# Refraction of Oblique Shock Wave on a Tangential Discontinuity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}transforms into a system with parameters f

_{2}are called shock wave processes, where f

_{1}and f

_{2}are some gas dynamic variables before and behind the gas dynamic discontinuity.

**M**= 1–5 and various combinations of ratios of specific heat capacities of two flows: 1.67 (monatomic gas), 1.4 (air), 1.11 (hydrocarbon fuel and air mixture), 1.25 (combustion products). Refraction phenomena in various engineering problems (hydrocarbon gaseous fuel and its combustion products, diatomic gas, fuel mixture of oxygen and hydrogen, etc.) are discussed.

## 2. A Brief Theory of an Oblique Shock Wave

_{2}/p

_{1}(shock intensity). Subscript 1 corresponds to flow quantities before shock, and subscript 2 corresponds to flow quantities behind shock.

_{1}and p

_{1}, and the flow quantities behind the shock are denoted by M

_{2}and p

_{2}.

_{n}and u

_{t}are the projections of the velocity vector onto the directions normal to the discontinuity plane and tangential to it (Figure 1).

^{2}on a normal shock when the angle of inclination of the shock is σ = 90°. Using the Mach number

**M**and the expression for the speed of sound ${a}^{2}=\gamma p/\mathsf{\rho}$ (γ is the adiabatic index, γ = c

_{p}/c

_{v}, c

_{p}is the specific heat capacity at constant pressure, c

_{v}is the specific heat capacity at constant volume), and also by expanding the velocity vector of the incoming flow into the components u

_{n}and u

_{t}. Taking into account the fact that u

_{1t}= u

_{2t}, after simple transformations from (1) and (3), one can obtain an equation that relates the intensity of the shock to the angle of its inclination:

**M**link J, β and σ. These equations are referred to as dynamic compatibility conditions on the oblique shock wave. These equations for a given freestream Mach number define a closed curve (heart-shaped curve) as shown in Figure 2. Since each

**M**number has its own curve, shock polars are also called isomachs. It is convenient that the polar begins at the origin of coordinates [0, 0]; therefore, it is usually constructed in the variables (lnJ, β).

## 3. Regular Refraction

_{1}is the incident shock wave, σ

_{2}is the refracted shockwave, ω

_{3}is the reflected rarefaction wave, σ

_{3}is the reflected shockwave, ν

_{3}is the reflected weak discontinuity, τ is the tangential discontinuity, ← is the left-moving discontinuity, → is the right-moving discontinuity, T is the refraction point, and

**M**is the Mach number. The subscript + denotes the flow quantities in original flow above τ, and subscript—denotes the flow quantities in the original flow under τ. For example,

**M**

_{+}and

**M**

_{−}denote Mach numbers in the relevant flow regions (above and below tangential discontinuity).

_{1}on the tangential discontinuity leads to an increase (Figure 3a) or decrease (Figure 3b) in its inclination angle. In the particular case in which the reflected discontinuity is a discontinuous characteristic, only the curvature of the incident shock wave changes. This refraction is called characteristic.

_{1}does not change. Then, the conditions of dynamic compatibility on τ are written in the form of the equality of the angles of flow turn:

_{σ}(γ

_{−},

**M**

_{−}, J

_{c}) = β

_{σ}(γ

_{+},

**M**

_{+}, J).

_{m}= (1 + ε)

**M**

^{2}− ε.

**M**

_{ω}, J

_{2}/J

_{1}is substituted for J, and ${\widehat{M}}_{1}$ is substituted for

**M**. Then, the Prandtl–Mayer function ω(

**M**) is found for

**M**

_{ω}and ${\widehat{M}}_{1}$.

## 4. Characteristic Refraction

**M**

_{+}) and shock polar J(

**M**

_{−}), which is realized at a strictly defined intensity of the initial oblique jump J

_{c}= J

_{c}(

**M**

_{+},

**M**

_{–}, γ

_{+}, γ

_{–}) at which Equation (9) is satisfied. The condition for the equality of the intensities of the shocks J

_{c}(

**M**

_{+}, γ

_{+}) = J

_{c}(

**M**

_{–}, γ

_{–}) reduces to the cubic equation

_{+}= γ

_{–}, ε

_{+}= ε

_{–}, Γ

_{+}= Γ

_{–}, and Equation (8) is simplified:

**M**< 2

^{1/2}, since one of them lies completely inside the other (Figure 5).

_{m}must also be satisfied. Then, both shocks, the original shock σ

_{1}and the refracted shock σ

_{2}, must be straight lines. Therefore, the second condition is that curves corresponding to characteristic refraction have an envelope

**M**

_{–}= ∞. Solutions of Equation (16) with limitations (17) and (18) are shown in Figure 6.

_{+}≠ γ

_{–}, it is necessary to solve the cubic Equation (8). Unlike the previous case, the solution can have one to three roots. Since z ≥ 0, it follows from (8) that when one of the roots is equal to zero (z = 0), the polars J(γ

_{–},

**M**

_{–}) and J(γ

_{+},

**M**

_{+}) have the same derivatives at the origin (Γ

_{–}= Γ

_{+}), since A

_{0}= 0. From the equality Γ

_{–}= Γ

_{+}, it follows that, between the numbers

**M**

_{–}and

**M**

_{+}, there must be a relation

_{+}≥ γ

_{–}. Therefore,

**M**

_{–}and

**M**

_{+}. These equations allow us to construct the domains of existence (Figure 7).

## 5. Domains of Existence

_{τ}, the reflected discontinuity r

_{3}can be either a compression shock σ

_{3}or a rarefaction wave ω

_{3}. If r

_{3}is a shock wave, then, at certain intensities J

_{1}of the incident shock, the solution of Equation (5) may not exist, since polar (3) emitted from point (1) corresponding to J

_{1}does not intersect with polar (1) (Figure 9). The shock wave structure corresponding to MRef is shown in Figure 10.

_{1}branches with the formation of a reflected shock σ

_{3}and a main shock σ

_{2}(the Mach stem). The main shock experiences characteristic refraction on the tangential discontinuity τ. The Mach stem is enclosed between two points, T and T

_{1}, from which tangential discontinuities originate, τ and τ

_{1}. Figure 11 shows the regions of existence of a shock wave structure with a different type of reflected discontinuity r

_{3}(Figure 1), which are formed during refraction of the shock σ

_{1}at the tangential discontinuity τ.

**M**

_{–}= 2 and

**M**

_{+}< 2, there are quite extensive regions with both the reflected shock wave σ

_{3}and the reflected rarefaction wave ω

_{3}. As

**M**

_{–}increases, the domains of existence of the reflected σ

_{3}at

**M**

_{+}<

**M**

_{–}are significantly reduced (Figure 12 and Figure 13).

_{1}(gray area in Figure 11, Figure 12 and Figure 13) is bounded from above by the intensity of the incoming shock J = J

_{s}. It should also be noted that the line corresponding to the characteristic refraction at

**M**

_{+}<

**M**

_{–}gradually shifts to the left and enters the MRef region. This is especially noticeable in Figure 13. The red arrow shows a thin line corresponding to the solution in the plane of the shock polar in Figure 14. This line marks the characteristic refraction.

_{4}is subsonic. Between these two lines lies a region with a reflected rarefaction wave. Consequently, depending on the initial conditions, either RRef with a reflected wave or MRef with a shock wave structure can be realized as shown in Figure 14b.

## 6. Conclusions

**M**= 2, the line ($\overleftarrow{R}=\overleftarrow{\mathsf{\nu}}$) corresponding to the characteristic refraction is the boundary separating the various types of reflected discontinuity, shock wave ($\overleftarrow{R}=\overleftarrow{\mathsf{\sigma}}$) or rarefaction wave ($\overleftarrow{R}=\overleftarrow{\mathsf{\omega}}$). At

**M**= 5, however, this boundary does not separate two types of reflection, but is a line of a jump-like change in intensity. The vertical boundary at

**M**

_{-}=

**M**

_{+}shows a line corresponding to a weak tangential discontinuity, where there is equality of flow velocities and adiabatic indices on both sides. The shock region to the left of this boundary practically disappears with an increase in the Mach number to 5. With an increase in the Mach number, the point of intersection of this vertical boundary with the characteristic refraction line shifts significantly upward, approaching the intensity boundary J = J

_{s}, at which the flow behind the incoming shock of the compaction moves at a speed equal to the speed of sound. This leads to the fact that the area of reflection in the form of a rarefaction wave to the right of the vertical boundary is significantly reduced.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**M**

_{–}= 2 are shown in Figure A1.

**Figure A1.**Domains of existence of reflected discontinuities for refraction from a gas consisting of combustion products into a diatomic gas. The Mach number of the flow from which the shock comes is

**M**

_{–}= 2.

**M**

_{–}= 3 are shown in Figure A2.

**Figure A2.**Domains of existence of reflected discontinuities for refraction from a gas consisting of combustion products into a diatomic gas. The Mach number of the flow from which the shock comes is

**M**

_{–}= 3.

**M**

_{–}= 5 are shown in Figure A3.

**Figure A3.**Domains of existence of reflected discontinuities for refraction from a gas consisting of combustion products into a diatomic gas. The Mach number of the flow from which the shock comes is

**M**

_{–}= 5.

**M**

_{–}= 2 are shown in Figure A4.

**Figure A4.**Domains of existence of reflected discontinuities for refraction from a gas consisting of combustion products into a monatomic gas. The Mach number of the flow from which the shock comes is

**M**

_{–}= 2.

**M**

_{–}= 3 are shown in Figure A5.

**Figure A5.**Domains of existence of reflected discontinuities for refraction from a gas consisting of combustion products into a monatomic gas. The Mach number of the flow from which the shock comes is

**M**

_{–}= 3.

**M**

_{–}= 5 are shown in Figure A6.

**Figure A6.**Domains of existence of reflected discontinuities for refraction from a gas consisting of combustion products into a monatomic gas. The Mach number of the flow from which the shock comes is

**M**

_{–}= 5.

**M**

_{–}= 2 are shown in Figure A7.

**Figure A7.**Domains of existence of reflected discontinuities for refraction in a flow of diatomic gas. The Mach number of the flow from which the shock comes is

**M**

_{–}= 2.

**M**

_{–}= 3 are shown in Figure A8.

**Figure A8.**Domains of existence of reflected discontinuities for refraction in a flow of diatomic gas. The Mach number of the flow from which the shock comes is

**M**

_{–}= 3.

**M**

_{–}= 5 are shown in Figure A9.

**Figure A9.**Domains of existence of reflected discontinuities for refraction in a flow of diatomic gas. The Mach number of the flow from which the shock comes is

**M**

_{–}= 5.

**M**

_{–}= 2 are shown in Figure A10.

**Figure A10.**Domains of existence of reflected discontinuities for refraction from a monatomic gas into a gas consisting of combustion products. The Mach number of the flow from which the shock comes is

**M**

_{–}= 2.

**M**

_{–}= 3 are shown in Figure A11.

**Figure A11.**Domains of existence of reflected discontinuities for refraction from a monatomic gas into a gas consisting of combustion products. The Mach number of the flow from which the shock comes is

**M**

_{–}= 3.

**M**

_{–}= 5 are shown in Figure A12.

**Figure A12.**Domains of existence of reflected discontinuities for refraction from a monatomic gas into a gas consisting of combustion products. The Mach number of the flow from which the shock comes is

**M**

_{–}= 5.

**M**

_{–}= 2 are shown in Figure A13.

**Figure A13.**Domains of existence of reflected discontinuities for refraction from a diatomic gas into a gas consisting of combustion products. The Mach number of the flow from which the shock comes is

**M**

_{–}= 2.

**M**

_{–}= 3 are shown in Figure A14.

**Figure A14.**Domains of existence of reflected discontinuities for refraction from a diatomic gas into a gas consisting of combustion products. The Mach number of the flow from which the shock comes is

**M**

_{–}= 3.

**M**

_{–}= 5 are shown in Figure A15.

**Figure A15.**Domains of existence of reflected discontinuities for refraction from a diatomic gas into a gas consisting of combustion products. The Mach number of the flow from which the shock comes is

**M**

_{–}= 5.

**M**

_{–}= 3.27 corresponding to Chapman–Judge detonation are shown in Figure A16.

**Figure A16.**Domains of existence of reflected discontinuities for refraction from a hydrogen/oxygen mixture at Mach number

**M**

_{–}= 3.27 corresponding to Chapman–Judge detonation.

**M**

_{–}= 5.46 corresponding to Chapman–Judge detonation are shown in Figure A17.

**Figure A17.**Domains of existence of reflected discontinuities for refraction from a propane/air/combustion product mixture at Mach number

**M**

_{–}= 5.46 corresponding to Chapman–Judge detonation.

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**Figure 2.**Shock polar (Busemann curve). The point l corresponds to the shock with the maximum possible flow turning angle, and the point s is the sonic point.

**Figure 3.**Regular refraction of oblique shock wave σ

_{1}on tangential discontinuity τ: (

**a**) refraction with reflected rarefaction wave, (

**b**) refraction with reflected shock wave, (

**c**) characteristic refraction, where the reflected discontinuity is the discontinuous characteristic.

**Figure 4.**Three types of regular refraction on shock polar: (

**a**) refraction with reflected rarefaction wave, (

**b**) refraction with reflected shock wave, (

**c**) characteristic refraction.

**Figure 6.**The values of the intensity J

_{c}of the incident shock for characteristic refraction. The dashed line shows the envelope corresponding to the case when the refracted shock has the maximum intensity J

_{2}= J

_{m}.

**Figure 7.**The domains of existence of characteristic refraction at the interface between the hot hydrocarbon fuel mixture (γ = 1.1) and the products of its combustion (γ = 1.25), where domains 1, 2 and 3 are the first, second and third roots of Equation (8).

**Figure 8.**Dependence of the intensity J

_{c}of the incident shock corresponding to the characteristic refraction on the Mach numbers

**M**

_{–}and

**M**

_{+}in flows separated by a tangential discontinuity. The dotted line shows the dependencies corresponding to the case when the refracted shock has the maximum intensity J

_{2}= J

_{m}.

**Figure 9.**Regular refraction when the flow behind the reflected flow is supersonic (

**a**) and subsonic (

**b**).

**Figure 11.**Domains of existence of various types of hydrocarbons formed during refraction of a shock with intensity J propagating from a flow with a Mach number

**M**

_{−}= 2 into a flow with a Mach number

**M**

_{+}. The grey area corresponds to irregular refraction with a supersonic flow

**M**

_{1}> 1 behind a shock σ

_{1}, and white area corresponds to irregular refraction with a subsonic flow behind a shock σ

_{1}.

**Figure 12.**Domains of existence of various types of hydrocarbons formed during refraction of a shock with intensity J, propagating from a flow with a Mach number

**M**

_{–}= 3 into a flow with a Mach number

**M**

_{+}.

**Figure 13.**Domains of existence of various types of hydrocarbons formed during refraction of a shock with intensity J, propagating from a flow with a Mach number of

**M**

_{–}= 5 to a flow with a Mach number of

**M**

_{+}.

**Figure 14.**Ambiguity of the solution on the polar plane: irregular refraction (A), characteristic refraction (B): (

**a**) solutions on the plane of shock polar, (

**b**) shock wave structure corresponding to the Mach refraction MRef.

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**MDPI and ACS Style**

Bulat, P.; Melnikova, A.; Upyrev, V.; Volkov, K.
Refraction of Oblique Shock Wave on a Tangential Discontinuity. *Fluids* **2021**, *6*, 301.
https://doi.org/10.3390/fluids6090301

**AMA Style**

Bulat P, Melnikova A, Upyrev V, Volkov K.
Refraction of Oblique Shock Wave on a Tangential Discontinuity. *Fluids*. 2021; 6(9):301.
https://doi.org/10.3390/fluids6090301

**Chicago/Turabian Style**

Bulat, Pavel, Anzhelika Melnikova, Vladimir Upyrev, and Konstantin Volkov.
2021. "Refraction of Oblique Shock Wave on a Tangential Discontinuity" *Fluids* 6, no. 9: 301.
https://doi.org/10.3390/fluids6090301