RD-V: 0505 Shock Tube

The transitory response of a perfect gas in a long tube separated into two parts using a diaphragm is studied. The problem is well-known as the Riemann problem. The numerical results based on the finite element method with the Lagrangian and Eulerian formulations, are compared to the analytical solution.

This famous experiment is interesting for observing the shock-wave propagation. Moreover, this case uses the representation of perfect gas and compares the different formulations: The ALE uses Lagrangian or Eulerian.

rad_ex_13_shock-tube
Figure 1.
The first part of the study deals with the modeling description of perfect gas with the hydrodynamic viscous fluid LAW6. The purpose is to test the different formulations:
  • Lagrangian (mesh points coincident to material points)
  • Eulerian (mesh points fixed)

The propagation of the gas in the tube can be studied in an analytical manner. The gas is separated into different parts characterizing the expansion wave, the shock front and the contact surface. The simulation results are compared with the analytical solution for velocity, density and pressure.

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Input Files

Refer to Access the Model Files to download the required model file(s).

Model Description

The shock tube problem is one of the standard problems in gas dynamics. It is a very interesting test since the exact solution is known and can be compared with the simulation results. The Finite Element method using the Eulerian and Lagrangian formulations was used in the numerical models.

A shock tube consists of a long tube filled with the same gas in two different physical states. The tube is divided into two parts, separated by a diaphragm. The initial state is defined by the values for density, pressure and velocity, as shown in Figure 2 and Figure 3. All the viscous effects are negligible along the tube sides; it is also assumed that there is no motion in the beginning.

rad_ex_fig_13-1
Figure 2. Shock Tube Sketch

rad_ex_fig_13-2
Figure 3. Initial States with Discontinuities

The initial state at time t = 0 consists of two constant states 1 and 4 with p 4 > p 1 , ρ 4 > ρ 1 , and v 4 = v 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaisdaaeqaaOGaeyypa0JaamODamaaBaaaleaacaaIXaaa beaakiabg2da9iaaicdaaaa@3D00@ (table).

Table 1. Initial Conditions in the Shock Tube
  High Pressure Side (4) Low Pressure Side (1)
Pressure p 500000 [ Pa ] 20000 [ Pa ]
Velocity v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2baaaa@375A@ 0 [ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@ 0 [ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@
Density ρ 5.7487 [ k g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaai4AaiaacEgaaeaacaGGTbGaaiyBamaaCaaaleqabaGa ai4maaaaaaaakiaawUfacaGLDbaaaaa@3D0B@ 0.22995 [ k g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaai4AaiaacEgaaeaacaGGTbGaaiyBamaaCaaaleqabaGa ai4maaaaaaaakiaawUfacaGLDbaaaaa@3D0B@
Temperature T 303 [ K ] 303 [ K ]

Just after the membrane is removed, a compression shock runs into the low pressure region, while a rarefaction (decompression) wave moves into the high pressure part of the tube. Furthermore, a contact discontinuity usually occurs.

Model Method

The hydrodynamic viscous fluid LAW6 is used to describe compressed gas.

The general equation describing pressure is:(1) p = C 0 + C 1 μ + C 2 μ 2 + C 3 μ 3 + ( C 4 + C 5 μ ) E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbGaey ypa0Jaam4qamaaBaaaleaacaaIWaaabeaakiabgUcaRiaadoeadaWg aaWcbaGaaGymaaqabaGccqaH8oqBcqGHRaWkcaWGdbWaaSbaaSqaai aaikdaaeqaaOGaeqiVd02aaWbaaSqabeaacaaIYaaaaOGaey4kaSIa am4qamaaBaaaleaacaaIZaaabeaakiabeY7aTnaaCaaaleqabaGaaG 4maaaakiabgUcaRmaabmaabaGaam4qamaaBaaaleaacaaI0aaabeaa kiabgUcaRiaadoeadaWgaaWcbaGaaGynaaqabaGccqaH8oqBaiaawI cacaGLPaaacaWGfbaaaa@5235@

With μ = ρ ρ 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey ypa0ZaaSaaaeaacqaHbpGCaeaacqaHbpGCdaWgaaWcbaGaaGimaaqa baaaaOGaeyOeI0IaaGymaaaa@3EDA@

Where,
p
Pressure
C i
Hydrodynamic constants
E n
Internal energy per initial volume
ρ
Density
Perfect gas is modeled by setting all coefficients:(2) C 0 = C 1 = C 2 = C 3 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIWaaabeaakiabg2da9iaadoeadaWgaaWcbaGaaGymaaqa baGccqGH9aqpcaWGdbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Jaam 4qamaaBaaaleaacaaIZaaabeaakiabg2da9iaaicdaaaa@41AE@ And

C 4 = C 5 = γ 1

Where, γ is the gas constant.

Then the initial internal energy, per initial volume is calculated from initial pressure:(3) E 0 = p 0 γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS baaSqaaiaaicdaaeqaaOGaeyypa0ZaaSaaaeaacaWGWbWaaSbaaSqa aiaaicdaaeqaaaGcbaGaeq4SdCMaeyOeI0IaaGymaaaaaaa@3E63@

Under the assumption γ = c o n s t . = 1.4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHZoWzcq GH9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaiaac6cacqGH9aqp caaIXaGaaiOlaiaaisdaaaa@41AF@ (valid for low temperature range), the hydrodynamic constants C 4 = C 5 = 0.4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaaisdaaeqaaOGaeyypa0Jaam4qamaaBaaaleaacaaI1aaa beaakiabg2da9iaaicdacaGGUaGaaGinaaaa@3E0E@ .

Gas pressure is described by:(4) p = ( C 4 + C 5 μ ) E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbGaey ypa0ZaaeWaaeaacaWGdbWaaSbaaSqaaiaaisdaaeqaaOGaey4kaSIa am4qamaaBaaaleaacaaI1aaabeaakiabeY7aTbGaayjkaiaawMcaai aadweaaaa@40BE@ (5) p = ( 0.4 + 0.4 ρ ρ 0 ρ 0 ) E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbGaey ypa0ZaaeWaaeaacaaIWaGaaiOlaiaaisdacqGHRaWkcaaIWaGaaiOl aiaaisdadaWcaaqaaiabeg8aYjabgkHiTiabeg8aYnaaBaaaleaaca aIWaaabeaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaaaaakiaa wIcacaGLPaaacaWGfbaaaa@4800@ (6) E 0 = p 0 0.4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS baaSqaaiaaicdaaeqaaOGaeyypa0ZaaSaaaeaacaWGWbWaaSbaaSqa aiaaicdaaeqaaaGcbaGaaGimaiaac6cacaaI0aaaaaaa@3D3E@

Parameters of material LAW6 are provided in Table 2.

Table 2. Material Properties of Gas in LAW6
  High Pressure Side (4) Low Pressure Side (1)
Initial volumetric energy density (E0) 1.25x106 [ J m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaciOsaaqaaiaac2gacaGGTbWaaWbaaSqabeaacaGGZaaa aaaaaOGaay5waiaaw2faaaaa@3C01@ 5x104 [ J m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaciOsaaqaaiaac2gacaGGTbWaaWbaaSqabeaacaGGZaaa aaaaaOGaay5waiaaw2faaaaa@3C01@
C 4 and C 5 0.4 0.4
Density ρ 5.7487 [ k g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaai4AaiaacEgaaeaacaGGTbGaaiyBamaaCaaaleqabaGa ai4maaaaaaaakiaawUfacaGLDbaaaaa@3D0B@ 0.22995 [ k g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaai4AaiaacEgaaeaacaGGTbGaaiyBamaaCaaaleqabaGa ai4maaaaaaaakiaawUfacaGLDbaaaaa@3D0B@

Analytical Approach

The shock tube problem has an analytical solution of time before the shock hits the extremity of the tube. 1

rad_ex_fig_13-3
Figure 4. Schematic Shock Tube Problem with Pressure Distribution for Pre- and Post-diaphragm Removal

Evolution of the flow pattern is illustrated in Figure 4. When the diaphragm bursts, discontinuity between the two initial states breaks into leftward and rightward moving waves, separated by a contact surface.

Each wave pattern is composed of a contact discontinuity in the middle and a shock or a rarefaction wave on the left and the right sides separating the uniform state solution. The shock wave moves at a supersonic speed into the low pressure side. A one-dimensional problem is considered.

rad_ex_fig_13-4
Figure 5. Shock Diagram, Expansion Waves and Contact Surface
There are four distinct zones marked 1, 2, 3 and 4 in Figure 5.
  • Zone 1 is the low pressure gas which is not disturbed by the shock wave.
  • Zone 2 (divided in 2 and 2' by the contact surface) contains the gas immediately behind the shock traveling at a constant speed. The contact surface across which the density and the temperature are discontinuous lies within this zone.
  • The zone between the head and the tail of the expansion fan is noted as Zone 3. In this zone, the flow properties gradually change since the expansion process is isentropic.
  • Zone 4 denotes the undisturbed high pressure gas.

Equations in Zone 2 are obtained using the normal shock relations. Pressure and the velocity are constant in Zones 2 and 2'.

The ratio of the specific heat constant of gas γ is fixed at 1.4. It is assumed that the value does not change under the temperature effect, which is valid for the low temperature range.

The analytical solution to the Riemann problem is indicated at t=0.4 ms. A solution is given according to the distinct zones and continuity must be checked. Evolution in Zones 2 and 3 is dependent on the constant conditions of Zone 1 and 4. The analytical equations use pressure, velocity, density, temperature, speed of sound through gas and a specific gas constant. Equations in Zone 2 are obtained using normal shock relations and the gas velocity in Zone 2 is constant throughout. The shock wave and the surface contact speeds make it possible to define the position of the zone limits.
Zone 4 Zone 1
Pressure p p 4 = 500000 [ Pa ] p 1 = 20000 [ Pa ]
Velocity v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2baaaa@375A@ v 4 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaisdaaeqaaOGaeyypa0JaaGimaaaa@3A0E@ [ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@ v 1 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaaaa@3A0B@ [ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@
Density ρ ρ 4 = 5.7487 [ k g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaai4AaiaacEgaaeaacaGGTbGaaiyBamaaCaaaleqabaGa ai4maaaaaaaakiaawUfacaGLDbaaaaa@3D0B@ ρ 1 = 0.22995 [ k g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaai4AaiaacEgaaeaacaGGTbGaaiyBamaaCaaaleqabaGa ai4maaaaaaaakiaawUfacaGLDbaaaaa@3D0B@
Temperature T T 4 = 303 [ K ] T 1 = 303 [ K ]
Speed of sound through gas:(7) a = p γ ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGHbGaey ypa0ZaaOaaaeaadaWcaaqaaiaadchacqGHflY1cqaHZoWzaeaacqaH bpGCaaaaleqaaaaa@3F1C@
Specific gas constant:(8) R = a 2 T γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbGaey ypa0ZaaSaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamiv aiabgwSixlabeo7aNbaaaaa@3EEF@
High Pressure Side (4) Low Pressure Side (1)
α a 4 =348.95 [ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@ a 1 = 348.95 [ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@
R 287.049 [ J kgK ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada WcaaqaaiaabQeaaeaacaqGRbGaae4zaiabgwSixlaabUeaaaaacaGL BbGaayzxaaaaaa@3DB3@
Table 3. Zone 2
Analytical Solution Results at t = 0.4 ms
Pressure p p 4 p 1 = p 2 p 1 ( 1 ( γ 1 ) ( a 2 a 1 ) ( p 2 p 1 1 ) 2 γ [ 2 γ + ( γ + 1 ) ( p 2 p 1 1 ) ] ) 2 γ γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai aadchadaWgaaWcbaGaaGinaaqabaaakeaacaWGWbWaaSbaaSqaaiaa igdaaeqaaaaakiabg2da9maalaaabaGaamiCamaaBaaaleaacaaIYa aabeaaaOqaaiaadchadaWgaaWcbaGaaGymaaqabaaaaOWaaeWaaeaa caaIXaGaeyOeI0YaaSaaaeaadaqadaqaaiabeo7aNjabgkHiTiaaig daaiaawIcacaGLPaaacqGHflY1daqadaqaamaaliaabaGaamyyamaa BaaaleaacaaIYaaabeaaaOqaaiaadggadaWgaaWcbaGaaGymaaqaba aaaaGccaGLOaGaayzkaaGaeyyXIC9aaeWaaeaadaWccaqaaiaadcha daWgaaWcbaGaaGOmaaqabaaakeaacaWGWbWaaSbaaSqaaiaaigdaae qaaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaaaeaadaGcaaqaaiaa ikdacqaHZoWzdaWadaqaaiaaikdacqaHZoWzcqGHRaWkdaqadaqaai abeo7aNjabgUcaRiaaigdaaiaawIcacaGLPaaacqGHflY1daqadaqa amaaliaabaGaamiCamaaBaaaleaacaaIYaaabeaaaOqaaiaadchada WgaaWcbaGaaGymaaqabaaaaOGaeyOeI0IaaGymaaGaayjkaiaawMca aaGaay5waiaaw2faaaWcbeaaaaaakiaawIcacaGLPaaadaahaaWcbe qaamaalaaabaGaeyOeI0IaaGOmaiabeo7aNbqaaiabeo7aNjabgkHi Tiaaigdaaaaaaaaa@74EA@ p 2 = 80941.1 [ Pa ]
Velocity v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2baaaa@375A@ v 2 = a 1 γ ( p 2 p 1 1 ) ( 2 γ ( γ + 1 ) p 2 p 1 + ( γ 1 ) / ( γ + 1 ) ) 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaWGHbWaaSbaaSqa aiaaigdaaeqaaaGcbaGaeq4SdCgaamaabmaabaWaaSGaaeaacaWGWb WaaSbaaSqaaiaaikdaaeqaaaGcbaGaamiCamaaBaaaleaacaaIXaaa beaaaaGccqGHsislcaaIXaaacaGLOaGaayzkaaWaaeWaaeaadaWcaa qaamaaliaabaGaaGOmaiabeo7aNbqaamaabmaabaGaeq4SdCMaey4k aSIaaGymaaGaayjkaiaawMcaaaaaaeaadaWccaqaaiaadchadaWgaa WcbaGaaGOmaaqabaaakeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaaaa kiabgUcaRmaalyaabaWaaeWaaeaacqaHZoWzcqGHsislcaaIXaaaca GLOaGaayzkaaaabaWaaeWaaeaacqaHZoWzcqGHRaWkcaaIXaaacaGL OaGaayzkaaaaaaaaaiaawIcacaGLPaaadaahaaWcbeqaamaalaaaba GaaGymaaqaaiaaikdaaaaaaaaa@5CFB@ v 2 = 399.628 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaikdaaeqaaOGaeyypa0JaaG4maiaaiMdacaaI5aGaaiOl aiaaiAdacaaIYaGaaGioaaaa@3E85@ [ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@
Density ρ ρ 2 = ρ 2 R T 2 ρ 2 = 0.5786 [ k g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaai4AaiaacEgaaeaacaGGTbGaaiyBamaaCaaaleqabaGa ai4maaaaaaaakiaawUfacaGLDbaaaaa@3D0B@
Temperature T T 1 T 2 = p 2 p 1 ( ( γ + 1 ) ( γ 1 ) + p 2 p 1 1 + ( p 2 p 1 ) ( ( γ + 1 ) ( γ 1 ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai aadsfadaWgaaWcbaGaaGymaaqabaaakeaacaWGubWaaSbaaSqaaiaa ikdaaeqaaaaakiabg2da9maalaaabaGaamiCamaaBaaaleaacaaIYa aabeaaaOqaaiaadchadaWgaaWcbaGaaGymaaqabaaaaOWaaeWaaeaa daWcaaqaamaaliaabaWaaeWaaeaacqaHZoWzcqGHRaWkcaaIXaaaca GLOaGaayzkaaaabaWaaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGL OaGaayzkaaaaaiabgUcaRmaaliaabaGaamiCamaaBaaaleaacaaIYa aabeaaaOqaaiaadchadaWgaaWcbaGaaGymaaqabaaaaaGcbaGaaGym aiabgUcaRmaabmaabaWaaSGaaeaacaWGWbWaaSbaaSqaaiaaikdaae qaaaGcbaGaamiCamaaBaaaleaacaaIXaaabeaaaaaakiaawIcacaGL PaaacqGHflY1daqadaqaamaaliaabaWaaeWaaeaacqaHZoWzcqGHRa WkcaaIXaaacaGLOaGaayzkaaaabaWaaeWaaeaacqaHZoWzcqGHsisl caaIXaaacaGLOaGaayzkaaaaaaGaayjkaiaawMcaaaaaaiaawIcaca GLPaaaaaa@6387@ T 2 = 487.308 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaS baaSqaaiaaikdaaeqaaOGaeyypa0deaaaaaaaaa8qacaaI0aGaaGio aiaaiEdacaGGUaGaaG4maiaaicdacaaI4aaaaa@3E7C@ [ K ]
Shock wave speed:(9) v s = a 1 γ + 1 2 γ ( p 2 p 1 1 ) + 1 = 663.166 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaadohaaeqaaOGaeyypa0JaamyyamaaBaaaleaacaaIXaaa beaakmaakaaabaWaaSaaaeaacqaHZoWzcqGHRaWkcaaIXaaabaGaaG Omaiabeo7aNbaadaqadaqaamaalaaabaGaamiCamaaBaaaleaacaaI YaaabeaaaOqaaiaadchadaWgaaWcbaGaaGymaaqabaaaaOGaeyOeI0 IaaGymaaGaayjkaiaawMcaaiabgUcaRiaaigdaaSqabaGccqGH9aqp caaI2aGaaGOnaiaaiodacaGGUaGaaGymaiaaiAdacaaI2aaaaa@501C@
[ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@
Therefore, x 2 1 = 0.4 v s + 500 = 765.266 [ mm ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4bWaaS baaSqaamaaliaabaGaaGOmaaqaaiaaigdaaaaabeaakiabg2da9iaa icdacaGGUaGaaGinaiaadAhadaWgaaWcbaGaam4CaaqabaGccqGHRa WkcaaI1aGaaGimaiaaicdacqGH9aqpcaaI3aGaaGOnaiaaiwdacaGG UaGaaGOmaiaaiAdacaaI2aGaaGjbVpaadmaabaGaciyBaiaac2gaai aawUfacaGLDbaaaaa@4D20@
Table 4. Zone 2'
Analytical Solution Results at t = 0.4 ms
Pressure p p 2 = p 2 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaaIYaaabeaakiabg2da9iaadchadaWgaaWcbaGaaGOmaaqa baGccaGGNaaaaa@3B75@ p 2 ' = 80941.1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaaIYaaabeaakiaacEcacqGH9aqpcaaI4aGaaGimaiaaiMda caaI0aGaaGymaiaac6cacaaIXaGaaGPaVlaadcfacaWGHbaaaa@41F9@ [ Pa ]
Velocity v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2baaaa@375A@ v 2 = v 2 ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaikdaaeqaaOGaeyypa0JaamODamaaBaaaleaacaaIYaGa ai4jaaqabaaaaa@3BE0@ v 2 ' = 399.628 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaikdacaGGNaaabeaakiabg2da9iaaiodacaaI5aGaaGyo aiaac6cacaaI2aGaaGOmaiaaiIdaaaa@3F30@ [ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@
Density ρ ρ 2 ' = ρ 3 ( x 4 / 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaaikdacaGGNaaabeaakiabg2da9iabeg8aYnaaBaaaleaa caaIZaaabeaakmaabmaabaGaamiEamaaBaaaleaacaaI0aGaai4lai aaiodaaeqaaaGccaGLOaGaayzkaaaaaa@41F6@ ρ 2 ' = 1.5657 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaaikdacaGGNaaabeaakiabg2da9iaaigdacaGGUaGaaGyn aiaaiAdacaaI1aGaaG4naiaaysW7caqGRbGaae4zaiaab+cacaqGTb GaaeyBamaaCaaaleqabaGaae4maaaaaaa@459F@ [ k g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaai4AaiaacEgaaeaacaGGTbGaaiyBamaaCaaaleqabaGa ai4maaaaaaaakiaawUfacaGLDbaaaaa@3D0B@
Temperature T ρ 2 ' = r 2 ' R T 2 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaaikdacaGGNaaabeaakiabg2da9iaadkhadaWgaaWcbaGa aGOmaiaacEcaaeqaaOGaamOuaiaadsfadaWgaaWcbaGaaGOmaiaacE caaeqaaaaa@4030@ T 2 ' = 180.096 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIYaGaai4jaaqabaGccqGH9aqpcaaIXaGaaGioaiaaicda caGGUaGaaGimaiaaiMdacaaI2aGaaGjbVlaabUeaaaa@40F3@ [ K ]

Surface contact speed: v c v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaadogaaeqaaOGaeyOeI0IaamODamaaBaaaleaacaaIYaaa beaaaaa@3B48@

Therefore, x 2 2 ' = 0.4 v s + 500 = 659.85 [ mm ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4bWaaS baaSqaamaaliaabaGaaGOmaaqaaiaaikdacaGGNaaaaaqabaGccqGH 9aqpcaaIWaGaaiOlaiaaisdacaWG2bWaaSbaaSqaaiaadohaaeqaaO Gaey4kaSIaaGynaiaaicdacaaIWaGaeyypa0JaaGOnaiaaiwdacaaI 5aGaaiOlaiaaiIdacaaI1aWaamWaaeaaciGGTbGaaiyBaaGaay5wai aaw2faaaaa@4B86@

Zone 3

Zone 3 is defined as:(10) a 4 X t v 3 a 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislca WGHbWaaSbaaSqaaiaaisdaaeqaaOGaeyizIm6aaSaaaeaacaWGybaa baGaamiDaaaacqGHKjYOcaWG2bWaaSbaaSqaaiaaiodaaeqaaOGaey OeI0IaamyyamaaBaaaleaacaaIZaaabeaaaaa@4320@

Where, x = X + 500 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4bGaey ypa0JaamiwaiabgUcaRiaaiwdacaaIWaGaaGimaaaa@3C54@

At v 3 a 3 = a 4 = 348.95 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaiodaaeqaaOGaeyOeI0IaamyyamaaBaaaleaacaaIZaaa beaakiabg2da9iabgkHiTiaadggadaWgaaWcbaGaaGinaaqabaGccq GH9aqpcqGHsislcaaIZaGaaGinaiaaiIdacaGGUaGaaGyoaiaaiwda aaa@4544@ [ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@ X = 348.95 t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHshI3ca WGybGaeyypa0JaeyOeI0IaaG4maiaaisdacaaI4aGaaiOlaiaaiMda caaI1aGaamiDaaaa@40F6@

x 4 3 ( t = 0.4 ) = 360.42 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHshI3ca WG4bWaaSbaaSqaamaaliaabaGaaGinaaqaaiaaiodaaaaabeaakmaa bmaabaGaamiDaiabg2da9iaaicdacaGGUaGaaGinaaGaayjkaiaawM caaiabg2da9iaaiodacaaI2aGaaGimaiaac6cacaaI0aGaaGOmaaaa @4697@ [ mm ]

At v 3 a 3 = v 2 a 2 ' = v 2 p 2 γ p 2 ' = 130.602 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaiodaaeqaaOGaeyOeI0IaamyyamaaBaaaleaacaaIZaaa beaakiabg2da9iaadAhadaWgaaWcbaGaaGOmaaqabaGccqGHsislca WGHbWaaSbaaSqaaiaaikdacaGGNaaabeaakiabg2da9iaadAhadaWg aaWcbaGaaGOmaaqabaGccqGHsisldaGcaaqaamaalaaabaGaamiCam aaBaaaleaacaaIYaaabeaakiabeo7aNbqaaiaadchadaWgaaWcbaGa aGOmaiaacEcaaeqaaaaaaeqaaOGaeyypa0JaaGymaiaaiodacaaIWa GaaiOlaiaaiAdacaaIWaGaaGOmaaaa@51B6@ [ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@ 348.95 t X 130.602 t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHshI3cq GHsislcaaIZaGaaGinaiaaiIdacaGGUaGaaGyoaiaaiwdacaWG0bGa eyizImQaamiwaiabgsMiJkaaigdacaaIZaGaaGimaiaac6cacaaI2a GaaGimaiaaikdacaWG0baaaa@496D@

x 3 2 ' ( t = 0.4 ) = 552.24 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHshI3ca WG4bWaaSbaaSqaamaaliaabaGaaG4maaqaaiaaikdacaGGNaaaaaqa baGcdaqadaqaaiaadshacqGH9aqpcaaIWaGaaiOlaiaaisdaaiaawI cacaGLPaaacqGH9aqpcaaI1aGaaGynaiaaikdacaGGUaGaaGOmaiaa isdaaaa@4743@ [ mm ]
Table 5. Zone 3
Analytical Solution Results at t = 0.4 ms
Pressure p p 3 p 4 = ( 1 ( γ 1 ) 2 v 3 a 4 ) 2 γ γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai aadchadaWgaaWcbaGaaG4maaqabaaakeaacaWGWbWaaSbaaSqaaiaa isdaaeqaaaaakiabg2da9maabmaabaGaaGymaiabgkHiTmaalaaaba WaaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGLOaGaayzkaaaabaGa aGOmaaaadaWcaaqaaiaadAhadaWgaaWcbaGaaG4maaqabaaakeaaca WGHbWaaSbaaSqaaiaaisdaaeqaaaaaaOGaayjkaiaawMcaamaaCaaa leqabaWaaSaaaeaacaaIYaGaeq4SdCgabaGaeq4SdCMaeyOeI0IaaG ymaaaaaaaaaa@4DE2@ p 3 = 500000 ( 1 0.2 ( v 3 ( X ) 348.95 ) ) 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaS baaSqaaiaaiodaaeqaaOGaeyypa0JaaGynaiaaicdacaaIWaGaaGim aiaaicdacaaIWaWaaeWaaeaacaaIXaGaeyOeI0IaaGimaiaac6caca aIYaWaaeWaaeaadaWcaaqaaiaadAhadaWgaaWcbaGaaG4maaqabaGc daqadaqaaiaadIfaaiaawIcacaGLPaaaaeaacaaIZaGaaGinaiaaiI dacaGGUaGaaGyoaiaaiwdaaaaacaGLOaGaayzkaaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaI3aaaaaaa@4E53@
Velocity v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2baaaa@375A@ v 3 = 2 γ + 1 ( a 4 + X t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaiodaaeqaaOGaeyypa0ZaaSaaaeaacaaIYaaabaGaeq4S dCMaey4kaSIaaGymaaaadaqadaqaaiaadggadaWgaaWcbaGaaGinaa qabaGccqGHRaWkdaWcaaqaaiaadIfaaeaacaWG0baaaaGaayjkaiaa wMcaaaaa@438E@ v 3 = 290.792 + 2.0833 X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaiodaaeqaaOGaeyypa0JaaGOmaiaaiMdacaaIWaGaaiOl aiaaiEdacaaI5aGaaGOmaiabgUcaRiaaikdacaGGUaGaaGimaiaaiI dacaaIZaGaaG4maiaadIfaaaa@44A1@
Density ρ ρ 3 ρ 4 = ( p 3 p 4 ) γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai abeg8aYnaaBaaaleaacaaIZaaabeaaaOqaaiabeg8aYnaaBaaaleaa caaI0aaabeaaaaGccqGH9aqpdaqadaqaamaalaaabaGaamiCamaaBa aaleaacaaIZaaabeaaaOqaaiaadchadaWgaaWcbaGaaGinaaqabaaa aaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcqaHZoWzaaaaaa@4507@ ρ 3 = 5.7487 ( p 3 ( X ) 500000 ) 1 1.4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCda WgaaWcbaGaaG4maaqabaGccqGH9aqpcaaI1aGaaiOlaiaaiEdacaaI 0aGaaGioaiaaiEdadaqadaqaamaalaaabaGaamiCamaaBaaaleaaca aIZaaabeaakmaabmaabaGaamiwaaGaayjkaiaawMcaaaqaaiaaiwda caaIWaGaaGimaiaaicdacaaIWaGaaGimaaaaaiaawIcacaGLPaaada ahaaWcbeqaamaaliaabaGaaGymaaqaaiaaigdacaGGUaGaaGinaaaa aaaaaa@4BF8@
Temperature T p 3 p 4 = ( T 3 T 4 ) γ γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai aadchadaWgaaWcbaGaaG4maaqabaaakeaacaWGWbWaaSbaaSqaaiaa isdaaeqaaaaakiabg2da9maabmaabaWaaSaaaeaacaWGubWaaSbaaS qaaiaaiodaaeqaaaGcbaGaamivamaaBaaaleaacaaI0aaabeaaaaaa kiaawIcacaGLPaaadaahaaWcbeqaamaaliaabaGaeq4SdCgabaGaeq 4SdCMaeyOeI0IaaGymaaaaaaaaaa@45AD@ T 3 = 303 ( p 3 ( X ) 500000 ) 1 3.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaS baaSqaaiaaiodaaeqaaOGaeyypa0JaaG4maiaaicdacaaIZaWaaeWa aeaadaWcaaqaaiaadchadaWgaaWcbaGaaG4maaqabaGcdaqadaqaai aadIfaaiaawIcacaGLPaaaaeaacaaI1aGaaGimaiaaicdacaaIWaGa aGimaiaaicdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaadaWccaqaai aaigdaaeaacaaIZaGaaiOlaiaaiwdaaaaaaaaa@48D5@
Continuity verifications:(11) v 3 ( X 3 2 ' ) = v 2 ' ( X 3 2 ' ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaiodaaeqaaOWaaeWaaeaacaWGybWaaSbaaSqaamaaliaa baGaaG4maaqaaiaaikdacaGGNaaaaaqabaaakiaawIcacaGLPaaacq GH9aqpcaWG2bWaaSbaaSqaaiaaikdacaGGNaaabeaakmaabmaabaGa amiwamaaBaaaleaadaWccaqaaiaaiodaaeaacaaIYaGaai4jaaaaae qaaaGccaGLOaGaayzkaaaaaa@458F@ (12) v 3 ( X 4 3 ) = v 4 ( X 4 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaaiodaaeqaaOWaaeWaaeaacaWGybWaaSbaaSqaamaaliaa baGaaGinaaqaaiaaiodaaaaabeaaaOGaayjkaiaawMcaaiabg2da9i aadAhadaWgaaWcbaGaaGinaaqabaGcdaqadaqaaiaadIfadaWgaaWc baWaaSGaaeaacaaI0aaabaGaaG4maaaaaeqaaaGccaGLOaGaayzkaa aaaa@4394@ (13) p 3 ( X 3 2 ' ) = p 2 ' ( X 3 2 ' ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaS baaSqaaiaaiodaaeqaaOWaaeWaaeaacaWGybWaaSbaaSqaamaaliaa baGaaG4maaqaaiaaikdacaGGNaaaaaqabaaakiaawIcacaGLPaaacq GH9aqpcaWGWbWaaSbaaSqaaiaaikdacaGGNaaabeaakmaabmaabaGa amiwamaaBaaaleaadaWccaqaaiaaiodaaeaacaaIYaGaai4jaaaaae qaaaGccaGLOaGaayzkaaaaaa@4583@ (14) p 3 ( X 4 3 ) = p 4 ( X 4 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaS baaSqaaiaaiodaaeqaaOWaaeWaaeaacaWGybWaaSbaaSqaamaaliaa baGaaGinaaqaaiaaiodaaaaabeaaaOGaayjkaiaawMcaaiabg2da9i aadchadaWgaaWcbaGaaGinaaqabaGcdaqadaqaaiaadIfadaWgaaWc baWaaSGaaeaacaaI0aaabaGaaG4maaaaaeqaaaGccaGLOaGaayzkaa aaaa@4388@ (15) ρ 3 ( X 3 2 ' ) = ρ 2 ' ( X 3 2 ' ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCda WgaaWcbaGaaG4maaqabaGcdaqadaqaaiaadIfadaWgaaWcbaWaaSGa aeaacaaIZaaabaGaaGOmaiaacEcaaaaabeaaaOGaayjkaiaawMcaai abg2da9iabeg8aYnaaBaaaleaacaaIYaGaai4jaaqabaGcdaqadaqa aiaadIfadaWgaaWcbaWaaSGaaeaacaaIZaaabaGaaGOmaiaacEcaaa aabeaaaOGaayjkaiaawMcaaaaa@4719@ (16) ρ 3 ( X 4 3 ) = ρ 4 ( X 4 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCda WgaaWcbaGaaG4maaqabaGcdaqadaqaaiaadIfadaWgaaWcbaWaaSGa aeaacaaI0aaabaGaaG4maaaaaeqaaaGccaGLOaGaayzkaaGaeyypa0 JaeqyWdi3aaSbaaSqaaiaaisdaaeqaaOWaaeWaaeaacaWGybWaaSba aSqaamaaliaabaGaaGinaaqaaiaaiodaaaaabeaaaOGaayjkaiaawM caaaaa@451E@ (17) T 3 ( X 3 2 ' ) = T 2 ' ( X 3 2 ' ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaS baaSqaaiaaiodaaeqaaOWaaeWaaeaacaWGybWaaSbaaSqaamaaliaa baGaaG4maaqaaiaaikdacaGGNaaaaaqabaaakiaawIcacaGLPaaacq GH9aqpcaWGubWaaSbaaSqaaiaaikdacaGGNaaabeaakmaabmaabaGa amiwamaaBaaaleaadaWccaqaaiaaiodaaeaacaaIYaGaai4jaaaaae qaaaGccaGLOaGaayzkaaaaaa@454B@ (18) T 3 ( X 4 3 ) = T 4 ( X 4 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaS baaSqaaiaaiodaaeqaaOWaaeWaaeaacaWGybWaaSbaaSqaamaaliaa baGaaGinaaqaaiaaiodaaaaabeaaaOGaayjkaiaawMcaaiabg2da9i aadsfadaWgaaWcbaGaaGinaaqabaGcdaqadaqaaiaadIfadaWgaaWc baWaaSGaaeaacaaI0aaabaGaaG4maaaaaeqaaaGccaGLOaGaayzkaa aaaa@4350@

Finite Element Modeling with Lagrangian and Eulerian Formulations

Gas is modeled by 200 ALE bricks with solid property TYPE14 (general solid).

The model consists of regular mesh and elements, the size of which is 5 mm x 5 mm x 5 mm.

rad_ex_fig_13-5
Figure 6. Mesh Used for Lagrangian and Eulerian Approaches

In the Lagrangian formulation, the mesh points remain coincident with the material points and the elements deform with the material. Since element accuracy and time step degrade with element distortion, the quality of the results decreases in large deformations.

In the Eulerian formulation, the coordinates of the element nodes are fixed. The nodes remain coincident with special points. Since elements are not changed by the deformation material, no degradation in accuracy occurs in large deformations.

The Lagrangian approach provides more accurate results than the Eulerian approach, due to taking into account the solved equations number.

For the ALE boundary conditions (/ALE/BCS), constraints are applied on:
  • Material velocity
  • Grid velocity

The nodes on extremities have material velocities fixed in X and Z directions. The other nodes have material and velocities fixed in X, Y and Z directions.

The ALE materials have to be declared Eulerian or Lagrangian with /ALE/MAT.

Results

Finite Element Results and Analytical Solution Comparison

Simulation results along the tube axis at 0.4 ms are shown in the following diagrams.

ex_13_pressureA
Figure 7. Pressure

ex_13_densityA
Figure 8. Density

ex_13_velocityA
Figure 9. Velocity
Lagrangian Formulation
Scale factor
Δ t s c a = 0.1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqGuoGaam iDamaaBaaaleaacaWGZbGaam4yaiaadggaaeqaaOGaeyypa0JaaGim aiaac6cacaaIXaaaaa@3E9B@
Eulerian Formulation
Scale factor
Δ t s c a = 0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqGuoGaam iDamaaBaaaleaacaWGZbGaam4yaiaadggaaeqaaOGaeyypa0JaaGim aiaac6cacaaIXaaaaa@3E9B@

Pressure Distribution

Pressure wave produced in the shock-tube at t = 0.4 ms
ex_13_eulerian_pressure ex_13_pressure-lang
Figure 10. Different Approaches and Animations Regarding Pressure, Density and Velocity
1 J. D. Anderson Jr., Modern Compressible Flow with Historical Perspective, McGraw Hill Professional Publishing, 2nd ed., Oct. 1989