4
830
1.3696E-01
0.736
5.4692E-03
1.282
32 × 4
3230
8.2773E-02
0.727
2.9407E-03
0.895
64 × 4
12960
4.3317E-02
0.934
1.7294E-03
0.766
1.3298E-02
8×4
218
2.2180E-01
16 × 4
830
9.0211E-02
1.30
3.6741E-03
1.826
32 × 4
3230
3.0010E-02
1.59
9.5966E-04
1.937
64 × 4
12960
8.2305E-03
1.866
2.8153E-04
1.769
1.3025E-02
8×4
218
2.2407E-01
16 × 4
830
9.5783E-02
1.226
3.9092E-03
1.742
32 × 4
3230
3.0862E-02
1.634
1.0275E-03
1.928
64 × 4
12960
8.4520E-03
1.868
3.0251E-04
1.764
1.3074E-02
8×4
218
2.2180E-01
16 × 4
830
9.0043E-02
1.301
3.6693E-03
1.828
32 × 4
3230
2.9730E-02
1.597
9.5322E-04
1.945
64 × 4
12960
7.4675E-03
1.993
2.7561E-04
1.790
1.3025E-02
8×4
218
2.2221E-01
16 × 4
830
9.0661E-02
1.293
3.7111E-03
1.811
32 × 4
3230
2.8860E-02
1.651
9.5408E-04
1.960
64 × 4
12960
7.2473E-03
1.994
2.7505E-04
1.794
1.3020E-02
8×4
218
2.2179E-01
16 × 4
830
9.0010E-02
1.301
3.6684E-03
1.828
32 × 4
3230
2.9701E-02
1.600
9.5266E-04
1.945
64 × 4
12960
7.4306E-03
1.999
2.7530E-04
1.791
1.3025E-02
8×4
218
2.2517E-01
16 × 4
830
9.2112E-02
1.290
3.8280E-03
1.747
32 × 4
3230
2.9385E-02
1.648
9.8738E-04
1.955
64 × 4
12960
7.6506E-03
1.941
2.8601E-04
1.788
23
1.2853E-02
MLP MLP-u2(new) Venkatakrishnan MLP-pw exact
Density
0.8
MLP MLP-u2(new) Venkatakrishnan MLP-pw exact
2.8
2.6
Internal Energy
1
0.6
2.4
2.2
0.4
2 0.2
1.8 0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
0.6
x
x
(a)
(b)
0.8
1
Figure 7: Flow field distributions along the centerline of Sod problem.
ing shock wave, contact discontinuity and expansion. The dimensionless initial condition across the middle discontinuity is (ρ , u, v, p)L = (1, 0, 0, 1) and (ρ , u, v, p)R = (0.125, 0, 0, 0.1). The results at non-dimensional time t = 0.2 is presented in Fig.7. 345
Compared with Venkatakrishnan limiter, MLP-type limiters are showing more accurate results, but oscillations could be found in the right side of the contact discontinuity. Supersonic expansion problem is used to test the performance in low density and pressure that approximate to zero. The dimensionless initial condition across the middle discontinuity is (ρ , u, v, p)L = (1, −2, 0, 0.4) and (ρ , u, v, p)R = (1, 2, 0, 0.4). The
350
results in non-dimensional time t = 0.15 is presented in Fig.8. Again, Venkatakrishnan limiter shows more dissipative results. MLP-type limiters are showing more accurate results, and the result calculated by MLP-pw limiter shows the most accurate approximation in the central low density region. In order to reveal the details of the limiting process, density limiting value con-
355
tour of each limiter for Sod problem is shown in Fig.9. In general, MLP-pw limiter is less diffusive in most of the regions, especially in continuous region and near the contact discontinuity. Correspondingly, oscillation has been found near the contact discontinuity. Similar behavior had occurred in the results of MLP limiter [29] and WBAP limiter [38], for which the characteristic-wise reconstruction is successfully
360
used. Therefore, the oscillation of MLP-pw limiter has not been deemed as a signifi-
24
MLP MLP-u2(new) Venkatakrishnan MLP-pw exact
1 0.9 0.8
MLP MLP-u2(new) Venkatakrishnan MLP-pw exact
1 0.9 0.8
Internal Energy
Density
0.7 0.6 0.5 0.4
0.7 0.6 0.5
0.3
0.4 0.2
0.3
0.1 0
0
0.2
0.4
0.6
0.8
0.2
1
0
0.2
0.4
0.6
x
x
(a)
(b)
0.8
1
Figure 8: Flow field distributions along the centerline of supersonic expansion problem.
cant drawback. MLP-u2(new) limiter introduces limitation in steady region which the waves have not yet reached. It is assumed that the ∆q¯V (l) in Eq.24 is too small due to the constant flow field, and thus the limiting function fV is sensitive to tiny numerical noise. 365
5.3. Capturing smooth oscillating waves: shock-density wave interaction The classical one-dimensional shock-density wave interaction problem first designed by Shu and Osher [53] is used to test the capability in capturing smooth wave in this subsection. In this case, a 3 Mach shock wave travels along a two-dimensional shock-tube, which mimics the one-dimensional condition, interacting with sine wave
370
in density. This case is especially interesting because the numerical schemes simulating this case need to tackle the shock wave, which requires numerical dissipation, and to capture smooth critical points, which expects low dissipation for accuracy reason. The two-dimensional [−5, 5] × [0, 0.5] computation domain is discretized by 25046 triangular volume cells. There are 401 boundary grid points in the horizontal direction and 21 boundary grid points in the vertical direction. The initial condition of the case is designed as follows (ρ , u, v, p) =
(3.8571, 2.6294, 0, 10.3333), if x ≤ −4,
(1 + 0.2sin(5x), 0, 0, 1), 25
if x > −4.
(53)
(a) MLP
(b) MLP-u2(new)
(c) MLP-pw
(d) Venkatakrishnan Figure 9: Density limiting value contours of Sod problem. Thirty equally spaced contour lines from ϕ = 0 (blue) to ϕ = 1 (red).
An one-dimensional grid of 5000 cells is used to calculate the reference solution, using MLP-pw limiter. Since MLP limiter and Venkatakrishnan limiter are both sensitive to 375
the parameter K, in order to show better behavior of these two limiters, K = 10 is set in this case for Venkatakrishnan, MLP and MLP-pw limiters. Still, Runge-Kutta scheme is used for temporal solution, with using CFL = 0.6, and HLLC scheme is applied for calculating numerical fluxes. The computational results at non-dimensional time t = 1.8 are shown in Fig.10.
380
It can be found that Venkatakrishnan limiter is dissipative and smears the wave significantly, especially for the oscillating wave structure in Fig.10(b). MLP and MLPu2(new) are outperforming Venkatakrishnan limiter, as expected. Considering this oscillating smooth wave is also showing strong variation in flow variables, MLP and MLP-u2(new) will be activated in this case. On the other hand, MLP-pw limiter u-
385
tilises weak-MLP limitation for smooth oscillating wave, and thus the accuracy is improved. Moreover, all the limiters, including MLP-pw, are oscillation-free in capturing shock wave. Therefore, the accuracy of MLP-pw does not come with the cost of shock stability, at least in this case. Fig.11 further shows the limiting value distribution of each limiter along the central
390
line of the flow field. It can be found that MLP-pw limiter mainly introduces dissipation in the vicinity of shock wave, and is near inactivated in the region of smooth wave
26
5 4.5
4.5
4
4
Density
Density
3.5 3 2.5
MLP MLP-u2(new) Venkatakrishnan MLP-pw Exact
2 1.5 1 0.5
-4
-2
0
3.5
MLP MLP-u2(new) Venkatakrishnan MLP-pw Exact
3
2.5
2
0.5
4
1
1.5
2
2.5
x
x
(a)
(b)
Figure 10: Flow field distributions along the centerline of shock-density wave interaction problem.
structure. Therefore, MLP-pw limiter is able to attain high accuracy as has been shown. However, other three limiters are showing significant limitation in smooth region. 5.4. Multi-dimensional shock wave: double shock reflection 395
Inviscid supersonic flow over a wedge-shaped forward step is simulated, in which two reflection shock waves and an expansion fan are formed sequently. Flow field is discretized by 58684 triangular cells. LU-SGS (Lower-Upper Symmetric-GaussSeidel) scheme [54] is used for temporal solution with CFL = 10. HLLC scheme is used for the computations of numerical fluxes. The Mach number of the uniform inflow
400
is 2. The density contours are shown in Fig.12. There is little difference can be found in the flow field, especially the MLP-type limiters are showing similar results. However, the density limiting contours shown in Fig.13 give more details of the computations. MLP limiter is invoking more limiting process in the flow field, compared with
405
MLP-pw limiter, with using the same given K. Especially, MLP limiter is prone to be activated in smooth regions where the flow is compressing. MLP-u2(new) limiter is less dissipative than MLP and MLP-pw using K = 0.1, but the limiting is happening in pre-shock regions, as in the shock tube problems. Similar behavior of MLP-u2(new) limiter can be found in the supersonic bump simulation in [30], but this phenomenon
27
1
1
0.8
φρ
φρ
0.8 0.6
0.6 0.4
MLP MLP-u2(new) Venkatakrishnan MLP-pw
MLP MLP-u2(new) Venkatakrishnan MLP-pw
0.2
-4
-2
0
2
0.4 0
4
0.5
1
1.5
2
2.5
3
3.5
x
x
(a)
(b)
Figure 11: The limiting values along the centerline of shock-density wave interaction problem.
410
is not necessary to be a disadvantage since the flow field before the shock wave is uniformly steady and constant. Venkatakrishnan limiter is significantly more dissipative, and one of the results is shown in Fig.13(h) representatively. The convergent histories are shown in Fig.14 to further investigate the limtiers. The computations using MLP-pw limiter are converged in similar time steps, with using
415
different K. These results are another proof showing the consistent performance of MLP-pw. MLP limiter using K = 0.1 is fail to converge in this case. MLP-u2(new) limiter is converged but requires extra time steps. Venkatakrishnan limiter also shows consistent converged results, only one of which is shown in the figure for clarity, but it is more dissipative, as shown in Fig.13(h). Furthermore, the convergence of MLP-pw
420
limiter does not come with the cost of extra numerical dissipation. Of course, since the flow field is simple and clear, the extra dissipation does not affect the results as significant as in the shock-density interaction problem, but this case still gives notable characteristics of each limiter. 5.5. Multiple discontinuities: a Mach 3 wind tunnel with a step
425
Unsteady simulations are performed in this subsection. Uniform inviscid flow of which the Mach number is set as 3 passes a step will cause evolutive shock waves, Mach stem and contact discontinuity [55]. Flow field is discretized by 78246 triangular
28
(a) MLP (K = 10)
(b) MLP-pw (K = 10)
(c) Venkatakrishnan (K = 10)
(d) MLP-u2(new)
Figure 12: Density contours of double shock reflection problem. Forty equally spaced contour lines from
ρ = 1.0 to ρ = 2.8.
(a) MLP-pw (K = 0.1)
(b) MLP (K = 0.1)
(c) MLP-pw (K = 1)
(d) MLP (K = 1)
(e) MLP-pw (K = 10)
(f) MLP (K = 10)
(g) MLP-u2(new)
(h) Venkatakrishnan (K = 1)
Figure 13: Density limiting value contours of double shock reflection problem. Thirty equally spaced contour lines from ϕ = 0 (blue) to ϕ = 1 (red).
29
2
0
MLP-u2(new) MLP (K=0.1) MLP (K=1) MLP (K=10) MLP-pw(K=0.1) MLP-pw(K=1) MLP-pw(K=10) Venkatakrishnan (K=1)
log10(L∞ Res)
-2
-4
-6
-8
-10
-12 0
2000
4000
6000
Time steps
Figure 14: Density residuals of the computations.
cells. Four-step Runge-Kutts scheme with CFL = 1.5 is used for temporal solutions. Since there is a vertical forward step in the flow field, stagnation zone will be formed 430
ahead of the step, and thus the numerical stability in the vicinity of the bow shock should be enhanced. Therefore, AUSMPW scheme [44] is used for the computations of numerical fluxes. The parameter K of Venkatakrishnan function fV is set as 10. The results at non-dimensional time t = 4 are presented in Fig.15, and the entropy increment contours are shown in Fig.16. The slip line captured by Venkatakrishnan
435
limiter is significantly smeared due to its higher dissipation, and a thick entropy layer can also be found on the step. MLP-type limiters show similar flow fields in this case. Thereinto, MLP-pw limiter captures a clear slip line formed at the reflection point on the step, and the captured slip line of the triple point also develops to vortex shape structure. MLP-u2(new) limiter presents a thinner entropy layer on the step, but this is
440
not a concluded evidence of advantage or disadvantage. The density limiting value contours are shown in Fig.17. Again, Venkatakrishnan limiter introduces extra dissipation in smooth flow field. MLP-type limiters are, mostly, restricting the limiting regions near shock waves. MLP-pw limiter has presented a clean limiting effect, which means only shock waves activate the limiter. Whereas,
445
MLP and MLP-u2(new) show extra limitation near slip line, which is unnecessary, at least for stability consideration.
30
(a) MLP
(b) MLP-pw
(c) Venkatakrishnan
(d) MLP-u2(new)
Figure 15: Density contours of the simulations of a Mach 3 wind tunnel with a step. Forty equally spaced contour lines from ρ = 0.5 to ρ = 4.0.
(a) MLP
(b) MLP-pw
(c) Venkatakrishnan
(d) MLP-u2(new)
Figure 16: Entropy increment contours of the simulations of a Mach 3 wind tunnel with a step. Sixty equally spaced contour lines from s = 0.05 to s = 2.05.
(a) MLP
(b) MLP-pw
(c) Venkatakrishnan
(d) MLP-u2(new)
Figure 17: Density limiting value contours of the simulations of a Mach 3 wind tunnel with a step. Thirty equally spaced contour lines from ϕ = 0 (blue) to ϕ = 1 (red).
31
5.6. Hypersonic inviscid flows around a cylinder 5.6.1. Basic considerations MLP-pw limiter is especially improved for the simulations of strong shock waves. 450
Therefore, the simulation of hypersonic inviscid flow around a cylinder is used here to examine the performance. It is particulary important that the presented MLP-pw limiter is capable to capture the shock wave using a relaxed limiting parameter. In this case, Mach number of uniform inflow is 8, and all the simulations use forward Euler scheme for the temporal solution, with using CFL = 0.8.
455
It should be noticed that van Leer scheme [56] is applied to calculate the numerical flux at cell interfaces. It is well known that some of the Riemann solvers might produce numerical shock instabilities, e.g. carbuncle phenomenon [57], in certain situations, and thus the numerical simulations can be fail or inaccurate. Various improvements of Riemann solver can be successfully applied for first-order simulations [45, 10]. How-
460
ever, for second-order spatial accuracy simulations, even well known robust Riemann solvers can be showing oscillations in flow field [58]. Therefore, the instability or oscillation in second-order hypersonic simulations can not be simply removed by improving the Riemann solver. On the other hand, first-order shock-wave simulations can be stable by using a ro-
465
bust upwind flux scheme or applying improvements or amendments on unstable numerical upwind fluxes, and thus, for a robust upwind flux scheme, e.g. van Leer scheme, numerical oscillation in second-order computation can be avoided by properly reducing accuracy in part of the flow field. Using an improved limiter can probably serve this purpose and does not globally reduce the accuracy to first-order. In other words, using
470
van Leer scheme will be helpful to investigate the performance of limiters without the interfering of an instability-prone numerical upwind fluxes. 5.6.2. Grids Three types of grid are used for the simulations. Grid 1 is quadrilateral grid, which has 120(circumferential) × 80(radial) cells. Grid 2 is an irregular triangular grid in
475
which 12156 cells are produced by Delaunay triangulation. Grid 3 is a regular triangular grid that produced by bisecting the quadrangles in Grid 1, and thus there are 19200 32
(a) Grid 1
(b) Grid 2
(c) Grid 3
Figure 18: Grids for the simulations of hypersonic flows around a cylinder. (Boundary nodes are halved in the figures for clarity)
triangle cells in this computational domain. 5.6.3. Computations on the quadrilateral grid The results calculated on quadrilateral grid are first investigated. Here, the conver480
gent histories are shown in Fig.19 at first. In this case, only MLP-pw limiter is showing converged results with using both two choices of K, i.e. K = 1 and K = 10. The other limiters are unable to converge in this case. It should be noticed again that the robust van Leer scheme is applied for calculating numerical fluxes, and as aforementioned, Kitamura, et al. [58] had also presented unstable results using robust upwind flux
485
schemes and second-order spatial discretization. In this subsection the effects of using different limiters will be introduced in detail. The numerical flow field contours are shown in Fig.20. MLP-pw limiter is able to show clear and stable bow shock with using different K. In fact, as shown in the isentropic vortex advection problem, MLP-pw limiter is relatively insensitive to the
490
parameter K. However, the other three limiters are unable to converge, and MLP and MLP-u2(new) also show oscillating flow field. The result of MLP limiter (K = 1), which is more dissipative, still has significant oscillation in the stagnation zone. Al-
33
2
0
MLP (K=1) MLP (K=10) MLP-pw (K=1) MLP-pw (K=10) Venkatakrishnan (K=1) Venkatakrishnan (K=10) MLP-u2(new)
log10(L∞ Res)
-2
-4
-6
-8
-10
-12
0
20000
40000
60000
Time steps
Figure 19: Density residuals of the computations on grid 1.
though Venkatakrishnan limiter shows oscillation-free flow field, it is fail to converge and more dissipative. 495
Then the density limiting value contours are shown in Fig.21. It could be found that only MLP-pw limiter has restricted the limitation in shock wave, even using a smaller K. Whereas, the other three limiters are showing limitation on smooth region. In fact, as already proved by the flow field contour, MLP-pw limiter can be stable and converged by using the relaxed limitation, i.e. K = 10. Therefore, more details will be
500
introduced in Fig.22. The density limiting value distribution along the central line and y = 0.2 cutting line of the flow field are shown. In Fig.22(a), MLP-pw limiter has only limited the region near the bow shock, which locates at x ≈ −1.4, and the other limiters show extra unnecessary limitation in smooth region. Along the central line, it is not clear that how
505
MLP-pw provides stricter limitation for capturing shock wave, but the Fig.22(b) gives more information. In Fig.22(b), it can be found that MLP-pw limiter gives the strictest limitation at the pressure jump, i.e. shock wave, even using K = 10. Although MLPu2(new) limiter also shows strict limitation, the limited position is somehow deviates from the centre of the numerical pressure discontinuity. Moreover, MLP limiter gives a
510
less effective limitation at the discontinuity, even using K = 1. Therefore, it seems that although MLP-pw has removed limitation from the most of the part of the flow field,
34
(a) MLP (K = 1) (b) MLP-u2(new) (c) MLP-pw (K = (d) Venkatakrish-
10)
nan (K = 1)
Figure 20: Inviscid hypersonic flow around a cylinder at M = 8 on quadrilateral grid. Thirty equally spaced contour lines from ρ = 1.2 (blue) to ρ = 5.8 (red).
(a) MLP (K = 1) (b) MLP-u2(new) (c) MLP-pw (K = (d) Venkatakrish-
1)
nan (K = 1)
Figure 21: Density limit value contour of each limiter. Thirty equally spaced contour lines from ϕ = 0 (blue) to ϕ = 1 (red).
35
1.2
80
1
70 1
0.8
60 0.6
φρ
φρ
0.4
40
0.6
0.2
0
MLP (K=1) MLP-pw (K=1) Venkatakrishnan (K=1) MLP-u2(new)
-0.2 -1.8
-1.6
-1.4
30
MLP (K=1) MLP-u2(new) MLP-pw (K=10) p
0.4
-1.8
-1.2
Pressure
50 0.8
-1.6
20
10
-1.4
-1.2
-1
0
x
x
(a) y=0
(b) y=0.2 (along with pressure distribution) Figure 22: Density limit values.
the necessary limitation is further enhanced. In general, MLP-pw limiter has shown improved stability and reduced unnecessary numerical dissipation in this case. 5.6.4. Computations on the irregular triangular grid 515
As in the last subsection, the convergent histories are shown at first. Again, MLPpw limiter shows good convergence with using both two choices of K. Whereas, it should be noted that the convergence of MLP limiter and Venkatakrishnan limiter is highly depending on the parameter K. With using K = 10, MLP limiter shows convergence that is as good as that of MLP-pw limiter, but it is showing deteriorated conver-
520
gence with using K = 1. Venkatakrishnan limiter has to use a strict parameter, K = 1, to be converged, but the convergence is still unsatisfactory. In this case, MLP-u2(new) shows converged result. Density contours are shown in Fig.24. Due to the irregularity of the grid, it is impossible to attain a smooth flow contour as in the last subsection. However, three
525
MLP-type limiters show slightly advantage in keeping symmetrical flow field, comparing with Venkatakrishnan limiter. Specifically, MLP-pw limiter is keeping similar results with using different K, but MLP limiter shows more unsymmetrical result with using K = 1. Density limiting value contours are shown in Fig.25. Venkatakrishnan limiter
36
MLP (K=1) MLP (K=10) MLP-pw (K=1) MLP-pw (K=10) Venkatakrishnan (K=1) Venkatakrishnan (K=10) MLP-u2(new)
2
0
log10(L∞ Res)
-2
-4
-6
-8
-10
-12
0
20000
40000
60000
80000
100000
120000
Time steps
Figure 23: Density residuals of the computations on grid 2.
(a) MLP (K = 10) (b) MLP-u2(new) (c) MLP-pw (K = (d) Venkatakrish-
10)
nan (K = 1)
Figure 24: Inviscid hypersonic flow around a cylinder at M = 8 on irregular triangular grid. Twenty equally spaced contour lines from ρ = 1.2 (blue) to ρ = 5.8 (red).
37
(a) MLP (K = 10) (b) MLP-u2(new) (c) MLP-pw (K = (d) Venkatakrish-
10)
nan (K = 10)
Figure 25: Density limit value contour of each limiter. Thirty equally spaced contour lines from ϕ = 0 (blue) to ϕ = 1 (red).
530
shows extra limitation on smooth regions, which is not helpful for improving stability and detrimental for computation accuracy. MLP-u2(new) limiter has successfully restricted the limitation along the shock wave, but the limiting effect is stronger than the other two MLP-type limiters according to the contour. MLP limiter and MLP-pw limiter show similar results in the figures, and the convergence and the flow contour are
535
also similar, with using K = 10. In general, MLP-pw limiter constantly presents stable and converged results in this case, in which irregular triangular grid is applied, and the effect of decreasing K is only increasing dissipation, which is still relatively lower than other limiters, without damaging the stability or convergence. 5.6.5. Computations on the regular triangular grid
540
Again, the convergent histories are shown at first, and MLP-pw limiter is showing consistent results in convergence. MLP limiter can not be converged with using both two K. It should be noted that Venkatakrishnan limiter attained the best convergence if K = 10. However, a contrary result is produced if K = 1, which means Venkatakrishnan limiter is very sensitive to the parameter. Moreover, Venkatakrishnan limiter is very
545
dissipative if K is reduced. MLP-u2(new) is also showing good convergence in this
38
2
0
MLP (K=1) MLP (K=10) MLP-pw (K=1) MLP-pw (K=10) Venkatakrishnan (K=1) Venkatakrishnan (K=10) MLP-u2(new)
log10(L∞ Res)
-2
-4
-6
-8
-10
-12
0
20000
40000
60000
80000
100000
120000
Time steps
Figure 26: Density residuals of the computations on grid 3.
case. The density contours are shown in Fig.27. The three MLP-type limiters are showing similar results. In fact, although the convergence is different, the numerical flow fields are generally similar in this case. All the results are showing minor unsymmetrical 550
distribution, and no oscillation has been found. Fig.28 presents the density limiting value contours. Venkatakrishnan limiter is more dissipative as expected. MLP-type limiters show similar limiting effect, but the limitation of MLP-u2(new) is stricter along the shock wave. In order to look into the details of the limiters, the density residual contours of
555
three un-converged results are shown in Fig.29. It is clear that residuals are produced locally in the flow field, and thus the computations are unable to converge. Therefore, it is necessary that MLP-pw limiter has improved the limitation near shock waves, properly. 5.7. General remarks
560
For limiters using the Venkatakrishnan function, there is a parameter K to be defined, which affects the numerical dissipation eventually. Therefore, some freedom or uncertainty are introduced before the computations. MLP-u2(new) limiter removes the necessity of tuning this parameter by using the function in Eq.24, and thus the function
39
(a) MLP (K = 1) (b) MLP-u2(new) (c) MLP-pw (K = (d) Venkatakrish-
10)
nan (K = 10)
Figure 27: Inviscid hypersonic flow around a cylinder at M = 8 on regular triangular grid. Twenty equally spaced contour lines from ρ = 1.2 (blue) to ρ = 5.8 (red).
(a) MLP (K = 10) (b) MLP-u2(new) (c) MLP-pw (K = (d) Venkatakrish-
10)
nan (K = 10)
Figure 28: Density limit value contour of each limiter. Thirty equally spaced contour lines from ϕ = 0 (blue) to ϕ = 1 (red).
40
(a) MLP (K = 1) (b) MLP (K = 10) (c) Venkatakrish-
nan (K = 1) Figure 29: Density residual contours of un-convergence results. Fifteen equally spaced contour lines from log10 (Resρ ) = −3 (blue) to log10 (Resρ ) = 0.5 (red).
can also tune the numerical dissipation adaptively. Whereas, since the original MLP 565
condition is applied in MLP-u2(new) limiter, the problem introduced in Fig.4 is still affecting the performance of MLP-u2(new). On the other hand, by applying weak/strict-MLP conditions, MLP-pw limiter has improved the stability and accuracy, with using the original Venkatakrishnan function. Especially, strict-MLP condition treats the potentially crucial areas which might cause
570
instability or oscillation, and weak-MLP condition reduces dissipation but still guarantees the maximum/minimum principle. Therefore, although MLP-pw limiter uses the tunable parameter to change its numerical dissipation, the accuracy and the convergence are usually satisfactory in the numerical cases. For example, in the isentropic vortex advection case, MLP-pw limiter shows high accuracy, using a small K. Even
575
using a relaxed K, MLP-pw is able to provide stable and converged results, at least in the given numerical cases.
6. Conclusions As a modification of the multi-dimensional limiting process (MLP) on unstructured grids which significantly improves the convergence and accuracy of the simulations on 41
580
diverse problems, the presented method is a combination of two novel modified limiting conditions, weak/strict-MLP condition. Maximum/minimum principles are satisfied by both two new conditions, and thus spurious oscillations will be well controlled. Especially, the strict-MLP condition strictly limits the reconstructed variables, and thus the monotonicity could be guaranteed. By using a pressure weight function that detects
585
shock waves, the strict-MLP condition is activated in the vicinity of shock waves and the weak-MLP condition is activated otherwise. Therefore, spurious oscillations are eliminated near shock waves, even in hypersonic simulations, and the numerical dissipation are reduced in continuous regions and near contact discontinuity. Furthermore, the convergence of the presented limiter, MLP-pw, is improved.
590
Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant 91541117, and 11602052; the Research Foundation of State Key Laboratory of Aerodynamics under Grant SKLA20160106. Xiao Liu at Dalhousie University gave advices to improve the writing of this article, who is sincerely appreciated.
595
References [1] A. Bowyer, Computing dirichlet tessellations, The Computer Journal 24 (2) (1981) 162–166. doi:10.1093/comjnl/24.2.162. [2] D. F. Watson, Computing the n-dimensional delaunay tessellation with application to voronoi polytopes, The Computer Journal 24 (2) (1981) 167–172.
600
doi:10.1093/comjnl/24.2.167. [3] L. Zhang, Z. Zhao, X. Chang, X. He, A 3D hybrid grid generation technique and a multigrid/parallel algorithm based on anisotropic agglomeration approach, Chinese Journal of Aeronautics 26 (1) (2013) 47–62. [4] D. Drikakis, D. Ofengeim, E. Timofeev, P. Voionovich, Computation of non-
605
stationary shock-wave/cylinder interaction using adaptive-grid methods, Journal of Fluids and Structures 11 (6) (1997) 665–692. 42
[5] F. Alauzet, P. Frey, P. George, B. Mohammadi, 3D transient fixed point mesh adaptation for time-dependent problems: Application to cfd simulations, Journal of Computational Physics 222 (2) (2007) 592–623. 610
[6] M. J. Harris, N. Qin, Geometric representation of flow features using the medial axis for mesh generation, AIAA Journal (2014) 1–14doi:10.2514/1.J053088. [7] E. Luke, E. Collins, E. Blades, A fast mesh deformation method using explicit interpolation, Journal of Computational Physics 231 (2) (2012) 586–601. [8] S.-J. Liu, Z.-Y. Lin, W.-D. Liu, W. Lin, F.-C. Zhuang, Experimental re-
615
alization of h2/air continuous rotating detonation in a cylindrical combustor, Combustion Science and Technology 184 (9) (2012) 1302–1317. doi:10.1080/00102202.2012.682669. [9] A. Mack, V. Hannemann, Validation of the unstructured DLR-TAU-code for hypersonic flows, in: 32nd AIAA Fluid Dynamics Conference and Exhibit, AIAA
620
Paper 2002-3111, St. Louis, MO, 2002. [10] F. Zhang, J. Liu, B. Chen, W. Zhong, A robust low-dissipation AUSM-family scheme for numerical shock stability on unstructured grids, Int. J. Numer. Meth. Fluids. [11] D. Anderson, A comparison of numerical solutions to the inviscid equations of
625
fluid motion, Journal of Computational Physics 15 (1) (1974) 1–20. [12] R. M. Beam, R. Warming, An implicit finite-difference algorithm for hyperbolic systems in conservation-law form, Journal of Computational Physics 22 (1) (1976) 87–110. [13] B. van Leer, Towards the ultimate conservative difference scheme. V. A second-
630
order sequel to Godunov’s method, Journal of Computational Physics 32 (1) (1979) 101–136. [14] A. Harten, High resolution schemes for hyperbolic conservation laws, Journal of Computational Physics 49 (3) (1983) 357–393. 43
[15] A. Harten, On a class of high resolution total-variation-stable finite-difference 635
schemes, SIAM J. Numer. Anal. 21 (1) (1984) 1–23. doi:10.1137/0721001. [16] P. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21 (5) (1984) 995–1011. doi:10.1137/0721062. [17] A. Jameson, Analysis and design of numerical schemes for gas dynamics, 1: Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multi-
640
grid convergence, International Journal of Computational Fluid Dynamics 4 (3-4) (1995) 171–218. doi:10.1080/10618569508904524. [18] C.-W. Shu, TVB uniformly high-order schemes for conservation laws, Mathematics of Computation 49 (179) (1987) 105–121. [19] A. Harten, B. Engquist, S. Osher, S. R. Chakravarthy, Uniformly high order ac-
645
curate essentially non-oscillatory schemes, iii, Journal of Computational Physics 71 (2) (1987) 231–303. [20] S. Spekreijse, Multigrid solution of monotone second order discretizations of hypersonic conservations laws, Mathematics of Computation 49 (179) (1987) 135– 155.
650
[21] T. J. Barth, D. C. Jespersen, The design and application of upwind schemes on unstructured meshes, in: 27th Aerospace Sciences Meetings, AIAA Paper 890366, Reno, NV, 1989. doi:10.2514/6.1989-366. [22] V. Venkatakrishnan, On the accuracy of limiters and convergence to steady state solutions, in: 31st Aerospace Sciences Meeting, AlAA Paper 93-0880, Reno,NV,
655
1993. doi:10.2514/6.1993-880. [23] G. D. van Albada, B. van Leer, J. Roberts, W. W., A comparative study of computational methods in cosmic gas dynamics, Astronomy and Astrophysics 108 (1) (1982) 76–84. [24] A. Nejat, C. Ollivier-Gooch, A high-order accurate unstructured finite volume
660
Newton-Krylov algorithm for inviscid compressible flows, Journal of Computational Physics 227 (4) (2008) 2582–2609. 44
[25] C. Michalak, C. Ollivier-Gooch, Accuracy preserving limiter for the high-order accurate solution of the Euler equations, Journal of Computational Physics 228 (23) (2009) 8693–8711. 665
[26] K. Kitamura, E. Shima, Simple and parameter-free second slope limiter for unstructured grid aerodynamic simulations, AIAA Journal 50 (6) (2012) 1415– 1426. doi:10.2514/1.J051269. [27] K. H. Kim, C. Kim, Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows: Part II: Multi-dimensional limiting pro-
670
cess, Journal of Computational Physics 208 (2) (2005) 570–615. [28] S.-H. Yoon, C. Kim, K.-H. Kim, Multi-dimensional limiting process for threedimensional flow physics analyses, Journal of Computational Physics 227 (12) (2008) 6001–6043. [29] J. S. Park, S.-H. Yoon, C. Kim, Multi-dimensional limiting process for hyper-
675
bolic conservation laws on unstructured grids, Journal of Computational Physics 229 (3) (2010) 788–812. [30] J. S. Park, C. Kim, Multi-dimensional limiting process for finite volume methods on unstructured grids, Computers & Fluids 65 (2012) 8–24. [31] P. Gerlinger, Multi-dimensional limiting for high-order schemes including turbu-
680
lence and combustion, Journal of Computational Physics 231 (5) (2012) 2199– 2228. [32] S. Do, Y. Ha, M. Kang, C. H. Kim, Application of a multi-dimensional limiting process to central-upwind schemes for solving hyperbolic systems of conservation laws, Journal of Scientific Computing 69 (1) (2016) 274–291.
685
[33] H.-M. Kang, K. H. Kim, D.-H. Lee, A new approach of a limiting process for multi-dimensional flows, Journal of Computational Physics 229 (19) (2010) 7102–7128.
45
[34] J. S. Park, C. Kim, Higher-order multi-dimensional limiting strategy for discontinuous Galerkin methods in compressible inviscid and viscous flows, Computers 690
& Fluids 96 (2014) 377–396. [35] J. S. Park, C. Kim, Hierarchical multi-dimensional limiting strategy for correction procedure via reconstruction, Journal of Computational Physics 308 (2016) 57– 80. [36] J. S. Park,
695
H. You,
C. Kim,
Higher-order multi-dimensional limit-
ing process for dg and fr/cpr methods on tetrahedral meshes, Computers & Fluids 154 (Supplement C) (2017) 322 – 334,
iCCFD8.
doi:https://doi.org/10.1016/j.compfluid.2017.03.006. [37] H. You, C. Kim, Higher-order multi-dimensional limiting strategy for subcell resolution, in: 23rd AIAA Computational Fluid Dynamics Conference, AIAA Paper 700
2017-4110, Denver, CO, 2017. [38] W. Li, Y.-X. Ren, G. Lei, H. Luo, The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids, Journal of Computational Physics 230 (21) (2011) 7775–7795. [39] W. Li, Y.-X. Ren, The multi-dimensional limiters for solving hyperbolic con-
705
servation laws on unstructured grids ii: Extension to high order finite volume schemes, Journal of Computational Physics 231 (11) (2012) 4053–4077. [40] J. Blazek, Computational Fluid Dynamics: Principles and Applications, 2nd Edition, Elsevier, 2005. [41] S. Godunov, A difference method for numerical calculation of discontinuous so-
710
lutions of the equations of hydrodynamics, Mat. Sb. 47(89) (3) (1959) 271–306. [42] N. T. Frink, P. Parikh, S. Pirzadeh, A fast upwind solver for the Euler equations on three-dimensional unstructured meshes, in: 29th Aerospace Sciences Meetings, AIAA Paper 91-0102, Reno, NV, 1991. doi:10.2514/6.1991-102.
46
[43] E. Shima, K. Kitamura, T. Haga, Green-Gauss/weighted-least-squares hybrid gra715
dient reconstruction for arbitrary polyhedra unstructured grids, AIAA Journal 51 (11) (2013) 2740–2747. doi:10.2514/1.J052095. [44] K. H. Kim, J. H. Lee, O. H. Rho, An improvement of AUSM schemes by introducing the pressure-based weight functions, Computers & Fluids 27 (3) (1998) 311–346.
720
[45] F. Zhang, J. Liu, B. Chen, W. Zhong, Evaluation of rotated upwind schemes for contact discontinuity and strong shock, Computers & Fluids 134-135 (2016) 11– 22. [46] Y. X. Ren, A robust shock-capturing scheme based on rotated Riemann solvers, Computers & Fluids 32 (10) (2003) 1379–1403.
725
[47] H. Nishikawa, K. Kitamura, Very simple, carbuncle-free, boundary-layerresolving, rotated-hybrid Riemann solvers, Journal of Computational Physics 227 (4) (2008) 2560–2581. [48] D. Wang, X. Deng, G. Wang, Y. Dong, Developing a hybrid flux function suitable for hypersonic flow simulation with high-order methods, Int. J. Numer. Meth.
730
Fluids 81 (5) (2016) 309–327. [49] K.
Kitamura,
A.
Hashimoto,
Simple
a
posteriori
slope
lim-
iter (post limiter) for high resolution and efficient flow computations,
Journal
of
Computational
Physics
341
(2017)
313
–
340.
doi:https://doi.org/10.1016/j.jcp.2017.04.002. 735
[50] A. JAMESON, W. SCHMIDT, E. TURKEL, Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes, in: 14th Fluid and Plasma Dynamics Conference, AIAA Paper 1981-1259, Palo Alto, CA, 1981. doi:10.2514/6.1981-1259. [51] E. Toro, M. Spruce, W. Speares, Restoration of the contact surface in the HLL-
740
Riemann solver, Shock Waves 4 (1) (1994) 25–34.
47
[52] G. A. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, Journal of Computational Physics 27 (1) (1978) 1–31. [53] C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory 745
shock-capturing schemes,ii,
J. Comput. Phys. 83 (1) (1989) 32–78.
doi:10.1016/0021-9991(89)90222-2. [54] S. Yoon, A. Jameson, Lower-upper symmetric-gauss-seidel method for the euler and navier-stokes equations, AIAA Journal 26 (9) (1988) 1025–1026. doi:10.2514/3.10007. 750
[55] P. Woodward, P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, Journal of Computational Physics 54 (1) (1984) 115–173. [56] B. van Leer, Flux-vector splitting for the Euler equations, in: E. Krause (Ed.), Eighth International Conference on Numerical Methods in Fluid Dynamics, Vol. 170 of Lecture Notes in Physics, Springer Berlin Heidelberg, Aachen, Germany,
755
1982, pp. 507–512. [57] K. Peery,
S. Imlay,
Blunt-body flow simulations,
in:
24th Join-
t Propulsion Conference, AIAA Paper 88-2904, Bellevue, WA, 1988. doi:10.2514/6.1988-2904. [58] K. Kitamura, P. Roe, F. Ismail, Evaluation of Euler fluxes for hypersonic flow 760
computations, AIAA Journal 47 (1) (2009) 44–53. doi:10.2514/1.33735.
48
