Supersonic (Mach 3) flow around cylinder 

Yes, you shouldn’t swim at Mach 3

@@muratcan4624 is that a challenge

Have bored, will surf

Not surprising, we have used slip boundaries on top and bottom to simulate flow around a disc in a channel. We could simulate flow around the disk without this wall effect - but it makes for a much less interesting video.

@@matthiasmaier8956 oh! Very well, thanks for the reply.

What Sukuna said to Gojo before Infinite Void shattered:

Yeah bruh. Me .

Yes, the high resolution comes from the fact that we use a very fine mesh, but also to a large degree from the fact that our method is high order (with very controlled artificial viscosity).

@@matthiasmaier8956 I see! That's what I thought! PS: Congratulations the results look really good.

Sure. Initial conditions are a uniform flow field with Mach 3. On the left (supersonic) inflow boundary we enforce the full initial state, on the right (supersonic) outflow boundary we do nothing. All other boundaries have a slip boundary condition where the normal component of the velocity is removed (and the total energy is updated).

But they do distort. This is most noticeable in the von Kármán vortex street when the vortices emerging from the cylinder hit a strong shock. For the ones closer to the boundary: I think the perturbation is just not that noticeable in the eyeball norm with the "schlieren plot" we're using here.

@@matthiasmaier8956 yes, I guess the central ones do; and the path of the ones on the outside is diverted (refracted?), like light hitting a density boundry at an angle; but I'd have expected the shockwave to have torn the vortices apart, not just subtly diverted them

Yes, the initial velocity field is not zero - and thus not terribly physically realistic. But it makes for a great numerical test whether the algorithm can handle the emerging low pressure bubble behind the disc. The simulated flow is inviscid (some residual graph viscosity is applied for numerical stability, though). But for practical purposes - apart from a viscous boundary layer - viscous flow with a Reynolds number of 10^6 - 10^7 would probably produce a very similar flow field.

😃

This is what would happen during the first few microseconds after you suddenly spawn a giant pillar in Jupiter's or Saturn's upper atmosphere before it gets crushed and sheared away by the insanely intense aerodynamic forces. Not saying it's impossible, but very unlikely to say the least :p What am I saying? Atmospheric reentry of a rod. There you go.

This is a second-order convex-limiting scheme, i.e., we first compute a low-order provisional update (using a "graph viscosity" constructed with an upper bound on the maximal wavespeed). We then construct a high-order update (by lowering the graph viscosity using an entropy-viscosity indicator) and finally limit the high-order update to preserve some crucial flow invariants. You will find all details written out here arxiv.org/abs/2007.00094 , the scheme is based on the following publication epubs.siam.org/doi/10.1137/17M1149961 that we have also extended to compressible Navier-Stokes arxiv.org/abs/2009.06022

Because we imposed slip boundary conditons that remove the normal velocity component and thus reflect shocks.

Yes, the dark lines visualize regions with large gradients in the density. So what you see are shocks and contact lines.

The upper and lower boundaries have reflecting slip boundary conditions imposed.

No, but degrees of freedom located on the boundary are post-processing to enforce slip boundary conditions.

You mean the upper and lower boundaries? Those had been chosen to be slip boundary conditions on purpose - meaning the simulations is an obstacle in a channel/tube. It is possible to replace these boundary conditions by something else to simulate an obstacle in a uniform outer flow.

If you linearly extrapolate (just 2-3 points) to populate the state variables of the upper and lower boundaries, it would behave as if it were free flying

@@matthiasmaier8956 You mean to say the slip boundary condition assumes inviscid flow, right? Edit: Description says compressible Euler, right. So this is just the Euler equations on a ridiculously fine grid with very small timesteps. Still super satisfying to watch

@@OSRS2ndBase It's not a problem to replace the boundary conditions by some (dynamic) outflow condition to simulate a cylinder in free flow. But that looks significantly less spectacular than the channel flow simulated here 🙂

No, this is a fixed, uniformly refined mesh. The animation is showing the norm of the gradient of the density on an exaggerated exponential scale (a "schlieren plot" of some sorts).

*oops* I have put the usage instructions back in place: github.com/conservation-laws/ryujin/blob/development/INSTALLATION.md github.com/conservation-laws/ryujin/blob/development/USAGE.md

I used paraview to convert the vtu files to pngs and then ffmpeg to make an mpg movie out of it.

@@matthiasmaier8956 how many pngs? 1 million?

@@rfadlika only about 2700 PNGs (30 snapshots per second of the movie)

What you see are numerical artefacts that are caused by a change in mesh geometry and are visible because the schlieren plot is shown on a logarithmic scale. Regarding the reflection on the lower boundary: I cannot spot a non-reflected shock.

@@matthiasmaier8956 thanks. i think the first shockwave looks nonreflected because it's strictly perpendicular to the boundary, so it merges with its reflection

The computation did run on a 2 processor (2 x 14 cores, Intel Xeon E5-2697) machine and took about 50h.

Matthias Maier thank you so much !

The normal shock ahead of the sphere causes the flow to become subsonic but it reaccelerates again over and below the cylinder until it becomes supersonic again, prompting the formation of these normal shocks. They are similar to the lambda shocks that form on a wing in the transonic regime (for a sufficiently high Mach number) since they are linked to the separated BL as you mentioned. They cause flow separation but are also formed due to the aforementioned BL separation. This is a chicken and egg situation... Unless you say that the curvature of the cylinder causes an adverse pressure gradient that accelerates the flow in the upstream direction and towards the wall, which has two effects: causes the BL to separate and the flow to accelerate, which simultaneously (?) creates both flow features.

Also since this is a Euler equation-resolving scheme, the Reynolds number is supposed to be infinite. As you increase the Reynolds number, the separation point is supposed to go further downstream (assuming the BL is turbulent. This is due to a more energetic BL which will be less sensitive to adverse pressure gradient effects)

@@professionalprocrastinator8103 but why is there a BL if it is an Euler flow ? Without taking into account the viscosity, we should observe the d’Alembert paradox with a symmetrical pressure coefficient distribution around the cylinder, but since we can see flow separation in the wake it must imply that some kind of viscosity is taken into account in this simulation, isn’t it ?

@@hugoboulenc5084 An Euler solver solves the Euler equations which by definition should not contain viscous dissipation as you mentioned. However from a numerical point of view your solver would struggle to converge since it would likely not reach a stable solution. So artificial viscosity is introduced in order to stabilize the solution and smooth out strong gradients which might cause your run to crash. The main difficulty is to segregate non-physical from physical gradients.

@@professionalprocrastinator8103 So what is this really? Solving the Euler equation with enough artificial dissipation to obtain a solution that looks like one from solving the N-S equations? Why not do a DNS on the N-S equations? The video is super cool though. The instabilities along the slip-line look fabulous.

The mesh had an almost uniform resolution of about h=0.0018 (the computational domain had a length of 4 and height of 2).

This is a direct numerical simulation (DNS) with a custom software based on deal.II. The postprocessing was done with Paraview.

From my guess because of the molecules that gather in front of the object and can‘t disperse fast enough creating a high pressure zone.

I believe what we are seeing is a bow shock wave. It'll appear as soon as the flow around an obstacle becomes supersonic and the angle of deviation is too great for an attached oblique shock to form. it should be noted that shock waves still exist past Mach 1 and its "pressure wall", Mach 1 is just a special regime where a build-up of pressure occurs.

The air slows down as it hits the sphere

Because it is moving at more than mach 1. Then you always have a shockwave in front of you.

@@jackmclane1826 nope XD there is no SW in front of you if you are moving faster than M1

Yes, the computations are relatively expensive. This particular one with about 2 million gridpoints probably ran for about a day on a computer node with 32 cores.

@@matthiasmaier8956 Wow, thanks

Wow, I expected it to be much more expensive.

if you have to ask, you can't afford it

Yes. We have done computations for the Onera OAT15a airfoil (mainly in 3D). Here is a video of a 2D simulation: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-yM2rT3teakE.html

Euler.

Not a stupid question at all. These are the compressible Euler equations (meaning no viscosity). If you would add viscosity to the system, you would end up with the compressible Navier-Stokes equations.

Yes DNS. :-)

Direct Navier-Stokes? Particle based CFD?

@@rfadlika DNS of the compressible Euler equations (no particle based simulation)