《Table 1.Law of the fluid-thickness effect on bubbles and spikes.》

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《Finite-thickness effect of the fluids on bubbles and spikes in Richtmyer–Meshkov instability for arbitrary Atwood numbers》

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In order to validate the finite-thickness effect，theory results are compared with the numerical simulation.For the two dimensional Euler equations of inviscid and compressible fluid dynamics，fifth-order finite difference WENO schemes with the third-order Runge–Kutta time discretization are used.In our numerical simulation，we set up the problem as follows:the computational domain is[0，0.25]?×?[0，2]；the location of the initial shock is at y?=?0.01，and after the shock，the gas density isρ0?=?0.2，the velocity in the x-direction is u?=?0，the one in the positive y-direction is v?=?2.700 31 and the pressure is p?=?3.5.The shock with Mach number 1.0 moves in the positive y-direction，and goes successively through the gas with densityρ1?=?0.1，the fluids with densitiesρ2?=?2.0 andρ3?=?3.0，and the gas with densityρ1?=?0.1.The planar interface between the gas of densityρ1and the fluid of densityρ2is at y?=?0.05，and the perturbed interface between the fluids of densitiesρ2andρ3is at y?=?0.3，and another planar interface between the fluid of the densityρ3and the gas of the densityρ1 is at y?=?0.55.Before the shock，the pressure keeps constant p?=?1 and the fluids and gas keep rest.The perturbed interface is set as y=0.3+0.001 cos（8px），and then the perturbed wavelength isλ?=?0.25.In this case，the shock travels two fluids with the same thicknessδ?=?λ?=?0.25.The reflective boundary conditions are imposed for the x?=?0 and x?=?0.25boundaries，and the free second-order boundary conditions are set at y?=?0 and y?=?2 boundaries.The mesh step is set as the uniform h?=?10-4.When an incident shock collides with the perturbed interface，it bifurcates into a transmitted shock，which moves in the direction of the incident shock，and a reflected wave，which moves in the opposite direction of the incident shock.When the reflected wave and transmitted shock leave the perturbed interface，the fluids are compressed and their densities become rl=5.33and rh=7.76.That is to say，the Atwood number is A?=?0.186.Because the whole perturbed interface move in the positive y-direction，it is inconvenient for us to track the amplitudes of the bubbles and spikes.Here，width of mixing area full of two fluids of densities rhand rlis defined as Wbs=∣Ab-As∣.From figure 7，one finds that results from the second-order theory and the numerical simulation are in good agreement.The numerical simulation validates the second-order theory provided in this paper.