Journal of Energy and Power Engineering 8 (2014) 121-136 D DAVID PUBLISHING

The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction

Vladimir Uskov1 and Mikhail Chernyshov2 1. Department of Plasma and Dynamics, Baltic State Technical University, Saint Petersburg 190005, Russia 2. Department of Extreme Phenomena in Materials and Blast Safety, Saint Petersburg State Polytechnical University, Saint Petersburg 195251, Russia

Received: March 04, 2013 / Accepted: July 02, 2013 / Published: January 31, 2014.

Abstract: The interaction of the oblique stationary shock with the preceding Prandtl-Meyer expansion or compression wave is studied theoretically and numerically. Two analytical models are proposed for the flow description, which, though being approximate, gives us the solution of the problem with very high accuracy level. Owing to proposed analytical models, distinctive features of the flow in the interaction region, curvilinear shock inflection, the reflected expansion/compression wave type change, the degeneration of the resulted waves (for example, of the oblique shock into weak discontinuity), occurrence of the subsonic pockets downstream the interacting shock, are discovered and characterized analytically.

Key words: Supersonic flow, shock-expansion interaction, analytical models and solutions.

Nomenclature    1  1  Pressure function of the isentropic flow J i The strength of the i-th stationary shock of  Shock inclination angle (relative to streamlines Prandtl-Meyer wave (i.e., the relation of the static pressures upstream) downstream and upstream it)  Flow angle relating to chosen horizontal direction j i-th stationary shock i  i i-th weak tangential (contact) discontinuity (the

K  Geometrical curvature of the shock slipstream)  Prandtl-Meyer wave direction index M  Slope angle of the straight characteristic line N i i-th “basic non-uniformity of the flow” n Natural coordinate (the direction normal to the  Shock inclination angle (relative to chosen streamline) horizontal direction) p Static pressure  Wave type index ( 1 for the compression wave

p0 Full (stagnation) pressure and   1 for the expansion one)   Prandtl-Meyer function ri i-th Prandtl-Meyer compression or expansion wave s Natural coordinate (streamline direction) M i , p i , etc. Flow parameters downstream the i-th shock or x , y Horizontal and vertical directions Prandtl -Meyer wave  Mach angle M , N i Flow parameters downstream the curvilinear interacting shock  i Flow deflection angle on the i-th shock or Prandtl-Meyer wave 1. Introduction  The ratio of gas specific heats  Flow symmetry factor The problems of interactions between  The function of specific heats ratio, Prandtl-Meyer waves and oblique stationary shocks can be treated as classical in the theory of Corresponding author: Mikhail Chernyshov, Dr. Sc. discontinuities interactions in gas dynamics. Owing to (Engr.), professor, research fields: gas dynamics, shock interaction blast inhibitors. E-mail: [email protected]. the numerical methods, these problems can be solved

122 The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction in each separate case. But, in spite of the long history of shock generation, is the length scale OA = 1 . j studies [1], full theoretical analysis of the solution has Initially, the straight shock 2 becomes curvilinear not been executed yet. A lack of exact analytical at its intersection with the last characteristic OB of r relations as well as the complexity of the flow (the the wave 1 . The reflected weak discontinuity BB 1 interaction is not localized at a single point and even in comes out at the point B , if the flow behind the shock j any finite region) is the main obstacles in the way of 2 is supersonic. This discontinuity is a border theoretical study. between the uniform flow region 2 behind the shock,

With the problem to be solved now, an initially and the reflected perturbations in the region r3 . r uniform gas stream with Mach number M 0 (Fig. 1) Due to contiguity theorem, the region 3 is either a turns along the plane wall, and simple (non-centered, rarefaction or compression Prandtl-Meyer wave.

Figs. 1a and 1c) or centered (with the only common Weak tangential (contact) discontinuity τ 1 (“weak” point of all straight acoustic characteristics, Figs. 1b means the discontinuity of not flow parameters, but of and 1d) Prandtl-Meyer wave r1 originates. their first spatial derivatives) originates from point B j τ The oblique stationary shock 2 of the same downstream the shock. The discontinuity 1 is the direction is situated downstream the isentropic wave. lower border of the substantially non-isentropic flow The supersonic gas flow in the region 1 between the region 5 which can also be named the vortex layer, or r j wave 1 and the shock 2 can be characterized by its the slipstream of finite width. Another weak tangential τ Mach number, M 1 and flow angle, θ1 . Here, we admit discontinuity 2 which originates from the point C that the flow angle θ1 = 0 (i.e., the flow direction is of intersection between the shock and the first straight horizontal in the chosen coordinate system). The characteristic of the wave r1 is the upper border of the distance between the point O of the intersection of the vortex region 5. The weak discontinuity CC1 also last Prandtl-Meyer straight characteristic OB with the goes out from the point C and encloses the reflected streamlined surface (in particular, O is the center of the wave r3 if the flow downstream the shock is really rarefaction wave in Fig. 1b), and the corner point A of supersonic at the point C .

Fig. 1 Interaction of the Prandtl-Meyer wave with the proceeding oblique shock: (a) the asymptotic degeneration of the shock inside the expansion fan; (b) the shock interaction with the centered wave; (c) the shock strengthening under the compression wave r1 influence; (d) the case of the subsonic flow downstream the shock j2.

The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction 123

If the Prandtl-Meyer expansion wave is too strong, then their spatial derivatives at the non-uniform stream of a the curvilinear shock degenerates into one more weak perfect inviscid gas, we introduce so-called in Ref. [2] discontinuity inside the expansion fan (Fig. 1a). The “basic non-uniformities” given from two-dimensional discontinuity τ 2 does not exist there, and non-interacted flow equations in natural coordinates: 2 part r4 of the wave r serves instead of the shock j4 M −1 ∂θ 1 + + N sinθ = 0 , which comes through the expansion fan otherwise. γM 2 ∂n 4

The oblique shock j4 (Figs. 1b-1d) or the 2 γM N 2 = − ∂ ln p ∂n , expansion wave r4 (Fig. 1a) of the same direction as ∂p ∂s = 0 . j 0 the wave r1 and the shock 2 (direction index χ = 1) Here, N1 = ∂ ln p ∂s expresses the flow as well as the reflected isentropic wave r3 of the non-isobaricity, N = ∂θ ∂s is the geometric opposite direction ( χ = −1 ) result due to the 2 curvature of streamlines, the quantity N 3 = ∂ ln p0 ∂n interaction considered. If the flow downstream the characterizes the flow vorticity (i.e., its non-isentropic shock is subsonic, then the Prandtl-Meyer wave r3 features), and N 4 = δ y expresses the type of flow does not exist, and the results of interaction sufficiently symmetry ( N 4 ≡ 0 for the plane flow considered here). depend upon the perturbation induced downstream. We Also, here s and n are natural coordinates counted do not consider the latter type of interaction, except of along and normally the sreamlines, correspondingly; several complementary remarks. M is the local Mach number; p is the static pressure; We should emphasize that the wave r3 is also partly p0 is the full (stagnation) pressure of the flow; y is a result of the refraction of the perturbations reflected the distance to the axis of symmetry in axis-symmetric from the shock in the region BCC 2 on the vortex flow flow; δ = 0 and δ = 1 for the plane flow and for the layer. At this refraction, the perturbations reflected axis-symmetric one, correspondingly; γ is the ratio of from the non-isentropic layer 5 are generated in their specific heats (γ = 1.4 everywhere if not specifically turn. These reflected perturbations distribute along the mentioned otherwise). characteristics of the first family, overtake the shock The geometrical curvature Kσ of the shock situated upstream them, influence its features and conventionally remarked here as the additional shape, and make it curvilinear even after point C. That non-uniformity ( N 5 ≡ Kσ ). We admit that the is why the shock j4 can not be formally called straight, curvature K σ > 0 for the shocks convex downwards and the flow behind it (in the region 4) is not exactly (for instance, the shocks BE in Fig. 1b, and BC in uniform and isentropic. Fig. 1c), and K σ > 0 for shocks convex upwards (e.g., The main goals of the present study are to define and the shock EC in Fig. 1b, and the shock after the point analyze the shape and the other features of the E in Fig. 1a). interacting shock, to find out the type of the resulted perturbations and their transition criteria, as well as 2.2 Prandtl-Meyer Flow Non-uniformities and the some special features of the considered interaction, and Envelope Line of Its Straight Characteristics the influence of the ratio of gas specific heats on the The variation of gas stream parameters at problem solution. Prandtl-Meyer wave can be described by using the 2. Materials and Methods isentropic flow functions. So the wave intensity, i.e., the relation J1 of the static pressures behind the wave 2.1 Non-uniformities of the Two-Dimensional Gas Flow and after this one:

To characterize not only flow parameters, but also J1 = π (M 1 )(π M 0 ) (1)

124 The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction

− γ ()γ − 1 where, π ()M = ()1 + 0 .5 (γ − 1 )M 2 is the characteristic of the corresponding reflected wave sector. isentropic pressure function. The Prandtl-Meyer wave We also designate Mach number of the flow just after deflection angle is: the voluntary point of the interacting shock as ) ) ˆ β1 = χ()ω()()M 0 −ω M (2) M ≡ M (ϕ ), the flow deflection angle at the interaction Here, χ = 1 is the wave direction index, and: shock as β , and the shock strength (relation of static ω()M = 1 ε arctan ε (M 2 −1) − arctan M 2 −1 (3) pressures downstream the shock and upstream it) as J . The Mach lines of the Prandtl-Meyer wave r have is the Prandtl-Meyer function, such as: 1 a discriminant curve. This discriminant curve can be a dω()M ()1− ε M 2 −1 = (4) smooth envelope (curve O O′ in Figs. 1a and 2a) for dM M ()1+ ε ()M 2 −1 0 a simple (non-centered) wave r1 . When the part of the ε = γ −1 γ +1 ()() straight Mach lines intersects at the single point, the Analogously, the strength of the reflected wave r3 discriminant curve has a sharp bend (Fig. 2b); when all and flow deflection angle are: Mach lines have the common point, the Prandtl-Meyer J = p p = π M π M (5) 3 3 2 ( 3 )(2 ) wave becomes centered, and the discriminant curve

β 3 = χ()ω()M 3 − ω ()M 2 (6) degenerates into the sole point—the center of either the where, χ = −1; p2 , M 2 , p3 and M 3 are pressures rarefaction wave (the point O ≡ O0 ≡ E1 in Fig. 1b) or and Mach numbers of the flow in regions 2 and 3 (Fig. the compression one. If one part of the Prandtl-Meyer 1b), consequently. wave realizes the flow rarefaction, and the another part Later we have to deal with not only the whole waves realizes the flow deflection, the envelope line consists r of two separate sections (Fig. 2c). r1 and 3 but with their parts the has just interacted. Such is the sector of the wave r1 bounded (b) by the characteristics OB and E1 E in Fig. 1b. The (a) (ϕ ) (ϕ ) flow deflection angle β1 and the strength J1 of this part of the wave are expressed by the formulas: ()ϕ β1 = χ()ω()M −ω (M1 ) (7) ()ϕ J1 = π ()(M π M 0 ) (8) Here, M ≡ M (ϕ) is the Mach number at the characteristic E1 E determined by the angle ϕ of its inclination:

ϕ = θ 0 + χ[]ω(M 0 )−ω ()M +α ()M (9) (c) where, α(M ) = arcsin(1 M ) is the Mach angle. In a similar manner, the strength of the corresponding sector of the reflected wave r3 after the interaction between the shock and the above-mentioned sector of the wave r1 is determined as:

()ϕ ()ϕ J 3 = π (M 3 ) π ()M 2 (10) Fig. 2 Discriminant curves of Prandtl-Meyer waves: (a) the The flow deflection angle is smooth envelope of the expansion wave; (b) the expansion ()ϕ ()ϕ wave with a centered sector; (c) the transition of the β 3 = χ()ω()M 3 − ω()M 2 (11) expansion wave into the compression one and the ()ϕ where， χ = −1, and M 3 is Mach number at the last corresponding discontinuity of the envelope curve.

The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction 125

ρ The shape of the envelope of the straight Mach lines f given in the voluntary direction q inside ρ family is defined by any given streamline of this wave Prandtl-Meyer wave depends on the angle between q unambiguously. Let the only streamline equation and gradient vector grad f which is normal to the y ()x (for example, the section AB of the straight characteristics: 0 0 ρ impermeable surface in Fig. 2a) to be given, as well as ∂f ∂q = q ⋅grad f (16) the initial Mach number M 0 and flow angle θ 0 . An The Eqs. (9), (13) and (16), and the well-known equation of one-parameter family of the straight functions of the isentropic flow allow us to calculate all characteristics looks like: flow parameters derivatives along the direction of the

y = y0 ()x0 + tanϕ(x0 )(⋅ x − x0 ) (12) oblique shock coming through Prandtl-Meyer fan of where, x0 is the parameter. characteristics situated upstream. The inclination angle ϕ of any straight Mach line 2.3 Dynamic Compatibility Conditions at Stationary and Mach number M on this one are defined by ′ Shocks y 0 ()x 0 through Eq. (9) and the following: ′ Among the numerous variables for shock description, y0 = tan[]θ 0 + χ()ω()()M 0 −ω M (13) we choose the shock strength J (i.e., the relation of The equation y Γ ()x Γ for the envelope of the the static pressures downstream and upstream the straight family of Mach lines is derived from Eqs. (3), shock) as the main parameter. We denote below that J (4), (9), (12) and (13) at the condition ∂y ∂x 0 = 0 is the strength of the shock inside the Prandtl-Meyer [3]: fan, J 2 = p2 p1 is the strength of the shock j2 , and χ 1−ε M 2 −1 ()( ) J j xΓ − x = 3 cosϕ 4 is the strength of the resulting shock 4 . Here, p1 M N 2 is the pressure in zone 1, and p2 is the pressure in χ()1−ε (M 2 −1) y − y = sin ϕ zone 2 of the flow (Fig. 1). Γ M 3 N 2 The shock strength here is the main quantity binding 2 ()1−ε (M −1) flow parameters before and behind the shock. For r = 3 N 2 M example, Mach numbers M 2 (downstream the shock j Here, r = C0C1 is the distance from the voluntary 2 ) and M 1 (upstream the shock) relate between point C1 at a straight characteristic to the themselves as it follows: corresponding point C at the envelope line. In 0 J + ε M 2 − 1−ε J 2 −1 x ≡ x y ≡ y ( 2 )()1 ()2 particular, at 0 , 0 , N 2 is the curvature of M 2 = (17) J 2 ()1+ εJ 2 the given streamline, and the envelope y Γ ()x Γ can be easily built. The flow deflection angle β 2 also depends upon The two first basic non-uniformities of the the shock strength: Prandtl-Meyer flow are inversely proportional to the ⎡ J M − J ⎤ m ()1 2 ()()1 − ε J 2 −1 (18) β2 = arctan⎢ ⎥ distance r to the envelope along the characteristic line: ⎣⎢ J 2 + ε J m ()M1 + ε − (1 − ε )()J 2 −1 ⎦⎥

2 2 ()1+ ε M −1 Here, J m (M ) = (1+ ε )M −ε is the strength of the N1 =ψ (14) rM normal shock in the flow with Mach number M . The ()1−ε (M 2 −1) relations analogous to Eqs. (17) and (18) are correct for N 2 = χψ 3 (15) rM the shock j4 : (here ψ = 1 for the compression wave, and ψ = −1 ()()J + ε M 2 − 1−ε ()J 2 −1 for the expansion one). M = 4 0 4 (19) 4 J ()1+ εJ The spatial derivative of any variable flow parameter 4 4

126 The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction

⎡ J − J 1− ε J −1 ⎤ m0 4 ()()4 (20) β4 = arctan⎢ ⎥ ⎣⎢ J 4 + ε J m0 + ε − ()()1− ε J 4 −1 ⎦⎥

(here, J m0 ≡ J m ()M 0 ) and also for any point at the interacting part BC of the shock considered: ) ()J + ε M 2 − ()1− ε ()J 2 −1 M = (21) J ()1+ εJ

⎡ J M − J 1− ε J −1 ⎤ m () ()() (22) β = arctan⎢ 2 ⎥ ⎣⎢ J + ε ()1+ ε M − ()()1− ε J −1 ⎦⎥ The angle σ of shock inclination relative to the flow velocity vector upstream the given point is also bound with shock strength ( J ) and local Mach number upstream ( M ): (a) J = ()1+ ε M 2 sin 2 σ −ε (23) In the coordinate system admitted here, Eq. (23) determines the dependence of shock slope angle ξ (Fig. 1) on the shock strength: (ϕ ) ξ = σ + β1 (24) ′ that is an equivalent to the quantity y σ ()x σ where

y σ (x σ ) is the shock shape looked for.

Except of J m ()M , we must emphasize here the following special shock strengths: the strength of the shock declining the flow to a maximum angle possible at a given Mach number: (b) 2 Fig. 3 The parametric solution of the interaction problem M 2 − 2 ⎛ M 2 − 2 ⎞ ⎜ ⎟ 2 considered: (a) general solution; (b) the region of small Jl ()M = + ⎜ ⎟ + ()1+ 2ε ()M −1 + 2 2 ⎝ 2 ⎠ Mach numbers. and the strength of the shock reducing the flow velocity Several forms of that unambiguous conditions have downstream it to the critical speed: been published already (e.g., in Refs. [4, 5]; the 2 M 2 −1 ⎛ M 2 −1⎞ ⎜ ⎟ 2 universal form for strong discontinuities in the solution J*()M = + ⎜ ⎟ + ε ()M −1 +1 2 ⎝ 2 ⎠ of two-variables quasi-linear system is given in Ref.

Dependencies J * (M ) , J l (M ) and J m (M ) are [6]). given in Fig. 3 (curves 1, 2 and 3, consequently). At all To express the correlation between the basic ) j = 1...3 Mach numbers M >1 , the inequality non-uniformities downstream ( N j , ) and N 1 < J * ()M < J l ()M < J m ()M is correct. upstream ( i , i = 1...5 ) the shock with the strength Eqs. (17)-(22) and a lot of similar others express J occurred in the flow with Mach number M , we use zero-order dynamic compatibility conditions (i.e., they the differential dynamic compatibility conditions in the bind flow parameters, not their spatial derivatives). To following form [2]: connect the spatial derivatives of the flow parameters ) 5 N j = C j ∑ A ji N i (25) downstream and upstream the shock, we should use i=1 first-order (differential) dynamic compatibility conditions. where, C j and Aji are factors depending on M , J ,

The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction 127 shock direction χ (χ = 1 everywhere at the present the shock, as seen from Eq. (25), it is enough to know study), and the flow angle θ (only at axis-symmetric the geometrical curvature of the shock ( N 5 ≡ Kσ ). flow). The factors we are interested in look as: 2.4 Interaction of the Oblique Shock with Preceding C = γ b J −1 , 1 ( ()) Overtaking Weak Discontinuities 3 С2 = ()1 − ε J ( b ()J −1 ), O C The last ( OB ) and the first ( 0 , Figs. 1b-1d) straight characteristics of the wave r are the weak 1 J + ε 2 , 1 A11 = − 3 []ap ()()Jm − J − bp Jm − J + cp γ ()Jm + ε discontinuities, i.e., the discontinuities of the basic

non-uniformities N1 and N 2 . Since these χs 2 A12 = − []aθ ()J + ε + bθ ()J + ε + cθ , 1 + ε non-uniformities are equal to zero in zone 0 and 2 correspond to Eqs. (14) and (15) inside Prandtl-Mayer A15 = −χ()1−ε ()aω J − bω J − cω , wave, we can determine the discontinuities [N1 ] and χs 2 A = []f + ()J −1 c f γ , [N ] at the characteristic line O0 C : 21 J 1 m 2 2 ()1+ ε M 2 −1 c ⎡ J m + ε ⎤ []N1 =ψ (26) A22 = − ⎢ f1 + ()J m − J f 2 ⎥ , rM J ⎣ 1−ε ⎦ (1−ε )(M 2 −1) N =ψ (27) A = a J ⋅ []2 3 25 rM ⋅{}2m + J ()()(1+ ε []J − J J +1+ 2ε )()()− 1+ εJ J −1 m On the contrary, the flow is non-uniform (also where: corresponds to Eqs. (14) and (15)) before the Mach line 1 2 a = []()()J m − J J + ε , OB and uniform behind it. So the discontinuities of b = a 2 + ()1+ εJ 2 , the basic non-uniformities at the last characteristic of

1 2 the Prandtl-Meyer wave are: c = []()()J + ε J m + ε , ()1+ ε M 2 −1 1 2 []N1 = −ψ (28) s = []()()J m − J J m + ε , rM 2 a p = ()(1−ε 3J m − 4 −ε ), (1−ε )(M −1) []N 2 = −ψ 3 (29) 2 2 2 rM bp = ()3 − 2ε Jm − 2()()1− ε + 3ε Jm − 4 + 6ε − 3ε , The distance r for centered or non-centered wave, c p = ()()()J m −1 1+ εJ m J m + ε , as it was shown in Subsection 2.2, can be easily a = 1−ε J + ε − 4 1+ ε , Θ ()[]m () determined owing to the given equation of the 2 bΘ = 2()()()1+ ε ()1− 3ε J m + ε − 2()1−ε , streamlined surface. 2 It is worthy of notice from Eqs. (26)-(29) that the cΘ = ()()()1+ ε 1−ε J m + ε , relation: aw = 3 + ε , [N1 ]− χΓ()M []N 2 = 0 b = 3J − 2 − 3ε , w m expresses the dynamical conditions of compatibility at cw = ()1+ 4ε J m +1; weak discontinuities of gas flow parameters (here 2 2 2 ; χ = 1 for wave r with the f1 = mJ + 2μ()()()J + ε ()m + 1+ ε J m − J J , Γ(M ) = γM M −1 1 2 straight characteristics of the first family). f 2 = 2m −γJ ()()()1+ εJ m J + ε − ()1−ε ()J m − J , The problem of the interaction of the stationary m = ()()()1,+ εJ []b − 1+ ε J 1+ εJ shock with the preceding overtaking weak −1 μ = −()()()1+ εJ m []1+ ε J m + ε . discontinuity has the following solution [2] in variables To define the flow non-uniformities downstream accepted here:

128 The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction

) []K C ()()ΓA + A + C Γ ΓA + A shock with the remaining part of the wave does not σ = 1 11 12 2 ) 21 22 (30) []N 2 C1 A15 + C2 ΓA25 change the shock section EC . So the assumption of ) ) zero shock curvature at the point of its exit out of the Here, Γ ≡ Γ ()M , Γ ≡ Γ (M ) , M is the Mach ) Prandtl-Meyer fan is equivalent to the possibility of number upstream the point of the intersection, and M is the Mach number behind the shock at this point (for using the relation analogous to Eq. (30) at all inner ) M = M points of the interacting shock: example, 1 and M = M 2 at point B ). ) Eqs. (27), (29) and (30) allow us not only to define C1 (ΓA11 + A12 )(+ C 2 Γ ΓA21 + A22 ) K σ = ) ⋅ N 2 (31) the discontinuities of shock curvature at points B and C1 A15 + C 2 ΓA25 C , but also to determine the shock curvature just after Considering Eqs. (15), (21) and (31), we conclude the point B quite exactly ( Kσ = []Kσ , Eq. (29)). that the dimensionless curvature of the interacting Declaring that the curvature of the shock is equal j4 shock depends only on wave type ψ = ±1, the shock to zero right after the point C , we can determine the strength J and the local Mach number M upstream: ) shock curvature of the interacting shock BC just C (ΓA + A )(+ C Γ ΓA + A ) rK =ψ ⋅ 1 11 12 2 ) 21 22 × before this point ( K = −[]K , Eq. (27)). But, as it was σ C A + C ΓA σ σ 1 15 2 25 shown in Section 1, this assumption is not absolutely ()1− ε ()M 2 −1 × . exact and needs some proof for its introduction. M 3

3 2 '' ' 2 Since K σ = y σ (1 + (y σ ) ) , where, y σ (x σ ) is 2.5 Problem Solution Based on the Assumption of Zero the shape of the shock, Eq. (31) allows us to determine Curvature of the Resulting Shock the shape of the shock in the interaction region. Eqs. The calculations carried out by the second-order (17)-(23) binding shock shape and strength and Eqs. method of characteristics at utmost asymptotic refining (2)-(4), (9) and (12)-(16) which allow us to calculate of the numerical grid have shown that the shock j4 is flow parameters derivatives at all points and in all of an extremely small curvature (in particular, at point directions inside the Prandtl-Meyer fan lead to the C ). For example, in case of the interaction with the following equations complementary to Eq. (31): centered expansion fan, its curvature is equal to dJ dM = 2(J + ε ) f3 M , (32) −6 2 ⋅10 at M 1 = 1.5 , J1 = 0.8 and J 2 = 1.4 ; to 2 ()1− ε M −1 J m ()M − J −5 −6 M = 3 J = 0.4 J = 4 f 3 = 1+ ⋅ ⋅ f 4 , 3 ⋅10 at 1 , 1 and 2 ; to 4 ⋅10 1+ ε ()M 2 −1 J + ε at M 1 = 5 , J1 = 0.3 and J 2 = 10 . The intersection ) C (ΓA + A )(+ C Γ ΓA + A ) with the weak discontinuity O C diminishes the f = 1 − 1 11 12 2 21 ) 22 , 0 4 M sin()σ − α ⋅ ()C A + C ΓA shock geometric curvature approximately by 1,200 1 15 2 25 times in the first case, by 800 times in the second case, where, to exclude trigonometric expressions, and by 4,500 times in the third one. So we can conclude ()J + ε ()M 2 −1 − J ()M − J sin σ −α = m () 2 that the curvature of the shock j4 is very small M 1+ ε compared with the curvature of the interacting shock (the intermediate calculations are contained in Ref. BC and can be accepted to be equal to zero. Then the [7]). curvature of the shock BC at point С can be easily Taking into account that the derivative dJ dM determined from Eqs. (27) and (30): K σ = −[]K σ . does not depend on upstream flow non-uniformities, Let us now consider the shock which has interacted we must conclude that Eq. (32) is most convenient to not with the whole expansion fan, but with its part solve the problem. Neither initial data ( M 1 , J 2 , etc.) bounded by the straight characteristics OB and E1 E nor local non-uniformities, but only local Mach (Figs. 1a and 1b). The following interaction of this number M and shock strength J participate in this

The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction 129 equation. The integral curves of Eq. (32) are shown in expansion-compression polars. Figs. 3a and 3b as slim lines. The shock strength Like Eq. (30) in the previous model, Eqs. (33) and decrease under the influence of the rarefaction wave (34) can be generalized for the inner points of the corresponds to the motion downward these curves; the interacting curvilinear shock: (ϕ )()ϕ increase of the strength J at the interaction with the J1 J 2 J 3 = J (35) compression wave corresponds to the motion upward. (ϕ ) (ϕ ) β1 (M (ϕ), J 1 )+ β 2 ()M 1 , J 2 + Mach number at sonic line 1 ( J = J * ()M there) or on (36) ()ϕ ()ϕ abscissa axis ( J =1 ; that corresponds to the shock + β 3 ()M 2 , J 3 = β ()M ()ϕ , J degeneration) can be considered as the only parameter. Here, M (ϕ ) is Mach number at a straight In fact, the assumption of zero curvature of the characteristic coming into the given point of the shock; resulted shock implied the neglect to the influence of (ϕ ) J 1 = π (M 1 ) π (M (ϕ )) is the strength of the the perturbations refracted at the entropy layer on the Prandtl-Meyer wave part which have just interacted shock parameters. These assumptions concerning the (ϕ )()ϕ with the shock; J 3 = π (M 3 ) π ()M 2 is the strength moving shock waves in non-steady flow underlie of the corresponding part of the reflected wave r3 ; Chester, Chisnell and Whitham’s approximate (ϕ ) M 3 is the Mach number at the last characteristic of analytical methods [8]. Analogous assumptions (ϕ ) ()ϕ this reflected part; β 1 and β 3 are the flow concerning shocks in steady flow were introduced in deflection angles at the named parts of the waves r1 Ref. [9] and other studies. and r3 . Shock strengths J 2 and J , corresponding 2.6 The Solution Based on the Static Pressure Equality deflection angles, and the Mach numbers are tied here and Collinearity of Flow Directions Downstream the by Eqs. (17), (18), (21) and (22). Corresponding flow Resulting Waves parameters at Prandtl-Meyer wave sectors are bound by Eqs. (7), (8), (10) and (11). J The straightness of the shock 4 in Fig. 1 is The system Eqs. (35) and (36) determines the local possible only at the absence of the perturbations which shock strength J and deflection angle β in any overtakes the shock from behind. The equality of static inner point of the interacting shock. pressures and the collinearity of the streams on both Differentiating Eqs. (35) and (36), we can get the sides of vortex layer 5 is a necessary (not sufficient, expressions analogous to Eq. (31) for the shock though) condition for it. These conditions can be curvature, and the expressions analogous to Eq. (32) formulated as the system binding wave strengths J i for the shock strength variation. But, unlike in Eqs. (31) β and deflection angles i (i = 1...4 ): and (32), the initial parameters ( M 1 and J 2 ) are J1J 2 J 3 = J 4 (33) present in these expressions which become rather more β ()()()M , J + β M , J + β M , J = 1 0 1 2 1 2 3 2 3 (34) complicated [7]. So the solutions of Eqs. (33) and (34) = β 4 ()M 0 , J 4 do not compose the one-parameter family of curves at

Shock strengths J 2 and J 4 , corresponding the (M , J ) -plane. The system Eqs. (33) and (34) is deflection angles, and Mach numbers are tied together definitely not an integral of Eq. (32). with Eqs. (17)-(20), and the same flow variables Conditions Eqs. (33) and (34) were, in fact, upstream and downstream Prandtl-Meyer waves are introduced in Ref. [10] to solve the problem of the bound with Eqs. (1), (2), (5) and (6). interaction between the shock and Prandtl-Meyer wave

Using Eq. (33) in logarithmic form ( Λ i = ln J i ), it is of opposite direction [11]. Eqs. (33)-(36) are formally not difficult to solve the system Eqs. (33) and (34) analogous to the compatibility conditions in the case of owing to the well-known technique of shock and two overtaking shocks intersections. But Eqs. (33) and

130 The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction

(34) give us, as a rule, only an approximate solution of The errors in the proposed models are small enough the problem because we neglect the influence of relatively to both the shock strength total variation refracted perturbations on flow downstream the shock ( J 2 − J 4 ) and the pressure variation in the reflected wave j r r J = 0.4 4 and the wave 3 . 3 . For example, J 3 = 1.014 at M 1 = 3 , 1 and

J 2 = 4 , and the errors of the resulting shock strength 3. Results and Discussion according to the above-mentioned models are −4 −5 3.1 The Accuracy of the Methods Proposed ΔJ 4 =1.8 ⋅10 and ΔJ 4 = 7 ⋅10 , correspondingly. The model [1] based on the neglect to the reflected The results achieved due to the “differential” model waves at all ( J ≡ 1 ) is qualitatively less accurate Eqs. (31) and (32) and the “integral” one Eqs. (35) and 3 ( ΔJ ≈ 0.02 ÷ 0.4 for the cases shown in Figs. 4a-4d). (36) demonstrate a very high accuracy. So, owing to very high order of smallness of the The errors ΔJ = J − J of the shock strength e neglected perturbations reflected from the vortex layer computation resulting at the application of Eqs. (31) and and influencing the shock from behind, the solutions (32) and Eqs. (35) and (36) are shown in Fig. 4 relative proposed here are accurate enough to study the type of to the series of incoming centered expansion wave the reflected wave r3 and other special features of the strengths J1 given shock interacts with. Here, J e ’s are interaction considered. the strengths of the interacting shock at the same M 1 ,

J 2 and J1 ’s calculated by second-order method of 3.2 The Inflections of the Interacting Shock characteristics with asymptotic grid refining and accepted The relation Eq. (31) determines the geometric here as being exact. Curves 1 in Fig. 4 show us the errors curvature of the interacting shock in an explicit form of the model based on Eqs. (31) and (32), and curves 2 though this formula can not be applied to the shock —of the model given by Eqs. (35) and (36). with the subsonic flow downstream. In the critical case ) ( J = J * (M ), M →1 ) the shock curvature is finite, and

Eq. (31) with the coefficients A2i computed at

J = J * (M ) confirms it. As it follows from Eq. (31), rather strong shocks interacting with the expansion wave are convex downwards (for example, the section BE of the shock (a) (b) in Fig. 1b). The loss of the shock strength under the further influence of the expansion wave leads to shock inflection, and the curvature becomes negative (such is the section EC ). The negative curvature of the shock degenerating inside the expansion fan into a weak discontinuity as shown in Fig. 1a asymptotically strives

(c) (d) to zero. If the shock interacts with the compression Fig. 4. The errors in the approximate analytical solutions wave, the sign of its curvature changes in a reverse Eqs. (31) and (32) and Eqs. (35) and (36) at the interaction of a given shock with a series of the centered expansion order. waves: (a) M 1 = 1.5 , J 2 =1.4 , J1 = 0.8...1.0 ; (b) The location of the shock inflection point is

M 1 = 2 , J 2 = 2.5 , J1 = 0.5...1.0 ; (c) M 1 = 3 , determined by Eq. (31) at the condition K σ = 0 . The

J 2 = 4 , J1 = 0.4...1.0 ; (d) M 1 = 5 , J 2 =10 , shock curvature direction changes when shock

J1 = 0.3...1.0 . parameters cross curve 4 in Fig. 3a. Curve 4 goes out

The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction 131

the point r at the “critical” (“sonic”) line 1. The finite the wave r3 can be analyzed based on Eqs. (35) and limit of the shock curvature determined from Eq. (31) (36). The second-family characteristic where the wave ) r at M →1 (J → J * (M )): 3 changes its type corresponds to the extremum of ΓA ()()M , J + A M , J current strength J (ϕ ) of the reflected wave part: K = 21 * 22 * ⋅ N 3 σ A ()M , J 2 25 * J (ϕ ) (J (ϕ ) )→ extr (37) leads to the equations 3 1 1 M 6 5 9 M 4 12 1 2 M 2 16 1 0 with the limitations Eqs. (35) and (36) binding (ϕ ) ()− ε r − (− ε )r + ( − ε )()r − − ε = J1 and (ϕ ) in an implicit form. The solution of Eq. (37) for Mach number M r =1.676 and J 3 3 2 J r − 2J r − 3εJ r − ()1−ε = 0 depends only on the local Mach number M and the shock strength J : for the shock strength J r = J * ()M r = 2.361 at the 3 point r where the inflection line starts. Inflection line 2i ∑ Di M = 0 (38) i = 0 4 has the horizontal asymptote ( J = J A = 6.360 ) at J 2 2 2 M → ∞ where, A is the root of the following D3 = J (1+ ε ) − 4ε (J + ε ) , equation: 2 D2 = 4ε (1− ε )(J + ε )(J −1)− 6 i 2 2 2 ∑ Bi J A = 0 , − 2()1− ε J ()()()J −1 − 4 1− 2ε J + ε , i=0 D1 = (1− ε )⋅ 2 4 B = ε ()1 − ε , 2 2 2 2 6 ⋅ []4()1− 2ε ()J −1 ()()()()J + ε + 4 J + ε + 1− ε J J −1 , 2 2 3 B = 2ε ()1−ε ()1− 5ε + 3ε − 3ε , 2 5 D = −4(1−ε ) (J + ε )(J 2 −1). B = 1−16ε + 47ε 2 − 48ε 3 + 71ε 4 − 0 4 Eq. (38) determines the curves ϕ i ( i = 1,2 ) at − 64ε 5 + 9ε 6 +16ε 7 , 2 3 4 5 (M , J )-plane (Figs. 3a and 3b). If r1 is the expansion B3 = −6 + 26ε − 36ε +100ε −158ε − 30ε − Prandtl-Meyer flow then the regions I and II − 72ε 6 −16ε 7 +16ε 8 , correspond to the reflection of compression waves, and B = ()1− 2ε ⋅ 2 2 3 4 5 6 7 the region III to the reflection of rarefaction waves. The ⋅ ()1−10ε + 42ε + 40ε +161ε +122ε + 44ε +16ε , contrary conclusions are correct if r1 is the B = −8ε 2 ()1− 7ε + 6ε 2 − ε 3 + 7ε 4 +14ε 5 + 4ε 6 , 1 compression wave. 2 B =16ε 4 ()1 − ε − ε 2 . ϕ 0 Curves i emerge from the points Fi at the axis M = 1.245 When we try to determine the shock curvature from J =1. The corresponding Mach numbers ( F1 , M = 2.540 Eqs. (35) and (36), the shock inflection depends not F2 ) are determined by the relation: only on the local parameters ( M and J ) but also on M F = 2(1± ε ) (1± 2 ε ) 1,2 the initial data for the incoming shock ( M 1 and J 2 ). But the results of the applications of the two proposed These special Mach numbers were d discovered models can be barely distinguished. So for M 1 = 3 earlier in the problems of overtaking shock-shock and J 2 = 6 the shock inflection occurs at M = 2.808 anshock—weak discontinuity interactions. They occur and J = 4.557 according to Eq. (31). Determining the also in the study of the systems which consist of shock curvature from Eqs. (35) and (36), it was Prandtl-Meyer waves and subsequent overtaking calculated that the inflection point corresponds to shocks and provide extreme static pressure at fixed M = 2.806 and J = 4.556 . total flow deflection [12, 13].

The curve ϕ1 reaches the “sonic” line 1 at the point 3.3 The Reflected Wave and the Change of Its Type s ( M s = 1.305 , J s = J * (M s ) = 1.466 ) with the Change of flow types (expansion or compression) in coordinates determined by the equations:

132 The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction

4εM 6 + (3+ 2ε − 9ε 2 )M 4 + We have to note the following correlation of the s s 2 2 2 basic non-uniformities immediately after every inner + 4()1−8ε + 6ε M s −16()1−ε = 0, 3 2 2 2 point of the curved shock: 4εJ s + 3()1−ε J s − ()5 − 2ε + 5ε J s + ) ) ) N + ΓN = 0 (40) + ()1− 3ε −ε 2 −ε 3 = 0. 1 2 which is analogous to the relation correct everywhere The coordinates ( M u = 2.089 , J u = 1.989 ) of the inside Prandtl-Meyer wave r3 . The Eq. (40) is the point u where the curve ϕ 2 has a vertical tangent are determined by more complex expressions (for example, special case of generalized Chester-Whitham theorem can be presented as a root of an eighth-order proven in Ref. [8]. J u ()ε ) Analyzing Eq. (39) for , we transformed it to algebraic equation). To find out its coordinates it is N1 = 0 easier to solve the system which consists Eq. (38) and the same Eq. (38) which was achieved earlier r the following equation: concerning wave 3 change of type. The coincidence 3 of the solutions subsequent to “differential” model (for E M 2i = 0 , ∑ i u the points situated immediately after the shock) and i=0 2 2 “integral” model (for the reflected wave) reveals that E3 = 2()1−ε J −8ε , 2 2 2 3 the flow type seems not to change along the E2 = −2()3()1− ε J + 2()1− 4ε − ε + 2ε J + 2ε ()3 − 5ε , E = 2()1 − ε ⋅ second-family reflected characteristics, even in zone 5. 1 ⋅ []2()1− ε J 3 + 3 (1− 3ε )J 2 + ()()1− ε 5 + 8ε J − 2 (1− 4ε ), Flow type change in the reflected wave can be 2 2 illustrated by the influence of the expansion wave on E0 = 4()1− ε ()1− 2εJ − 3J . the shock with initial “sonic” strength As it is seen in Fig. 3a, one of the integral curves of J 2 = J * = 8.282 at M 1 = 3 (the integral curve KK 1 Eq. (32) has point v of contact with the curve ϕ 2 . To in Fig. 3). The sector KK 2 of this curve corresponds achieve its coordinates ( M v = 2.282 , J v = 3.434 ), we to the flow expansion in the reflected wave (from ϕ 2 have to equate the slopes of the curve and the (ϕ ) M 2 =1 to M 3 =1.0208 ). The flow compression in integral curve of Eq. (32). The thick integral curve 5 of the reflected wave begins at M = 2.860 and Eq. (32) descending from the point ( M = 2.670 , J = 6.543 (point K ). Shock inflection point K J = J (M ) = 6.440 ) at the “sonic” line 1 to the point 2 3 * ( M = 2.572 , J = 4.258 ) is situated in this sector. The ( M = 1.477 , J =1 ) contacted the curve ϕ 2 at the Mach number in the reflected wave diminishes to point v . All the integral curves situated to the right of M (ϕ ) =1.0007 at the end of the compression sector curve 5 correspond to at least one reflected wave type 3 K K ( M = 2.089 and J = 2.016 at point K ). change point. 2 4 4 But the flow at the reflected compression wave does The curve ϕ 2 asymptotically reaches the line not decelerate to the critical speed because the new J = M 2 −1 at M → ∞ . K expansion sector begins after the point 4 till the very Using the “differential” curved shock model Eqs. J →1 (31) or (32) and differential conditions (25) of the shock degeneration into the weak discontinuity ( at M = 1.630 ; point K1 ). The Mach number in the dynamic compatibility at the interacting shock as well reflected wave strives there to 1.0016 , and the integral as Eqs. (14) and (15) we went to the following strength J of the reflected wave strives to 0..9981 correlation of flow non-uniformities upstream and 3 The change of the reflected wave type under the downstream the shock: ) ) influence of the compression wave on the oblique N 1 N 2 = = shock occurs in an inverse order. N 1 N 2 (39) C C []A ()()Γ A + A − A Γ A + A Comparing the data on the reflected wave obtained = − 1 2 15 21 22 ) 25 11 12 C 1 A15 + C 2 Γ A 25 here with the analytical results in Ref. [13] where the

The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction 133

sequence of the Prandtl-Meyer wave and the shock is line ϕ1 (M b = 2.506 , J b = 4.611 ), the points d considered without the interaction between them, we ( M d = 1.464 , J d = J * ()M d =1.804 ) and e r must conclude in the following way. The wave 3 ( M e = 3.013 , J e = J * (M e ) = 8.358 ) where the reflected as a result of the interaction of the wave r1 curves hi finish at “sonic” line 1. The integral curves s (flow deflection angle β1 ) with the shock 2 (flow of Eq. (32) outgoing the sector be of the curve h2 β deflection angle 2 ) always strives to reduce have two intersection points with the curve ϕ 2 . This gradually the pressure and the total deflection angle means that the reflected wave r3 which consists of downstream it and, consequently, these parameters two expansion sectors and a compression one can be behind the resulting wave s4 (r4 ), to the parameters divided mentally into two waves of unit strength. after the sole shock (wave) in a flow with Mach 3.5 Mach Numbers Downstream the Shocks, Subsonic number M which deflects the stream on an angle 0 Flow Pockets equal to β1 + β 2 . Basing on Ref. (38), we can consider this conclusion as proven analytically. Two contradictory factors influence on the variation ) of the Mach number M immediately downstream the 3.4 The Degeneration of the Resulting Shocks and interacting shock. On the one hand, this curvilinear Waves shock weakens inside the expansion fan, its strength J diminishes, and that leads to a Mach number increase The condition J 4 = 1 is introduced as supplementary to Eqs. (33) and (34) corresponds to behind the shock. But, on the other hand, the Mach the shock degeneration at the very first (upstream) number upstream the shock decreases, and it leads to a decrease in the Mach number downstream. characteristic of the Prandtl-Meyer fan r1 . The total Computations show that Mach number just deflection angle in the incoming waves β1 + β 2 ≠ 0 generally in this case. The overall angle of flow downstream the shock increases as a rule at large and moderate Mach numbers upstream and diminishes at deflection far from the streamlined surface strives to small Mach numbers. β1 + β 2 asymptotically mainly due to the ) The equation dM dM = 0 analyzed together with disturbances occurring at wave r3 reflection from Eqs. (21) and (32) which bind the shock strength and this surface. Mach numbers on its sides determines curve 6 (i.e., the Inasmuch as some sectors of the reflected wave r3 line kn in Fig. 3b) of the local Mach numbers maxima realize the expansion flow, and some sectors the just behind the interacting shock. Here M k = 1.127 , compression one, they also mutually compensate each J k =1. The coordinates of the point n (M n =1.257 , other sometimes. Supposing both J 3 = 1 and J 4 = 1 J n =1.376 ) are the only real roots of the equations in Eqs. (33) and (34), we define the special type of the (1− ε )M 6 − (4 − 5ε )()(M 4 + 7 − 8ε M 2 − 5 − 4ε )= 0 , interaction between the expansion wave and the n n n 3 oblique shock. These conditions determine the curves J n − (1+ ε )J n −1 = 0 . hi (i =1, 2 ) in Figs. 3a and 3b. When the shocks A local decrease in the downstream Mach number which correspond to these curves degenerate to weak that corresponds to the region Okn can lead to the discontinuities, they leave mutually annihilating flow transition downstream the shock through the ) disturbances in the reflected wave contemporarily. critical speed ( M =1). It can become earlier than the

The curves hi begin at the above-mentioned shock wave degenerates into a weak discontinuity. So points Fi . We can indicate the coordinates of the the integral curves of Eq. (32) situated in the area marginal left point a of the curve h2 (M a = 2.230 , between “sonic” line 1 and dotted integral line 7, or

Og M g = 1.391 J g =1.640 J a = 2.081 ), the point b of its intersection with the ( , ) both start and finish at

134 The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction the “sonic” line (to the right of point g and to the left on the solutions derived here, it seems enough to study of it, correspondingly). Considering only the the variation of the special curves and the points plotted supersonic flow downstream the shock, we can above. ϕ conclude that the integral curve starting from point n Starting points Fi of the curves i merge at M = M = 2 finishes at the same point at once. γ →1 ( F1 F2 then). A distance between

When the shock j2 interacts with the preceding these curves increases with the growth of γ . So the expansion wave r1 , the shock strength diminishes, and region II which corresponds to the expansion wave full pressure losses at the interacting shock become reflection at shock interaction with incoming smaller. The Mach number immediately behind the expansion wave widens then. interacting shock is always larger than the Mach The first of the start points Fi moves to smaller M =1.245 number at the corresponding characteristic of the Mach numbers region (so F1 at γ = 1.4 , M =1.189 M → 2 3 γ → ∞ reflected wave. So, when the flow immediately after F1 at γ = 3 , F1 at ). ϕ the shock sometimes decelerates to a critical and even The same is correct for the whole 1 (F1s ) line (so subsonic velocity, it means that the supersonic flow in M s →1.318 and J s →1.434 at γ →1 ; the reflected mainly compression wave r3 have M s =1.305 and J s =1.466 at γ =1.4 ; M →1.272 decelerated to the critical velocity sooner. M s =1.288 and J s =1.529 at γ = 3 ; s

According to the model Eqs. (33) and (34), the and J s →1.618 at γ → ∞ ). So region I where the critical flow deceleration in the reflected compression reflected wave r3 is of another type than incoming wave is possible for incoming shocks which wave r1 becomes narrower. correspond to the dashed regions “A” and “B” adjacent The curve kn of the maximum downstream Mach to the curves 8a (Fig. 3b) and 8b (Fig. 3a) if the numbers also exists at all 1< γ < ∞ moving a little to incoming expansion wave r1 is strong enough. Points the larger Mach numbers (so M n →1.253 and d and e at the “sonic” line whose coordinates are J n →1.325 at γ →1 ; M n =1.257 and M = 1.464 calculated in Subsection 3.4 ( d , J n =1.376 at γ = 1.4 ; M n =1.263 and J n =1.567 M = 3.013 e ) are the endpoints of “A” and “B” regions, at γ = 3 ; M n →1.272 and J n →1.618 at γ → ∞ ). correspondingly. The region of possible flow Points n and s coincide at γ → ∞ . deceleration immediately downstream the shock Another reflected wave type change curve ϕ 2 bounded by the integral curve Og and “sonic” line 1 moves sufficiently to the right. So the coordinates of its is situated inside the region “A” of flow deceleration in extreme left point u are the following: M u =1.790 the reflected wave. and J u =1.436 at γ =1.2 ; M u = 2.089 and

Possible solutions for the strong incoming shocks J u =1.989 at γ = 1.4 ; M u = 2.539 and J u = 2.980 (at J (M ) < J < J (M ) , so M <1 , Fig. 1d) at γ = 1.66 . The Mach number M also increases * 1 2 m 2 F2 ignore the influence of the other disturbances from the ( M =1.876 at γ = 1.2 , M = 2.540 at γ = 1.4 , F2 F2 subsonic flow downstream the shock to this one. So it M =16.325 at γ = 1.66 ) and strives to infinity at F2 seems that the solutions for the strong shocks cannot γ = 5 3 . The lower branch of the curve ϕ 2 has a be mentioned here due to their low practical horizontal asymptote ( J =1 ) in the last case. The importance. further increase of the specific heats ratio leads to a corresponding shift of the curve ϕ to the right (so 3.5 The Influence of Gas Specific Heats Ratio on the 2 M = 6.027 and J = 16.098 at γ = 3 ; M → ∞ Problem Solution u u u and J u → ∞ at γ → ∞ ). The curve h2 and the To characterize the influence of specific heats ratio region 3b corresponding to shock degeneration and

The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction 135 critical flow deceleration in the reflected waves also differential dynamic compatibility conditions on a shift to the right. The rules determining the direction of curved shock in a generally non-uniform flow, two shock curvature remain the same for other γ ’s. Mach reliable analytical solutions for overtaking numbers and shock strengths corresponding to the Prandtl-Meyer wave—oblique shock interaction were shock inflection increase a little: for example, obtained.

M r →1.659 and J r → 2.206 at γ →1 ; When used together, these solutions allows us to

M r =1.676 and J r → 2.361 at γ = 1.4 ; estimate both the flow parameters and their spatial

M r → 3 and J r → 3 at γ → ∞ . The inflection derivatives downstream the shock as well as the curve 2 always has the horizontal asymptote J = J A distinctive features of the interacting shock and

(e.g., J A →13.061 at γ → ∞ ). reflected Prandtl-Meyer wave. So the geometrical So the increase in specific heats ratio widens the curvature and the inflection points at the interacting expansion wave reflection region (as r1 is the shock were determined analytically, in particular. expansion wave) and shifts the majority of the special A simple expression determining the curves and points plotted here to the region of larger rarefaction/compression flow type change immediately Mach numbers and shock strengths. behind the shock as well as in the reflected wave was The dependence of some specific Mach numbers derived also. It was proven that the reflected upon the specific heats ratio in the practically important Prandtl-Meyer wave can change its type, not once range (1 < γ < 2 ) is shown in Fig. 5. One can see, for somewhere. The criteria of the full mutual M example, that some specific small Mach numbers ( k , compensation of the different sectors of the reflected M M M M M n , s , g , d , r ) are weakly dependent on wave and the criteria of the shock degeneration were γ in the range shown there. also derived and studied. 4. Conclusions Non-monotonic variation of the Mach numbers just downstream the curvilinear interacting shock was Owing to the expressions derived for the differential discovered, and the possibility of subsonic flow parameters of Prandtl-Meyer flowfield and the pockets was revealed. The influence of the ratio of gas

specific heats on the deduced solutions was found more quantitative than qualitatively remarkable. The most evident applications of the presented results are: the optimal design of supersonic inlets; the design of optimally shaped aerodynamic bodies for supersonic flight; the analysis of supersonic jet technology apparatus because of the sequential interchange of compression and expansion flow regions divided by the shocks in the “barrels” of supersonic jet flows.

References [1] R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Fig. 5 Variation of the specific Mach numbers in the Waves, Wiley Interscience, New York, 1948. practically important range of γ ’s. According to the [2] A.L. Adrianov, A.L. Starykh, V.N. Uskov, Interaction of citation order: M (curve 1), M F and M (curves 2 r 1 F2 Stationary Gas-Dynamic Discontinuities, Novosibirsk, and 3, correspondingly), M (curve 4), M (5), M (6), s u v Nauka, 1995. (in Russian) M (7), M (8), M (9), M (10), M (11), M a b d e k n [3] V.N. Uskov, M.V. Chernyshov, Conjugation of M (12) and g (13). Prandtl-Meyer wave with quasi-one-dimensional flow

136 The Interaction of Prandtl-Meyer Wave with the Oblique Shock of the Same Direction

region, Mathematical Modeling 15 (6) (2003) 111-119. (in [9] A.L. Adrianov, A Model of the Flow Downstream the Russian) Curvilinear Stationary Shock, Krasnoyarsk, Computational [4] G. Emanuel, H. Hekiri, Vorticity jump across a shock in Center of RAS Siberian Branch, 1997. (in Russian) nonuniform flow, Shock Waves 21 (1) (2011) 71-72. [10] H. Li, G. Ben-Dor, Oblique Shock—Expansion Fan [5] G. Emanuel, H. Hekiri, Vorticity and its rate of change just Interaction—Analytical Solution, AIAA Journal 43 (2) downstream of a curved shock, Shock Waves 17 (1-2) (1996) 418-421. (2007) 85-94. [11] V.R. Meshkov, A.V. Omelchenko, V.N. Uskov, The [6] A.V. Omelchenko, On the correlation of derivatives at a interaction of the stationary shock with the expansion strong discontinuity, Computational Mathematics and wave of opposite direction, Vestnik Saint Petersburg Mathematical Physics 42 (8) (2002) 1246-1257. (in University: Mathematics 2 (9) (2002) 99-106. (in Russian) Russian) [12] A.V. Omelchenko, V.N. Uskov, An optimal [7] M.V. Chernyshov, The Interactions of the Elements of shock-expansion system in a steady gas flow, Journal of Shock-Wave Systems Between Themselves and with Applied Mechanics and Technical Physics 38 (3) (1997) Different Surfaces, Ph.D. Thesis, Saint Petersburg, Baltic 204-210. State Technical University, 2002. (in Russian) [13] A.V. Omelchenko, V.N. Uskov, Optimum overtaking [8] A.V. Omelchenko, The generalized Chester-Whitham compression shocks with restriction imposed on the total invariant, Technical Physics Letters 27 (11) (2001) flow-deflection angle, Journal of Applied Mechanics and 891-893. Technical Physics 40 (4) (1999) 638-646.