The Riemann solution to the shock tube problem

The objective of this section is to find the Riemann solution of the shock tube problem introduced in Section 6.4.1. The non-linear hyperbolic system of PDEs in Eq. (6.51) with the piecewise constant initial condition

U(x,0) =

UL(ρ4, ρ₄v₄, E₄) ifx≤x₀

UR(ρ1, ρ1v1, E1) if x > x0 (6.72) define the Riemann problem for the shock tube [87,88]. Thus, the Riemann solution gives the resulting wave pattern for a flow field with discontinuous initial data. The Riemann solution of the shock tube problem therefore follows the mathematical description given in Section 6.4.2, and provides a basic understanding of the governing equations in Eq. (6.51) because all properties (such as shock and rarefaction waves) appear as characteristics in the solution.

In the following, the mathematical description of the shock tube problem is simplified by considering an infinitely long tube (see Figure 6.10), neglecting viscous effects in the flow. Hence, the following only considers the incident

6.4. General shock tube theory 147 shock and expansion (i.e., rarefaction) waves and is not valid when reflections occur at the tube ends or at obstacles inside the tube. Even though this is an idealized case, it provides an understanding of the governing physics and phenomena in the shock tube. To study reflected wave patterns and pressures due to closed ends, it is often necessary with numerical simulations.

Rarefact ionwaves

Contact surface

Shock wave (4)

(E) (3) (2)

(1) x₀

O Distancex

Timet

t= 0

High pressure (driver) p₄, ρ₄, v₄, E₄

(4)

Low pressure (driven) p₁, ρ₁, v₁, E₁

(1) Diaphragm

Particle paths

Figure 6.10: Simplified shock tube problem considering an infinitely long tube.

The diaphragm is positioned atx0 and completely removed at t= 0. Under the assumption of no dissipative phenomena, the compressible flow in the shock tube is described by the 1D Euler equations in Section 6.4.2. The solution involves discontinuities, such as the shock wave and contact discontinuity, and smooth transition waves such as the rarefaction waves. This makes it convenient to separated the tube into four uniform regions with constant parameters (see Figure 6.10). The respective regions are separated by the waves centered and originating at the initial position of the diaphragm (t= 0,x=x₀). Regions 1 and 4 are given by the initial conditions in Eq. (6.72) and the two intermediate regions, occurring after the removal of the diaphragm, are denoted 2 and 3.

An important part of the solution is to identify these regions in thex−tplane.

Since the shock and the contact discontinuity propagate in uniform zones (i.e., assuming no dissipative phenomena), the slope of these curves represent the (constant) velocity of each wave acting as lines in thex−t plane. The expansion wave extends through the new zoneE, denoted the expansion fan, in which the flow parameters vary continuously since the gas is rarefied when passing throughEfrom 3 to 4. Remember that the shock wave and the contact discontinuity propagate to the right, while the expansion wave moves to the left (i.e., ρ₄, ρ₄v₄, E₄ > ρ₁, ρ₁v₁, E₁). As before, the subindices of the conserved variables refers to the respective regions (e.g. v4 refers to the velocity in region 4).

The jump relations over the shock wave discontinuity in terms of the Mach

numberMs(=vs/c₁) are already introduced in Eq. (6.45) and Section 6.3.3. A similar expression may also be found for the particle velocityv₂[125], i.e.,

v₂

wherev2andvs are constant (see Figure 6.10). These jump relations and the information propagated along the characteristics can then be used to find the Riemann solution.

First, recall from the previous section (Section 6.4.2) that the Riemann in-variants make it possible to associate the parameters in regions 3 and 4 by considering a pointP^∗inside region 3 and going back along the characteristics to regions where the solution is already known (i.e., the initial condition).

Figure 6.11a illustrates the characteristics passing through P^∗, where it is noticed thatC⁰ andC⁺ are the only characteristics intersecting the expansion fan to search for information in region 4.

x

Figure 6.11: Illustration of the Riemann solution to the shock tube problem: (a) characteristicsCⁱand (b)x−tdiagram.

The relation between these regions are therefore given by the corresponding invariantsr⁰ andr⁺ in Eq. (6.69) as wherev₄= 0 from the initial conditions and the solution is obtained by following the characteristics back to the curve on which initial data are prescribed to determiner⁰andr⁺ in Eq. (6.69). It is also noticed that the first invariantr⁰ is related to the isentropic relation presented in Section 6.2.5 and Eq. (6.34).

Moreover, recall that the contact surface is a jump in the density (see Section 6.4.1), while the pressure and velocity remain constant and continuous across

6.4. General shock tube theory 149 this discontinuity. Thus,

v₃=v₂ , p₃=p₂ (6.75)

The solution in regions 2 and 3 are then found by combining Eqs. (6.73), (6.74) and (6.75), i.e.,

and replacingp2/p1 according to Eq. (6.45a), the solution to the shock tube problem in Eq. (6.76) may be expressed as the following implicit equation

Ms− 1 with the only unknownMs. This non-linear equation can be solved by an iterative method (e.g. Newton-Raphson), and the value of Ms is then used in Eqs. (6.45), (6.73), (6.74) and (6.75) to determine all the parameters of the uniform regions 2 and 3. The result in Eq. (6.78) is also commonly expressed in the literature as [125]

p4 which, by using the jump relation in Eq. (6.45a), may be expressed in terms of the shock strength as [202]

p₄

Figure 6.12 shows the respective solutions in Eqs. (6.79) and (6.80) for the resulting Mach number Ms(Figure 6.12a) and shock strengthp2/p1(Figure 6.12b) at given initial conditions ofp4 andp1.

0 50 100 150 200

Figure 6.12: Riemann solution to the shock tube problem as a function of the initial conditionsp4/p1: (a) Mach number Ms in Eq. (6.79) and (b) the shock strength p2/p1 in Eq. (6.80). Note that these solutions are only valid fort < tbin Figures 6.6 and 6.7.

It is emphasized that the solutions in Eqs. (6.79) and (6.80) are only valid until the formation of the blast wave in Figures 6.6 and 6.7 (i.e., fort < t_b).

To complete the solution, it is necessary to determine the range of each region, i.e., to calculate the values of the abscissasx1,x2, x3 andx4in Figure 6.11b for a given timet. Starting with the expansion fan E (see Figure 6.11a), it is observed that this is left-bounded by theC⁻ characteristic starting from the point x0 and considered to belong to region 4 (i.e., the line of slope dx/dt = v4−c4 = −c4). The right bound of the expansion fan is the C⁻ characteristic starting from the same pointx₀, however, now considered to belong to region 3 (i.e., the line of slopedx/dt=v3−c3). Thus, according to Eq. (6.61), the values ofx4 andx3 are given as

x4=x0−c4t , x3=x0+ (v3−c3)t (6.81) Note that if v3 > c3, i.e., v3 is supersonic, the tail of the rarefaction wave propagates to the right although the front of the wave is left-running.

Now, considering a point (x, t) inside the regionE(x4≤x≤x₃), the properties of the expansion fan can be established. Since this point belongs to theC⁻ characteristic starting fromx0, Eq. (6.61) states that

dx

dt =x−x0

t =v−c (6.82)

Using theC⁺ characteristic from pointLand the corresponding invariantr⁺

6.4. General shock tube theory 151

(i.e., v₄+ 2c4/(γ−1) =v+ 2c/(γ−1)), it further observed that c4=c+γ−1

2 v (6.83)

Combining the last two equations and remembering the relations from Eq. (6.74) and Eq. (6.77), the solution inside the expansion fan is given by

v= 2 γ+ 1

c₄+x−x0

t

, c=c₄−(γ−1)v 2 , p=p4

c c₄

_γ^2γ₋₁

, ρ=ρ4

p p₄

_γ¹ (6.84)

Since the contact discontinuity is transported at constant velocity v₃=v₂, its location is given by

x2=x0+v2t=x0+v3t (6.85) wherev₂ is found from Eq. (6.73) sinceMsis known from Eq. (6.78). Finally, the shock wave propagating at constant velocityvsis located at

x₁=x₀+vst (6.86)

From this it is shown that the solution U(x, t) of the shock tube problem is only dependent on the ratiox/t, and is therefore commonly expressed in the x−tplane in the literature.