Linear stability of a fluid channel with a porous layer in the center

We perform a linear stability analysis of a Poiseuille flow in a channel inserted with one porous layer in the centre,and focus mainly on the efect of porous filling ratio.The spectral collocation technique is adopted to solve the coupled linear stability problem.We investigate the efect of permeability,σ,with fixed porous filling ratioψ=1/3 and then the efect of change in porous filling ratio.As shown in the paper,with increasingσ,almost each eigenvalue on the upper left branch has two subbranches atψ=1/3.The channel flow with one porous layer inserted at its middle(ψ=1/3)is more stable than the structure of two porous layers at upper and bottom walls with the same parameters.By decreasing the filling ratioψ,the modes on the upper left branch are almost in pairs and move in opposite directions,especially one of the two unstable modes moves back to a stable mode,while the other becomes more instable.It is concluded that there are at most two unstable modes with decreasing filling ratioψ.By analyzing the relation betweenψand the maximum imaginary part of the streamwise phase speed,Cimax,we find that increasing Re has a destabilizing efect and the efect is more obvious for small Re,whereψa remarkable drop in Cimax can be observed.The most unstable mode is more sensitive at small filling ratioψ,and decreasingψcan not always increase the linear stability.There is a maximum value of Cimax which appears at a small porous filling ratio when Re is larger than 2 000.And the value of filling ratioψcorresponding to the maximum value of Cimax in the most unstable state is increased with increasing Re.There is a critical value of porous filling ratioψ(≈0.24)for Re=500;the structure will become stable asψgrows to surpass the threshold of 0.24;When porous filling ratioψincreases from 0.4 to 0.6,there is hardly any changes in the values of Cimax.We have also observed that the critical Reynolds number is especially sensitive for smallψwhere the fastest drop is observed,and there may be a wide range in which the porous filling ratio has less efect on the stability(ψranges from 0.2 to 0.6 atσ=0.002).At larger permeability,σ,the critical Reynolds number tends to converge no matter what the value of porous filling ratio is.