in a methane-air flame i can get densities as high as 1.2 and as low as 0.2.

therefore i can expect density ratios of 5 or 6.

across a normal shock wave i can also expect a maximum density ratio of about 6.

why then do i consider my low speed flame incompressible, but my high speed shock wave compressible?

they seem to have the same density changes so what is the differnece between the two definitions?

obviously the difference is the speed but why does this make one compressible and one incompressible?

very simple: read the word 'compressibility' as what it literally means, i.e. the change of density due to change of pressure. To compress something means to apply pressure to increase its density. In compressible flow you have a strong dependence of density on pressure; in incompressible flow, such sensitivity is much weaker. What does compressibility have to do with your methane flame? Not much. In your flame, the density varies because of temperature gradients, not because of pressure gradients.

thankyou

that is a good answer.

We say that a flow is incompressible if the density of a fluid particle following its motion is nearly constant. In that case the continuity equation reduces to

div(u) = 0

If not then the flow is compressible and we have to use the full continuity equation. For ideal, non-reacting gas, if Mach < 0.3 then the flow can be considered incompressible.

In startified flows like the atmosphere or oceans, we can make the incompressiblility assumption even though the density is not uniform because the density of a fluid particle does not change following its motion.

Density can change either due to pressure changes or due to temperature changes. If either of these effects is large then we have to consider the flow as compressible and use the full continuity equation. To decide this, you have look at the non-dimensional form of continuity equation. For a discussion see

Flow Control, by Gad-el-Hak, Section 2.3