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We present a computational study of Edney type shock-shock interactions between an oblique shock and a bow shock in front of a circular cylinder. Both cold and hot hypersonic test cases were considered and the results revealed the main features characterizing the type III and type IV of interaction, without and with the jet impinging on the cylinder. The simulations also predicted qualitatively the complex flow structure observed in the experiments and the pressure and heat peak on the surface of the cylinder.

In the present work we analyze the effects of thermochemical relaxation phenomena on the interference between a cylindrical bow shock and a plane oblique shock. This problem has been extensively studied since the 60’s [

According to Edney [

In both cases the shock pattern is characterized by the occurrence of two triple

points caused by the interaction of the incident shock with the bow shock. It should be remarked that in a Type-III interference the amplification of the surface pressure and heat transfer is caused by the attached shear layer while in a Type IV the amplification is mainly due to the impingement of the supersonic jet originating at the lower triple point and terminating with a “nearly normal” jet shock in front of the body.

For cold hypersonic flows many works (both numerical and experimental) can be found in the literature. Wieting and Holden [

Considering now real gas effects, it is known that, when the free-stream specific kinetic energy is of the same order of magnitude of the dissociation energy, non-equilibrium phenomena must be taken into account [

The motivation of the present work is to assess the influence of non-equilibrium on Type-III and Type-IV shock/shock interactions using numerical modeling In particular, we are interested on the effects of non-equilibrium on the flow scales and unsteadiness, and on the dynamic and thermal loads. This represents a first step where we focus on the complex shock pattern in the case of ideal gas simulations and present preliminary results of the simulations for thermochemical nonequlibrium. The approach relies on a finite volume method for the solution of the conservation laws for a mixture of gases in non-equilibrium. In addition, in order to understand the effects of vibrational relaxation and chemical reactions we have also performed ideal gas simulations.

In this section we formulate the two dimensional conservation equations for a multispecies non-ionizing gas in thermochemical non-equilibrium, following the approach developed in Grasso and Capano [

∂ ∂ t ∫ S w d S + ∮ ∂ S f ⋅ n d l = ∫ S w ˙ d S (1)

where

w = [ ρ k , ρ u , ρ v , ρ E , ρ e v ] T

f = f E − f V

f E = [ ρ k u , ρ u u + p I , ρ u H , ρ u e v ] T (2)

f V = [ − J k , σ , σ ⋅ u − J q , − J q v ] T

w ˙ = [ ρ ˙ s , 0 , 0 , 0 , ρ e ˙ v ] T

and the mass flux of species k ( J k ), the internal ( J q ) and vibration ( J q v ) energy diffusion fluxes (due to conduction and species diffusion) and the stress tensor σ are

J k = − ρ D k ∇ Y k (3)

J q = j q t r + j q v + ∑ k h k J k = − ( η t + η r ) ∇ T − η v ∇ T v + ∑ k h k J k (4)

J q v = j q v + ∑ k ′ h k ′ v J k ′ = − η v ∇ T v + ∑ k ′ h k ′ v J k ′ (5)

σ = μ ( ∇ u + ∇ u T ) − 2 3 μ ∇ ⋅ u I (6)

where k ′ indicates diatomic species and Y k = ρ k / ρ is the species mass fraction, while the transport coefficients ( D k , η’s and μ ) are defined according to Chapmann-Enskog theory [

For atomic species the internal energy ( e k ) accounts only for the translational ( e k t ) and the enthalpy of formation ( Δ h k 0 ) contributions

e k = e k t + Δ h k o = 3 2 R k T + Δ h k o (7)

while for diatomic species the internal energy also accounts for rotation ( e k ′ r ) and vibration ( e k ′ v ) contributions and

e k ′ = e k ′ t + e k ′ r + e k ′ v + Δ h k ′ o = 3 2 R k ′ T + 1 2 R k ′ T + R k ′ θ k ′ v exp ( θ k ′ v / T v ) − 1 + Δ h k ′ o (8)

where θ k ′ v is the characteristic vibrational temperature. The vibrational energy ( e v ) and the total enthalpy (H) of the mixture are

e v = ∑ k ′ Y k ′ e k ′ , H = E + p ρ = ∑ k Y k e k + u ⋅ u 2 + p ρ (9)

The species production ( ρ ˙ k ) due to finite rate chemistry has been modeled by means of Park’s reaction mechanism whereby we have introduced a rate controlling temperature for the dissociation reactions defined as T d = T T v so as to account for the coupling between vibration and dissociation. The vibrational energy source ( e ˙ v ) is the sum of the the T-V energy exchange contribution ( S T V ) and to the energy removal due to vibration-dissociation coupling ( S V D ) that have been modeled as discussed in Grasso and Capano [

S T V = ∑ k ′ ρ k ′ e k ′ v * − e k ′ v τ k ′ v , S V D = ∑ k ′ ρ ˙ k ′ e k ′ v (10)

where e k ′ v * = R k ′ θ k ′ v / ( exp ( θ k ′ v / T ) − 1 ) is the equilibrium vibrational energy of molecular species, and τ k ′ v is the relaxation time obtained from the Millikan-White relation [

The basic numerical algorithm relies on a cell-centered finite-volume “patched” subdomain formulation and the governing equations are cast in the following fully discretized form:

S i , j d w i , j d t + ∑ β = 1 4 ( f ^ E − f ^ V ) β ⋅ n β Δ l β = S i , j w ˙ i , j (11)

where Δ l β and S i , j are, respectively, the size of face β and the cell area, while f ^ E and f ^ V represent, respectively, the numerical inviscid and viscous fluxes. The former is based on a second order upwind biased total variation diminishing scheme that accounts for non-equilibrium phenomena, following the general methodology described in Grasso and Capano [

For example, the numerical inviscid flux discretization at the generic cell face ( i + 1 2 , j ) , is cast in the following form:

( f ^ E ⋅ n ) i + 1 2 , j = 1 2 [ ( f E ⋅ n ) i , j + ( f E ⋅ n ) i + 1 , j + R i + 1 2 , j Φ i + 1 2 , j ] (12)

where Φ is obtained by characteristic decomposition in the direction normal to the cell face and the use of a minmod slope limiter. The right eigenvector matrix is defined as

R = [ δ k q 0 Y k Y k Y k u − c n y 0 u + c n x u − c n x v c n x 0 v + c n x v − c n y u ⋅ u / 2 − χ q / K c ( u ⋅ b ) 1 H + c n x H − c n y 0 0 1 e v e v ] (13)

where b ⋅ n = 0 and the pressure derivatives K and χ q are defined as

K = ∑ k Y k R k ∑ k Y k c v k t ; χ q = R q T − K e q t

and c v k t and c 2 are, respectively, the species translational-rotational specific heat and the frozen speed of sound that are defined as

c v k t = ∂ e k t ∂ T ; c 2 = ∑ q χ q Y q + K ( h − e v )

The values at cell interfaces are calculated by using a generalization of Roe’s averaging [

In addition, in order to avoid the carbuncle phenomenon which takes place in all Roe based schemes [

Finally, the time integration is performed by a three-stage Runge-Kutta algorithm where the source terms are treated by a point implicit algorithm by introducing a precondition matrix which is related to the partial Jacobian of the source term [

The objective of the present paper is to analyze Type-III and Type-IV shock-shock interactions with the main intent of studying the evolution of the shock wave pattern consequent to the variation of the shock impingement location (steady simulations). We assumed the free-stream conditions of the experimental work of Grasso et al., [

For the problem under investigation the predictions of the peak (thermal and dynamic) loads and their locations are mainly affected by the mesh spacing along the body (ξ), provided the minimum grid spacing in the normal-to-wall direction (η) is of the order of microns. We have then performed a grid independence study by changing the mesh spacing in the ξ-direction. In particular we have considered three grids consisting of 144 × 160 , 192 × 160 and 294 × 160 total number of cells (having a minimum normal-to-wall spacing of 3.34 × 10^{−6} m). The results of the analysis (not reported) showed that grid independency is achieved with the 192 × 160 grid, which has then been selected for all simulations.

In order to span the Type-III and Type-IV shock wave interference patterns, we have followed the same quasi-steady procedure adopted in the experiment by moving the incident shock generator in the vertical direction so as to change the location of the impingement location on the cylinder surface (for a fixed shock angle β = 23.57 deg ). In particular, the shock generator is displaced vertically once a steady (or pseudo steady) flow configuration (for the given shock impingement location) is attained.

In

in addition, we also report the sonic iso-Mach line. We observe that, as the shock impingement location moves upward, the jet shock inclination with respect to the body decreases. Consequently, the pressure and thermal loads increase as shown in

structure; note that such phenomenon is enhanced as the Mach number increases since the interaction takes place closer to the body where the shear layer is more spread thus leading to higher pressure gradients.

We finally present preliminary results of simulations carried out with air in thermochemical non-equilibrium. The stagnation quantities were taken from the experimental investigation of Carl et al. [

In this study we presented and discussed the results of numerical simulations of shock-shock interactions between an oblique shock and a bow shock in front of a circular cylinder. These interactions are categorized as types III and IV according to Edney experimental work. The simulations revealed the main features characterizing the two types of interaction and also predicted the pressure and heat peak on the cylinder. The study was conducted mainly for the cold hypersonic test case of Ref. [

The author wishes to thank Francesco Grasso for initiating this work.

The author declares no conflicts of interest regarding the publication of this paper.

Paoli, R. (2018) Numerical Simulations of Shock-Shock Interactions. Open Journal of Fluid Dynamics, 8, 392-403. https://doi.org/10.4236/ojfd.2018.84025