LINEBURG

R R R

Fρ , FρVn , and FE when plugged into equations (16.2), (16.5) and (16.6),

a regression rate for this conversion. When the regression rate is based on

That is, we want ghost node values of density, velocity, internal energy, and

some sort of chemical reaction, the interface can pass over a material exactly

pressure such that

once, changing it into another material. The same chemical reaction cannot

occur to a material more than once, and the reverse reaction is usually

ρG (Vn ’ D) = Fρ ,

G R

(16.7)

not physically plausible due to an entropy condition. However, for readily

ρG (Vn ’ D)2 + pG = FρVn ,

G R

(16.8)

reversible chemical reactions, the zero level set may pass over a material in

one direction (the reaction) and then pass back over the same material in ρG (Vn ’ D)2

G

ρG eG + + pG (Vn ’ D) = FE ,

G R

(16.9)

the opposite direction (the reverse reaction).

2

Shocks can be interpreted as the conversion of an uncompressed material

at each grid node, where the “G” subscript designates a ghost node value.

into a compressed material. Here D is the shock speed, and the ghost ¬‚uid

Adding the equation of state for the ghost ¬‚uid as

method can be used to follow a lead shock, but since shocks can pass over a

material more than once in the same direction, all subsequent shocks must

pG = (γ G ’ 1)ρG eG (16.10)

be captured or modeled by separate level set functions.

yields four equations for four unknowns, which can be arranged into a

G

quadratic equation for Vn ’ D, where

16.4 Shock Waves 2

γ G FρVn

R R

γ G FρVn R

2(γ G ’ 1)FE

G

’D = G ± ’

Vn (16.11)

R (γ G + 1)Fρ

R (γ G + 1)Fρ

R

(γ + 1)Fρ

Consider the representation of a lead shock by a level set function where the

positive values of φ correspond to the unshocked material and the negative

G

expresses the two solutions. Choosing one of these two solutions for Vn

values of φ correspond to the shocked material. Then the normal N points

allows us to obtain ρG from equation (16.7), pG from equation (16.8), and

from the shocked material into the unshocked material.

eG from equation (16.10). In addition, uG = Vn N .

G

In one spatial dimension, the normal velocity is de¬ned as Vn = V · N ,

In order to choose the correct solution (of the two choices) from equa-

and equations (15.4), (15.5) and (15.6) become

tion (16.11), we have to determine whether the ghost ¬‚uid is an unshocked

Fρ = ρ(Vn ’ D) (16.2)

(preshock) ¬‚uid or a shocked (postshock) ¬‚uid. Node by node, we use the

FρV = ρ(u ’ DN T )(Vn ’ D) + pN T real values of the unshocked ¬‚uid to create a shocked ghost ¬‚uid. Likewise,

(16.3)

we use the real values of the shocked ¬‚uid to create an unshocked ghost

ρ|u ’ DN |2 ¬‚uid. If the ghost ¬‚uid is a shocked ¬‚uid, then D is subsonic relative to

+ p (Vn ’ D)

FE = ρe + (16.4)

2 the ¬‚ow; i.e., Vn ’ cG < D < Vn + cG or |Vn ’ D| < cG . If the ghost

G G G

¬‚uid is an unshocked ¬‚uid, then D is supersonic relative to the ¬‚ow; i.e.,

where it is useful to de¬ne

|Vn ’ D| > cG . Therefore, the “±” sign in equation (16.11) should be cho-

G

FρVn = N FρV = ρ(Vn ’ D)2 + p G

sen to give the minimum value of |Vn ’ D| when a shocked ghost ¬‚uid is

(16.5)

16.5. Detonation Waves 193 194 16. Shocks, Detonations, and De¬‚agrations

G

constructed and the maximum value of |Vn ’ D| for an unshocked ghost den vel

¬‚uid. 80

3

For a simple nonreacting shock, the shock speed D can be de¬ned directly

70

from the mass balance equation as

60

ρ(1) u(1) ’ ρ(2) u(2)

D= (16.12) 50

2.5

ρ(1) ’ ρ(2)

40

in a node-by-node fashion. However, this simple de¬nition of the shock

speed will erroneously give D = 0 in the case of a standard shock tube 30

problem where both ¬‚uids are initially at rest. A somewhat better estimate 2

20

of the shock speed can be derived by combining equation (16.12) with the

10

momentum balance equation to obtain

0

2 2 1.5

ρ(1) u(1) p(1) ρ(2) u(2) p(2)

’ ’

+

0.4

0.2

D= , (16.13) 0 0.6 1

0.8 0 0.8 1

0.2 0.6

0.4

ρ(1) ’ ρ(2)

where the shock speed is now dependent on the pressure as well. Note that

equations (16.12) and (16.13) are only approximations of D. Clearly, these

temp press

5

x 10

approximations will lead to nonphysical values of D in certain situations.

In fact, D could be in¬nite or even imaginary. A more robust, but still 2

330

approximate, value for D can be obtained by evaluating D = Vn + c with

325

the Roe average of U (1) and U (2) (see, for example, LeVeque [105]), since 1.8

this is the exact shock speed for an isolated shock wave and never becomes 320

ill-de¬ned. Of course, the best de¬nition of the shock speed can be derived 315 1.6

by solving the Riemann problem for the states U (1) and U (2) , although this

310

generally requires an iterative procedure. The interested reader is referred

1.4

305

to the ongoing work of Aslam [9] for more details.

Figure 16.1 depicts a standard shock tube test case that was computed 300

1.2

using the level set method to track the location of the shock wave and the

295

ghost ¬‚uid method to accurately capture the boundary conditions across

1

290

that shock wave. Note the sharp (nonsmeared) representation of the shock

wave.

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Figure 16.1. Standard shock-tube test case using the ghost ¬‚uid method to keep

the shock wave sharp. The computed solution is depicted by circles, while the

16.5 Detonation Waves exact solution is drawn as a solid line.

Strong detonations and Chapman-Jouguet detonations can be approxi- where one can no longer set eo = 0 for both ¬‚uids. In detonations, the jump

mated as reacting shocks under the assumption that the reaction zone in eo across the reaction front indicates the energy release in the chemical

has negligible thickness. Again, assume that N points from the reacted reaction. Equation 16.11 becomes

material into the unreacted material.

2

Equations (16.7), (16.8) and (16.9) are still valid, while equation (16.10) γ G FρVn

R R

γ G FρVn R

2(γ G ’ 1) FE

G

’ eG

Vn ’ D = G ± ’

becomes o

R (γ G + 1)Fρ

R (γ G + 1) R

(γ + 1)Fρ Fρ

pG = (γ G ’ 1)ρG (eG ’ eG ), (16.14) (16.15)

o

16.6. De¬‚agration Waves 195 196 16. Shocks, Detonations, and De¬‚agrations

G press

6

where the “±” sign is chosen to give the minimum value of |Vn ’ D| for a x 10

G

reacted ghost ¬‚uid and the maximum value of |Vn ’ D| for an unreacted

ghost ¬‚uid. Equation (16.13) is used for the detonation speed D, although

one might want to use a Riemann solver; see, for example, Teng et al.

[163]. 5

Figure 16.2 shows an overdriven detonation wave traveling from left to

right. A solid-wall boundary condition is enforced on the left, creating a rar-

efaction wave that will eventually catch up with the overdriven detonation

and weaken it to a Chapman-Jouguet detonation. The circles depict the

pressure pro¬le calculated with 200 grid cells, while the solid line depicts 4

the computed pro¬le with 800 grid cells. Note that there is no numerical

smearing of the leading wave front, which is extremely important when one

attempts to eliminate spurious wave speeds for sti¬ source terms on coarse

grids; see, for example, Colella et al. [50].

3

16.6 De¬‚agration Waves 2

For a de¬‚agration wave, equations (16.7), (16.8), (16.9) and (16.14) are used

with the jump in eo equal to the energy release in the chemical reaction.

Equation (16.15) is used as well. However, since a de¬‚agration is subsonic,

G

the “±” sign is chosen to give the minimum value of |VN ’ D| for both the 1

reacted and the unreacted ghost ¬‚uids.

For a de¬‚agration, the Riemann problem is not well posed unless the

speed of the de¬‚agration is given. Luckily, there is abundant literature on

the G-equation for ¬‚ame discontinuities, and one can consult this literature

to obtain appropriate de¬‚agration speeds. For example, using a de¬‚agration 0

velocity from Mulpuru and Wilkin [116],

0 1 2 3 4 5 6 7 8

1.721

.1

p T

D = VN + 18.5 m/s (16.16) Figure 16.2. Overdriven detonation wave traveling from left to right with a

101000P a 298K

solid-wall boundary condition on the left. The circles depict the pressure pro¬le

calculated with 200 grid cells.

evaluated with the velocity, pressure, and temperature of the unreacted gas,

we compute the shock de¬‚agration interaction shown in Figure 16.3. Here a

left-going shock wave intersects a right-going de¬‚agration wave (unreacted

16.7 Multiple Spatial Dimensions

gas to the right), resulting in four waves: a shock, contact, de¬‚agration,

and rarefaction from left to right. All the waves are captured, except the

In multiple spatial dimensions we use equations (16.2), (16.5), and (16.14)

de¬‚agration wave, which is tracked with the level set function and resolved

along with

with the ghost ¬‚uid method. The circles depict the computed solution,

while the solid line depicts the exact solution. Note that the pressure drops

FρV ’ FρVn N T

slightly across a de¬‚agration wave, as opposed to the pressure rise across

= V T ’ Vn N T ,

FρVT = (16.17)

shock and detonation waves. Fρ

16.7. Multiple Spatial Dimensions 197 198 16. Shocks, Detonations, and De¬‚agrations

Combining

den vel

2.2

|V ’ DN |2 = |V |2 ’ 2DVn + D2 = |V |2 ’ Vn + (Vn ’ D)2

2

(16.18)

200

2

with

1.8

150

|V | 2 = V n + V T 1 + V T 2 ,

2 2 2

(16.19)

1.6

where VT1 and VT2 are the velocities in the tangent directions T1 and T2 ,

1.4

100

yields

1.2

|V ’ DN |2 = VT1 + VT2 + (Vn ’ D)2 ,

2 2

(16.20)

1

50

which can plugged into equation (15.6) to obtain

0.8

2 2

0.6 ρ(VT1 + VT2 ) ρ(Vn ’ D)2

+ p (Vn ’ D)

FE = ρe + + (16.21)

0

0.4 2 2

0.8

0.6 1

1 0.4

0

0 0.8 0.2

0.6

0.4

0.2

as a rewritten version of equation (15.6). We then write

2 2

ρ(Vn ’ D)2

Fρ (VT1 + VT2 )

ˆ

FE = FE ’ + p (Vn ’ D), (16.22)

= ρe +

2 2

press

temp 5

x 10

which (not coincidently) has the same right-hand side as equation (16.6). In

2000

R

fact, we eventually derive equation (16.15) again, except with FE replaced

3.8

1800

ˆR

by the identical FE .

3.6

The main di¬erence between one spatial dimension and multiple spatial

1600

dimensions occurs in the treatment of the velocity. The ghost cell veloc-

3.4

1400

ity V G is obtained by combining the normal velocity of the ghost ¬‚uid with

3.2

the tangential velocity of the real ¬‚uid using

1200

V G = Vn N + V R ’ Vn N ,

G R

3 (16.23)

1000

where V R ’ Vn N is the tangential velocity of the real ¬‚uid.

R

2.8

800

Figure 16.4 shows two initially circular de¬‚agration fronts that have

600 2.6

merged into a single front. The light colored region surrounding the de-

¬‚agration fronts is a precursor shock wave that causes the initially circular

400 2.4

de¬‚agration waves to deform as they approach each other. Figure 16.5

0 0.2 0.4 0.8 1

0.6

0.6 1

0 0.2 0.8

0.4

shows the smooth level set representation of the de¬‚agration wave.

Figure 16.3. Interaction of a left-going shock wave with a right-going de¬‚agration

wave, producing four waves: a shock, contact, de¬‚agration, and rarefaction from

left to right. The circles depict the computed solution using 400 grid cells, while

the solid line depicts the exact solution.

which is valid when Vn = D, i.e., except for the case of a contact discontinu-

ity. The necessary continuity of this expression implies the well-known fact

that tangential velocities are continuous across shock, detonation, and de-

¬‚agration waves. Note that tangential velocities are not continuous across

contact discontinuities unless viscosity is present.

16.7. Multiple Spatial Dimensions 199 200 16. Shocks, Detonations, and De¬‚agrations

1.6

density

interface location

100

100

1.4

90

90

80

80

1.2

70

70

60

60

1

50

50

40

40

0.8

30

30

20

20

0.6

10

10

10 20 30 40 50 60 70 80 90 100

20 40 60 80 100

0.4

Figure 16.5. Two initially circular de¬‚agration fronts that have recently merged

into a single front. Note the smooth level set representation of the de¬‚agration

wave.

Figure 16.4. Two initially circular de¬‚agration fronts that have recently merged

into a single front. The light colored region surrounding the de¬‚agration fronts

is a precursor shock wave that causes the initially circular de¬‚agration waves to

deform as they approach each other.

202 17. Solid-Fluid Coupling

oscillations until the shock is spread out over about six grid cells; see, e.g.,

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