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A modified multidimensional dissipation technique for +
AUSM on triangular grids S. Phongthanapanich
a
a
Department of Mechanical Engineering Technology, College of Industrial Technology, King Mongkut's University of Technology North Bangkok, Bangkok, Thailand Published online: 10 Feb 2015.
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To cite this article: S. Phongthanapanich (2015) A modified multidimensional dissipation technique for AUSM on triangular grids, International Journal of Computational Fluid Dynamics, 29:1, 1-11, DOI: 10.1080/10618562.2015.1010525 To link to this article: http://dx.doi.org/10.1080/10618562.2015.1010525
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International Journal of Computational Fluid Dynamics, 2015 Vol. 29, No. 1, 1–11, http://dx.doi.org/10.1080/10618562.2015.1010525
A modified multidimensional dissipation technique for AUSM + on triangular grids S. Phongthanapanich∗ Department of Mechanical Engineering Technology, College of Industrial Technology, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand
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(Received 29 September 2014; accepted 18 January 2015) The carbuncle phenomenon normally occurs numerically in the prediction of shock waves in flow computation. Most efforts to remedy this problem concern numerical treatment of the bow shock wave while many evidences declare that the carbuncle phenomenon problem may be unsolvable. This paper studies the numerical instability of the AUSM + scheme on twodimensional structured triangular grids. By examining several test cases, it is found that the scheme cannot satisfy robustness against shock-induced anomalies. A more stable version of the AUSM + scheme (so-called AUSM + δ scheme) is developed by applying the multidimensional dissipation technique to the numerical dissipation term in order to alleviate the shock instability. The dissipation mechanism against perturbations is investigated by applying a linearised discrete analysis to the odd–even decoupling problem. The recursive equations show that the AUSM + δ scheme is less sensitive to such anomalies than the original scheme. Finally, the scheme is further extended to achieve the second-order solution accuracy and evaluated by solving several test cases. Keywords: AUSM + scheme; carbuncle phenomenon; finite volume method; numerical instability; shock-capturing method
1. Introduction Computational fluid dynamics (CFD) has played an important role in many engineering analysis and design process due to its capability for analysing complex flow problems. The information provided by the CFD has been used to analyse the influence of flows in manufacturing equipment and/or machine (Delgadillo and Rajamani 2009; Kirk et al. 2009; Costes, Renaud, and Rodriguez 2012). CFD offers the possibility to model many complex situations and emphasises the necessity to understand the physics and mathematics of the underlying problems. The finite volume method is commonly used in many commercial CFD software as a standard tool for discretising the Euler and/or Navier– Stokes equations into a finite number of non-overlapping control volumes. The upwind schemes are proved to be stable and efficient for the solution of compressible inviscid flow according to their capability of capturing shock and contact discontinuities in variety of problems. Two of the most well-known methods used primarily in inviscid flux discretisation are the flux-vector splitting (FVS) and the flux-difference splitting (FDS) schemes. The FVS schemes (Steger 1981; VanLeer 1982; Hanel, Schwane, and Seider 1987) are known to be a fast, simple and robust for capturing of strong shock waves and rarefaction waves especially in hypersonic flows. However, many numerical simulations indicated that these type of schemes are too dissipative and may deteriorate the boundary layer profiles.
∗
Email:
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The FDS schemes (Roe 1981; Toro, Spruce, and Speares 1994), which employ Riemann solvers, are generally very robust and no explicit dissipation is required. They combine good shock-capturing mechanisms with the capability of correctly dealing with shear layers by introducing a small amount of an artificial dissipation into the solution. However, in certain cases, some existing schemes may produce spurious pressure glitches at sonic points (Moschetta and Gressier 2000), the shedding of spurious waves by slowly moving shocks (Roberts 1990), and the carbuncle phenomenon (Quirk 1994). The FDS scheme proposed by Roe (1981) is widely used due to its accuracy, quality and mathematical clarity. But the Roe’s scheme requires longer computational time as compared to FVS scheme due to its matrix calculations basis. The Roe’s scheme was found to produce unphysical solutions for the compressible Euler equations in certain problems. For example, the expansion shock that violates the entropy condition and shock anomalies associated with the carbuncle phenomenon on both quadrilateral or triangular grids (Quirk 1994; Phongthanapanich and Dechaumphai 2004). In addition, the scheme is not a positivity conservative scheme and can lead to negative densities or internal energy at high Mach numbers computations. The shock wave simulations in multidimensional compressible flows by the shock-capturing finite-volume methods usually exhibit the numerical instability such as
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carbuncle phenomenon (Quirk 1994; Robinet et al. 2000; Dumbser, Moschetta, and Gressier 2004) and it is most prevalent for axisymmetric blunt body flow simulation (MacCormack 2011). There are many explanations about this phenomenon, Pandolfi and D’Ambrosio (2001) concluded that the shock-capturing numerical schemes provide insufficient dissipation in the shock region, particularly in the direction parallel to the shock wave and may result in the formation of the carbuncle phenomenon. Robinet et al. (2000) suggested that the origin of the carbuncle phenomenon could lay in the physical instability of the surface of discontinuity itself. It is really difficult or nearly impossible to predict which numerical schemes are likely to be affected and/or in which circumstances carbuncles could appear. To overcome these problems, many researchers (Sanders, Morano, and Druguet 1998; Pandolfi and D’Ambrosio 2001; Phongthanapanich and Dechaumphai 2004, 2009) proposed entropy fix formulations to replace the near zero eigenvalues by some tolerances. The less-dissipative upwind schemes so-called the advection upstream splitting method (AUSM) was proposed by Liou and Steffen (1993) as a fast, simple, robust and accurate method. The idea is to combine the efficiency of the FVS and the accuracy of the FDS schemes together to give vanishing numerical diffusivity at the stagnation points. The AUSM-family schemes split the flux into convection and pressure parts that are treated separately. Many researchers have shown that the accuracy and efficiency of the AUSM scheme is comparable to the Roe’s scheme. However, the AUSM scheme produces, in certain cases, spurious pressure glitches at sonic points (sonic point glitch) and leads to unphysical expansion shocks when the flow undergoes rapid change of direction (Moschetta and Gressier 2000). Several attempts have been made in the following years to improve the original AUSM scheme (Liou 1996; Wada and Liou 1997; Kim, Lee, and Rho 1998; Kim, Kim, and Rho 2001; Liou 2006; Kitamura and Shima 2013). At present, the AUSM-family schemes are often used for hypersonic flow computations due to their robustness and simplicity. The AUSM + scheme proposed by Liou (1996) features the following properties: (1) exact resolution of 1D contact and shock discontinuities, (2) positivity preserving of scalar quantity, (3) free of carbuncle phenomenon, (4) free of oscillation at the slowly moving shock and (5) simplicity and easy extension to treat other hyperbolic systems. Kitamura, Roe, and Ismail (2009) presented a survey of numerical experiments from 12 different flux functions in one- and two-dimensional contexts including AUSMfamily schemes. They concluded that no universally stable scheme is free from one- and multidimensional shock instabilities. Recently, Ramalho, Azevedo, and Azevedo (2011) have shown that the AUSM + scheme produces the carbuncle phenomenon on triangular grid for Mach number 12.2 flow over a cylinder. This paper will confirm that
the scheme, in certain problems, is not free of the carbuncle phenomenon and the post-shock oscillation on twodimensional structured triangular grids. The main objective of this paper is to study the numerical instability known as the carbuncle phenomenon of the AUSM + scheme on two-dimensional structured triangular grids. A stable version of the AUSM + scheme (AUSM + δ scheme) is proposed in order to alleviate the numerical instability of the scheme and then compare the solutions with those obtained from the original scheme. The paper is organised as follows. A brief description on the governing equations and the numerical flux formulation of the AUSM + scheme is presented in Section 2. Some wellknown problems that exhibit the numerical shock instability and/or post-shock oscillations are examined in Section 3. In Section 4, the linearised analysis is applied to the odd–even decoupling problem with presence of perturbations that may cause the carbuncle phenomenon. The recursive equations are derived in order to study the damping mechanism of the scheme. A more stable AUSM + δ scheme is then proposed by applying the multidimensional dissipation technique to the numerical dissipation term, and its robustness and efficiency are examined by solving certain problems. Finally, the proposed scheme is further extended to achieve secondorder solution accuracy and then evaluated by several test cases in Section 5. 2. AUSM + scheme The two-dimensional compressible Euler equations can be expressed in conservation form as ∂Ey ∂Q ∂Ex + + = 0. ∂t ∂x ∂y
(1)
The vector of conserved quantities and the convective flux vectors are given by ⎛
⎛ ⎞ ⎞ ρu ρ ⎜ ρu ⎟ ⎜ 2 ⎟ ⎟; Ex = ⎜ ρu + p ⎟; Q=⎜ ⎝ ρv ⎠ ⎝ ρuv ⎠ ρE ρuH ⎞ ⎛ ρv ⎜ ρvu ⎟ ⎟ Ey = ⎜ ⎝ ρv 2 + p ⎠, ρvH
(2)
where ρ, u, v, p, E and H denote density, x-velocity, yvelocity, pressure, total energy and total enthalpy, respectively. However, the compact vector form of Equation (1) can be written as Qt + ∇ · F = 0,
(3)
International Journal of Computational Fluid Dynamics →
→
where F = Ex i + Ey j . And for a calorically perfect gas, the equation of state is given by p = ρ(γ − 1)[E − 0.5(u2 + v 2 )]
(4)
with γ = 1.4 for air. For all AUSM-family schemes, the inviscid flux is explicitly split into convective and pressure fluxes (Liou 1996) as F = F(c) + F(p) .
(5)
The numerical flux at the cell interface of the AUSM + scheme is calculated as 1 a1/2 [m1/2 (L + R ) 2 − |m1/2 |(R − L )] + P1/2 , p
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c + f1/2 = f1/2 = f1/2
(6)
where ⎛
L/R
⎞ ρ ⎜ ρu ⎟ ⎟ =⎜ ⎝ ρv ⎠ ; ρH L/R
⎛
0
⎞
⎜ p1/2 nx ⎟ ⎟ P1/2 = ⎜ ⎝ p1/2 ny ⎠ 0
(7)
with the common mass flux and the split Mach numbers defined by + − m1/2 = M(4,β) (ML ) + M(4,β) (MR ) (8) ⎧ 1 ⎪ ⎨± (M ± 1)2 ± β(M 2 − 1)2 |M| ≤ 1 ± 4 , M(4,β) (M) = ⎪ ⎩ 1 (M ± |M|) otherwise 2 (9)
where a common speed of sound (Liou 2006; Kitamura and Shima 2013) is defined by a1/2 = min(a˜ L , a˜ R )
(10)
with a˜ L/R =
2 a¯ L/R
¯ ±VL/R ) max(a,
(11)
and the critical speed a¯ is calculated via the isoenergetic H. condition by a¯ 2 = 2 γγ −1 +1 The split pressure is determined from + − (ML )pL + p(5,α) (MR )pR , p1/2 = p(5,α)
(12)
3
where ⎧ 1 ⎪ ⎪ ⎨ (M ± 1)2 (2 ∓ M) ± αM(M 2 − 1)2 4 ± p(5,α) (M) = ⎪ 1 ⎪ ⎩ (M ± |M|) M
|M| ≤ 1 otherwise.
(13) In this paper, the values of α and β are equal to 3/16 and 1/8, respectively.
3. Numerical instability test cases To study the robustness of the AUSM + scheme, various numerical computations are performed. The test problems include the M15 flow over a cylinder, the M6 slowly mov◦ ing normal shock, and the M5 shock reflection over a 46 wedge. The numerical flux formulation presented in previous section is discretised by means of the cell-centred finite volume method. For boundary conditions, free-stream values are specified as inflow condition, extrapolation from the inner computational domain is used for outflow condition and slip condition at a wall is specified for velocity. The computations were carried out with the first-order accuracy in both space and time with an explicit time evolution. By assuming the initial conditions of (ρ 0 , p0 , M0 ), the normalised conditions at each computation can be de√ termined by (ρ¯0 , |V¯0 |, p¯ 0 ) = ( ρρ0 , M0 γ , pp0 ). All examples are tested using the Courant number of 0.85.
3.1. M15 flow over a cylinder The carbuncle phenomenon refers to a spurious bump on the bow shock near the flow centre line ahead a cylinder. The phenomenon is highly grid-dependent (Pandolfi and D’Ambrosio 2001), but does not require a large number of grid points to appear. The first problem is presented for Mach 15 flow over a cylinder with radius of 1.5. Figure 1(a) shows the computational domain consisting of a structured triangular grid with 14 × 318 cells in the radial and circumferential directions, respectively. Figure 1(b) shows the steady-state density contours of the first-order accurate solution. The number of contour lines of this figure is 16 with the corresponding minimum and maximum values of 1.0 and 6.62, respectively. The AUSM + scheme produces a prominent carbuncle phenomenon ahead of the cylinder as shown by streamline contours in Figure 1(c). It should be noted that there is a circulation zone behind the kinked detached bow shock wave in the vicinity of the stagnation line, which disturbs the solution and may introduce some oscillations as shown in Figure 2. The numerical anomaly is generated in the subsonic region where the density has higher value and propagated in the upwind direction to perturb the shock wave structure before the carbuncle phenomenon is formed. Robinet et al. (2000) concluded that
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(a) Grid.
(b) Density.
(c) Streamline.
Figure 1. M15 flow over a blunt body. (a) Grid. (b) Density. (c) Streamline.
examine the carbuncle phenomenon by Quirk (1994). The grid at the mid-channel is perturbed alternately at odd and even points with a small magnitude of ±10−6 in order to initiate the instability. The computational domain consists of a uniform triangular grid with 800 × 20 equal intervals respectively along x − and y −directions. For some numerical schemes such as the Roe’s and Harten–Lax–van Leer contact (HLLC) schemes, this grid perturbation is sufficient to trigger numerical instabilities so-called carbuncle phenomenon (Quirk 1994; Phongthanapanich and Dechaumphai 2004). Figure 3 shows the density contours of the moving normal shock wave at the three locations (t = 0.12, 0.24, 0.36) along the duct, which reveals the carbuncle phenomenon as a broken moving normal shock wave. The number of contour lines of this figure is 16 with the corresponding minimum and maximum values of 1.17 and 6.82, respectively. As explained by Gressier and Moschetta (2000), the exact capturing of contact discontinuity and strict stability cannot be simultaneously satisfied in any upwind scheme. ◦
Figure 2. M15 flow over a blunt body: density contours and velocity vectors.
3.3. M5 shock reflection over a 46 wedge The kinked Mach stem generated from a Mach 5 shock ◦ moving over a 46 wedge (Quirk 1994) is the third test case used to examine the robustness of this scheme. The computational domain consists of a structured triangular grid with 200 × 200 cells in x − and y −directions, respectively. It has been known that the grid aspect ratio is significant in the establishment of the carbuncle phenomenon (Pandolfi and D’Ambrosio 2001) such that very elongated grid sizes along the normal to the shock promote instabilities. For this problem, the grid aspect ratio ( x: y) of cells around the right-ended of a wedge is approximately equal to 5:1 and there is potential to initiate the carbuncle phenomenon. Figure 4(a) shows the density contours of double Mach reflection simulation at time t = 0.11. The number of contour lines of this figure is 16 with the corresponding minimum and maximum values of 0.96 and 14.06, respectively. The AUSM + scheme clearly exhibits the kinked Mach stem, the broken upper-part of an incident shock wave and postshock oscillations as shown in Figure 4(b).
the carbuncle phenomenon is a purely numerical mechanism with no connection to the physics.
4. Numerical instability analysis
3.2. M6 slowly moving normal shock wave The problem of a Mach 6 planar shock wave moving through a two-dimensional duct (Quirk’s test) is chosen to
Figure 3.
M6 slowly moving normal shock: density contours.
To study the odd–even decoupling numerical instability mechanism, the linearised analysis developed by Pandolfi and D’Ambrosio (2001) is applied to the AUSM + scheme. The uniform flow conditions with normalised values of ρ 0 = 1, u0 = 0, v0 = 0, and p0 = 1 are assumed. The
International Journal of Computational Fluid Dynamics
5
5
4
Density
3 0
0.2
0.4
0.6
0.8
1 x
2
1
0
(b) Density at y = 0.95.
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(a) Density contours.
Figure 4.
◦
M5 shock reflection over a 46 wedge. (a) Density contours. (b) Density at y = 0.95.
ˆ p) ˆ are then prescribed and perturbation quantities (ρ, ˆ u, ˆ and p = the flow properties of ρ = 1 ± ρ, ˆ u = u0 ± u, 1 ± pˆ are defined in the two neighbouring cells. Then, the common mass flux is calculated as m1/2 = (0.25 + β) − (0.25 + β) = 0. By substitution of this common mass flux into Equation (6), the numerical flux at the cell interface is
fM+1/2
⎛ ⎞ 0 ⎜0⎟ ⎟ =⎜ ⎝ 1 ⎠. 0
K
ν K f , − fM−1/2 |V | + a M+1/2
(15)
fˆM+1/2
(17)
K+1 K = ρˆM ρˆM
uˆ K+1 = uˆ K M . M
(18)
K = pˆ M
Equation (18) shows that all perturbations are not damped nor amplified but all of them seem to be a part of the solutions. To alleviate numerical instability of the scheme, the multidimensional dissipation technique proposed by Phongthanapanich and Dechaumphai (2004) is adopted from the formulation suitable for applying the matrix-form of the Roe’s flux formulation to the scalar form of the AUSM + scheme. This original technique is first introduced for quadrilateral grid by Sanders, Morano, and Druguet (1998), and a more efficient version of this technique was presented by Pandolfi and D’Ambrosio (2001). In order to obtain a more stable AUSM + scheme (so-called AUSM + δ scheme), this paper proposes a modified version of the multidimensional dissipation technique applying to the |m1/2 | term as
(16)
2ν ˆ K . f |V | + a M+1/2
By substitution of Equations (14) and (16) into Equation (17), the perturbations at time step K + 1 of the AUSM + scheme are calculated from
K+1 pˆ M
where ν is the Courant number. By defining the numerical flux at the cell interface as ⎛ ⎞ 0 ⎜0⎟ ⎟ = fM+1/2 − ⎜ ⎝ 1 ⎠. 0
ˆK ˆ K+1 = Q Q M − M
(14)
Equation (14) shows that numerical flux of the AUSM + scheme does not respond to all perturbations. The numerical flux at the cell interface is then used to further study the odd–even decoupling problem. By starting from the set of the initial perturbations (ρˆ 0 , uˆ 0 , pˆ 0 ), result of the first-order accuracy at step K + 1 is calculated by QK+1 = QK M − M
Equation (15) becomes
|mδ1/2 | =
|m1/2 |2 + δ |m1/2 |
δ
|m1/2 | < 2δ , otherwise
(19)
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2
3
L
R
1
Cell interface of the numerical instability region.
where δ is calculated from Figure 7.
δ = κ max(η1 , η2 , η3 , η4 )
M15 flow past a blunt body (κ = 0.5).
(20) iterations, and the recursive equations are
max and ηi = max |λmax k,L − λk,R | and i = 1. . .4 (see Figure 5).
K+1 ρˆM = 1− uˆ K+1 = 1− M
k
are the maximum eigenvalues of the Jacobian The λmax k of fluxes. The constant value κ is usually less than 1 for the first-order scheme and more than or equal to 1 for the second-order scheme. After performing some numerical experiments, a good initial guess value of κ for the first- and second-order schemes should be 0.5 and 1.0, respectively. The perturbations at time step K + 1 of the AUSM + δ scheme is derived again as shown in Equation (21), and the responses to initial non-zero perturbations are also shown in Figure 6. It is evidenced that now the dissipation factor (δ) plays direct role in all perturbations. By applying the multidimensional dissipation technique to the scheme, all perturbations are damped to machine-zero after a few
K+1 pˆ M
0.01
0.01
100
150 K
0
50
100
150 K
(a) ρˆ0 = 0.01.
2νδ √ γ
(21)
0
0
50
100
150 K
^
^
^
u^ p^
u^ ^p
u^ ^p -0.02
-0.02
-0.02
K ρˆM
-0.01
-0.01
-0.01
Figure 6.
0
Perturbations
0.01
50
2
0.02
0
where = γ2γ+1 + u20 √2γγ −1 . (γ +1) Finally, the ability of Equation (19) to alleviate carbuncle phenomenon is evaluated by solving problems (3.1) and (3.3) again. Figure 7 shows that the carbuncle phenomenon near the centre line of a cylinder is now removed completely and no oscillations in downstream of detached bow shock wave for problem (2.1). For problem (2.2), there is no
0.02
0
2νδ √ γ
uˆ K M √ K = (1 − 2νδ γ )pˆ M ,
0.02
Perturbations
Perturbations
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Figure 5.
4
(b) pˆ0 = 0.01.
(c) ρˆ0 = pˆ0 = 0.01.
Odd–even decoupling problem of AUSM + δ = 0.5 . (a) ρˆ 0 = 0.01. (b) pˆ 0 = 0.01. (c) ρˆ 0 = pˆ 0 = 0.01.
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International Journal of Computational Fluid Dynamics
Figure 8.
M6 slowly moving normal shock (κ = 0.005).
Figure 9.
M5 shock reflection over a 46 wedge (κ = 0.5).
◦
Figure 12.
7
◦
M2 shock reflection over a 46 wedge (AUSM + ).
5. Numerical results carbuncle phenomenon but slight oscillations still exhibit with less serious than that of the original scheme as shown in Figure (8). Figure (9) shows that severely kinked Mach stem is recovered with slight oscillations in the downstream of the incident shock wave.
In this section, the robustness and the efficiency of the AUSM + δ scheme is further demonstrated with the second-order accuracy. A second-order spatial discretisation is achieved by applying the least-squares method with Venkatakrishnan’s limiter (Venkatakrishnan 1995). The second-order accurate Runge–Kutta time stepping method
(b) AUSM+δ with κ = 1.0.
(a) AUSM+ .
Figure 10.
M3 flow past a forward facing step. (a) AUSM + . (b) AUSM + δ with κ = 1.0.
(a) AUSM+ .
(b) AUSM+δ with κ = 0.02.
Figure 11.
M6 slowly moving normal shock. (a) AUSM + . (b) AUSM + δ with κ = 0.02.
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Figure 13.
S. Phongthanapanich
(a) κ = 1.0.
(b) κ = 0.1.
(c) κ = 0.5.
(d) κ = 20.
◦
M2 shock reflection over a 46 wedge (AUSM + δ with κ = 0). (a) κ = 1.0. (b) κ = 0.1. (c) κ = 0.5. (d) κ = 20.
(Shu and Osher 1988) is used for temporal discretisation. The selected test cases are (1) the M3 flow past a forward facing step, (2) the M6 slowly moving normal shock, (3) ◦ the M2 shock reflection over a 46 wedge and (4) the M5.09 shock diffraction around a corner.
5.1. M3 flow past a forward facing step The first problem is a supersonic flow at Mach 3 flows through a channel of length of 1 and height of 1 (Phongthanapanich and Dechaumphai 2004). A step of height 0.2 stands at location 0.6 downstream of the inlet. A transient detached bow shock wave forms up, hits the top wall, reflects back and further hits the bottom wall and reflects again. A structured triangular grid of x = y = 0.0125 is used for this problem. Figure 10(a) shows the density con-
tours of the AUSM + scheme. There is small circulation which disturbs the solution and may introduce some oscillations in the downstream of detached bow shock wave. The problem is tested again by the AUSM + δ scheme (κ = 1) in which oscillations are now suppressed and the spurious oscillations in vicinity of the triple points have been fully resolved as shown by Figure 10(b). The number of contour lines of these figures is 32 with the corresponding minimum and maximum values of 0.15 and 6.19, respectively.
5.2. M6 slowly moving normal shock wave The problem of a planar shock wave propagating through a two-dimensional duct is tested again. Figure 11(a) shows the density contours at three locations of the AUSM + scheme. This figure shows that the scheme produces
International Journal of Computational Fluid Dynamics
(b) AUSM+ -up.
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(a) AUSMPW+ .
Figure 14.
◦
M2 shock reflection over a 46 wedge. (a) AUSMPW + . (b) AUSM + -up.
serious carbuncle phenomenon. The post-shock oscillations are clearly seen by second-order computation due to its less numerical dissipation (Arora and Roe 1997). By applying the AUSM + δ scheme with a value of κ = 0.02, the carbuncle phenomenon is significantly removed as shown in Figure 11(b). It should be noted that by comparing Figure 11(a) and 11(b), the spurious oscillations may cause a normal shock wave moves with faster speeds. ◦
5.3. M2 shock reflection over a 46 wedge Next, a classical unsteady shock phenomenon with a shock reflection over a wedge (Takayama and Jiang 1997) is in-
(a) AUSM+ .
Figure 15.
9
◦
vestigated. This test case is given by a Mach 2 shock wave ◦ moving over a 46 wedge from left to right of the domain. The flow phenomenon consists of an incident shock wave, a Mach stem, a reflected shock wave and a slipstream. Many existing numerical schemes give severely kinked Mach stem (Phongthanapanich and Dechaumphai 2009). The AUSM + scheme provides a kinked Mach stem solution and spurious oscillations behind a Mach stem as shown in Figure 12. In this problem, the multidimensional dissipation technique is applied with a value of κ = 1. The carbuncle phenomenon and post-shock oscillations are removed completely as shown in Figure 13(a). Furthermore, this problem is used to demonstrate the effect of the κ
(b) AUSM+δ with κ = 1.0.
M2 shock reflection over a 46 wedge. (a) AUSM + . (b) AUSM + δ with κ = 1.0.
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S. Phongthanapanich
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parameter on the robustness of the proposed technique. Figure 13(b) and 13(c) show that the value of κ = 0.1 is not adequate to eliminate the carbuncle phenomenon. The Mach stem is recovered with slight oscillation by using a value of κ = 0.5. A higher value of κ = 20 works for this problem with the smearing of the shock waves solution. The scheme fails with a negative speed of sound when the value of κ is more than 24. To show the difficulty of the moving strong shock wave on the aligned-grid system, the AUSMPW + (Kim, Kim, and Rho 2001) and the AUSM + -up (Liou 2006) schemes are used to analyse this problem again. Figure 14(a) and 14(b) show that these schemes cannot satisfy robustness against shock-induced anomalies. The AUSMPW + scheme gives better solutions than the AUSM + -up scheme. The number of contour lines of all figures are 48 with the corresponding minimum and maximum values of 1.0 and 5.2, respectively. 5.4. M5.09 shock diffraction around a corner The last test case is a strong shock diffraction around a corner (Hillier 1991). The structure of the diffracted shock wave consists of expansion waves, a slipstream and a secondary shock wave. Figure 15(a) shows that the AUSM + scheme generates slight oscillations near a tip of an incident shock wave. The AUSM + δ scheme with a value of κ = 1 can eliminate oscillations completely as shown in Figure 15(b). The number of contour lines of these figures are 16 with the corresponding minimum and maximum values of 0.05 and 5.22, respectively. 6. Conclusions The numerical instability so-called the carbuncle phenomenon of the AUSM + scheme is investigated in this paper. Many numerical experiments have shown that the scheme has the potential to trigger the carbuncle phenomenon under certain conditions especially when the incident shock wave is aligned with the grid. The post-shock oscillations normally arise from the shock-capturing finitevolume methods. From the point of view of linearised analysis of the odd–even decoupling problem, all perturbations are not damp nor amplify but all of them are parts of the solutions. In order to overcome this problem, the multidimensional dissipation technique is applied to the AUSM + scheme (so-called AUSM + δ ) for healing carbuncle phenomenon and suppressing post-shock oscillations. Many test cases presented in this paper confirm that the AUSM + δ scheme is able to solve wide range of high-speed compressible flow problems accurately without carbuncle phenomenon, especially where strong physical discontinuities exist. However, several other aspects of the numerical instability discussed herein should be further investigated. The future challenge is how to design numerical schemes
that would intrinsically reject the creation of the carbuncle phenomenon without having to utilise some kind of artificial dissipation fix techniques.
Acknowledgements The author is grateful to the Department of Mechanical Engineering Technology, College of Industrial Technology, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand for supporting this research work.
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