Direct numerical simulation is used to study the interaction of a Mach 1.5 shock wave and various types of anisotropic turbulent flows. We compare the interaction of isotropic, axisymmetric and sheared turbulences (sometimes combined), with a specific interest for the sheared situation

Chassaing, Patrick

Crespo, Matthieu

Jamme, Stéphane

English

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DNS of the interaction between a shock waveand a turbulent shear flowMatthieu Crespo, Ste´phane Jamme, and Patrick ChassaingENSICA, 1 Place Emile Blouin 31056 Toulouse Cedex 5, Francemcrespo@ensica.fr, jamme@ensica.fr, chassain@ensica.fr1 Numerical considerationsWe solve the three-dimensional Navier-Stokes equations in non-dimensionalconservative form using a finite difference approach. The inviscid part isresolved using a fifth-order Weighted Essentially Non-Oscillatory scheme(WENO). Viscous terms are computed using a sixth-order accurate compactscheme. A third-order Runge Kutta algorithm is used to advance in time.Equations are solved on a cubic domain of size 2π in the three direc-tions and a grid of 176× 128× 128 points is used. The mean flow is alignedwith x. Periodic conditions are specified in the z direction, and non-reflectingboundary conditions with a sponge layer are used for the top and bottomboundaries along y as well as for the outflow where the flow is subsonic. Ateach time step, velocity, pressure, temperature, and density fields are speci-fied at the inflow. These fields are superpositions of a supersonic mean flowand turbulent fluctuations in velocity, pressure, temperature, and density.The mean velocity at the inflow varies linearly across streamlines while themean pressure is uniform. The mean temperature and density vary such asthe mean Mach number is uniform : U1(y) = U0 + S(y − ymin), V 1 = W 1 =0, P 1(y) = 1/(γM2r ), T 1(y) = M2r U21 /M21 , where the overbar denotes theconventional Reynolds average and the subscript 1 indicates the upstreamstate. A fluctuating field is then superposed onto the mean upstream flow andadvected through the shock. This field comes from a preliminary calculation.The anisotropy of the turbulent velocity field used in the inflow plane is eitherof axisymmetric type or typical of a turbulent shear flow.2 ResultsSeveral simulations were conducted with the following values of the referenceparameters: Rer =ρ∗ru∗rL∗rµ∗r= 94, Mr =u∗rc∗r= 0.1, P r = 0.7, where f∗r2 Matthieu Crespo, Ste´phane Jamme, and Patrick Chassaingrefers to a dimensional reference variable. The mean Mach number is fixed toM1 = 1.5, and the turbulence parameters in the inflow plane are the following :Reλ = Rer λu1rmsν = 47 and Mt =qc =√u′iu′ic = 0.173. Table 1 summarizesthe characteristics of the different runs. They differ by the nature of the meanflow (sheared or not), by the anisotropy of the upstream turbulent flow andby the amount of˜u′′2 immediately upstream of the shock wave.Table 1. Characteristics of the different runs (i = inflow ; b.s. = just before shock)Run S q22i(u′′2)b.s.  u′′2q2b.s.  v′′2q2b.s.  w′′2q2b.s.  u′′v′′q2b.s.RunSI 1.5 1.5 1.04 0.42 0.28 0.31 -0.14RunSA1 1.5 1.5 1.10 0.44 0.27 0.30 -0.12RunSA2 1.5 1.5 1.11 0.40 0.29 0.31 -0.17RunI 0 2 1.00 0.35 0.33 0.32 0.005RunA1 0 1.7 1.04 0.43 0.28 0.30 0.007Previous works lead to the conclusion that the amplification of the ki-netic energy behind the shock wave is strongly dependent of the upstreamanisotropic state, and that it is clearly determined by the amount of the lon-gitudinal normal Reynolds stress˜u′′2 upstream of the shock (see e.g. Jammeet al. [1]). The mean flow was uniform without shear stress is these studies.In the present work, we first compare three runs (RunSI, RunSA1 andRunSA2) where a mean shear has been introduced. The anisotropy of theturbulence is slightly different just before the shock for these three cases. Thenear-field amplification of q2/2 behind the shock wave is found to depend onthe amount of the correlation u˜′′T ′′ immediately upstream of the shock. Thiscorrelation is positive in the three cases, but its value is not the same. Themore u˜′′T ′′ is high upstream, the less q2/2 is amplified behind the shock.In order to get rid of the effect linked to the amount of˜u′′2, and trying toisolate the influence of the nature of the anisotropy of the incident turbulentflow itself, we conducted two more runs (RunI and RunA1) in which theamount of˜u′′2 is the same as in RunSI just before the shock, but not the valuesof the other components of the Reynolds stress tensor. Figure 1 shows that theaxisymmetric case displays a greater near-field amplification of˜u′′2 than theisotropic case, whereas the opposite is true for the sheared case. Concerningthe near-field behaviour of q2/2, both the axisymmetric and sheared casesshow a greater amplification than the isotropic case. Mahesh et al. [2] observeda decrease of q2/2 across a M1 = 1.2 shock for a sheared case, and theyattributed this trend to the fact that u˜′′T ′′ > 0 before the shock, which isknown to inhibit the amplification of the kinetic energy. In the present case(RunSI), we have u˜′′T ′′ > 0 upstream, but q2/2 is still more amplified in theDNS of the interaction between a shock wave and a turbulent shear flow 3(a) (b) (c)x0 1 2 3 4 50.911.11.21.31.41.51.6x0 1 2 3 4 50.60.70.80.911.11.21.31.41.51.6x0 1 2 3 4 50.60.70.80.911.11.21.31.41.51.6(d) (e) (f)x0 1 2 3 4 50.911.11.21.31.41.51.6x0 1 2 3 4 5-0.5-0.45-0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.0500.05x0 1 2 3 4 5-0.05-0.04-0.03-0.02-0.0100.010.020.030.040.05Fig. 1. Streamwise evolution of different turbulent statistics across the shock: (a)u′′2 ; (b) v′′2 ; (c) w′′2 ; (d) q2/2 ; (e) u′′v′′ ; (f) u′′T ′′. In (a), (b), (c), (d), curvesare normalized by their value immediately upstream of the shock wave. (—–) RunSI; (−−−) RunI ; (− · −) RunA1.near field compared to the isotropic situation. This difference with Maheshet al. [2] may be a consequence of the shock strength (M1 = 1.5 in our caseinstead of M1 = 1.2).The behaviour of u˜′′v′′ is found to be same as the one observed by Maheshet al. [2] : we notice a decrease of the magnitude of u˜′′v′′ across the shockwave.Moreover, a clear influence of the shear stress can be seen on the streamwisecomponent of the vorticity (not shown here). An increase of ω′21 in the nearfield behind the shock is indeed observed for the three cases, but this tendencyis much more pronounced for the sheared case. The vortex stretching by themean flow is found to be responsible for this increase of ω′21, which means thatthis term is enhanced in the sheared case.References1. S. Jamme, M. Crespo, and P. Chassaing. Direct numerical simulation of theinteraction between a shock wave and anisotropic turbulence. In 35th AIAAFluid Dynamics Conference and Exhibit, Toronto, Canada, AIAA Paper 2005-4886, 2005.2. K. Mahesh, P. Moin, and S.K. Lele. The interaction of a shock wave with aturbulent shear flow. Technical Report TF-69, June 1996.

Direct numerical simulation of the interaction between a shock wave and anisotropic turbulence.

The interaction of a shock wave with a turbulent shear ﬂow.

DNS of the interaction between a shock wave and a turbulent shear flow

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DNS of the interaction between a shock wave and a turbulent shear flow

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