<span style="text-decoration: underline">Background:</span>
A few months ago Ply1951Guy and I were talking about being able to simulate bullet flight accurately. Two primary topics of discussion were the dynamic simulation with a 6DOF model that's driven with the aerodynamic solution as well as a highly accurate aerodynamic solution. I'm working on that dynamic solution at this point, but Ply1951Guy has some preliminary results from the Aero solutions that I thought people might be interested in taking a look at.
Bryan Litz's book and Eaglet's calculator help us (as a community) get an estimate on a theoretical bullet calculation and Bryan did a huge amount of work to test a number of actual bullets in flight to get highly accurate BC's for each of the bullets we commonly shoot.
The question/discussion that always comes up is how accurate are the factory prediced BC's? Again back to Bryan's book with the data in the back, there's not much to argue with his presentation of data; the excellent work is a hit with everyone I've talked to about it and I really like it myself.
Ply1951Guy has developed a way to model the flow characteristics around the bullet during flight with a Compressible Computational Fluid Dynamic Simulation (C-CFD). It works on similar principles that Finite Element Analysis is based upon for solid objects where stress, strain, modal behavior, etc. is being investigated, it's simply a different set of partial differential equations to be solved.
The hitch is that it's compressible and highly shock dependent behavior. To our knowledge, nobody else is modeling bullets this way unless it's proprietary or classified research that has yet to be released to the general public. I will freely admit that while I understand the concepts of the model and the conceptualization of Finite Element Theory, if you want details on how the model is built you need to contact him. My specialty is in solid mechanics and mechanisms, not fluids...
The models I'm presenting here are the G7 standard bullet. They have been scaled to a 7mm projectile size and are being shown at Mach 2.76, which is 3000fps at standard conditions. Realistically this would be approximately a 140gr 7mm Boat Tail bullet with a 10 caliber ogive. Since everything we talk about is referenced off of this standard he started his simulations with that.
To validate the accuracy of the model he also built models of some other 7mm bullets that are included in Bryan's book and compared the doppler radar data to the results he obtained via his proprietary method. The predicted model is has less than 2% error compared to the standard table (1.91% if memory serves correctly) and the other bullets ranged between 1.5% and 1.9% lower drag than the radar measured.
We know what the source of this error is, it's in the fact that the bullet is simulated without rotation (not spinning in the air) but the complexity involved with that modeling is not realistic to include for reasonable solution times.
A common question that comes up is new shooters who've never experienced long range shooting, especially with quality optics want to understand what “the trace” is all about. This is where the background information above comes in.
<span style="text-decoration: underline">The Trace</span>
The trace is somewhat difficult to capture on a video and without a video that can be paused, backed up and played again for others to watch with instruction it can be difficult for someone to catch. Additionally I've seen a number of comments asking “why does the trace happen” and “what dictaes how strong the effect is” amongst others. The informed reader knows that this is because of the shock waves and density change around the bullet as it flies at supersonic speeds towards the target. The question arises though: “What am I actually looking at?”
Ply1951Guy put together some screen shots of various orientations and a kick-ass video showing the isosurfaces of constant density around the bullet at various points.
First is a cross sectional slice of the flow field around the bullet that shows the relative velocity of the air flowing around the bullet (keep in mind that the bullet doesn't know it's flying, it just knows what is acting on it from the outside, so the model treats it as a stationary object with the air moving by it instead).
The color gradient shows the local velocity of the air as it flows around the bullet and you can see the well defined shock waves formed from the supersonic flow around the bullet.
Keeping this in mind, now it's time to think about the way air acts as a shock wave moves through it. The density of the air is dependent upon several things, one of them is the pressure. The shockwaves create a pressure discontinuity where a large jump in pressure (and therefore density) happens as you move across the shock wave.
As the pressure increases the density increases. This means little if you have a large region like a "high pressure system" of weather. The change is very slow, the region is large and your eyes see nothing differently.
The bullet is a small, localized pressure difference that makes the light come through "clear" but that buttery ripple of the trace is where the high pressure wake of the bullet bends the light differently because the Refractive Index has changed in the small local region around the bullet as it flies downrange. The governing equations wouldn't add much but confusion to this synopsis, so I'll skip it. Here's the lookup for those who might be interested in the Elden and Updated Elden equations for the Refrative Index of Air
"An Updated Elden Equation for the Refractive Index of Air" by Birch and Downs
Density Plot, Side View
So that might help some, but Ply1951Guy really did quite the job putting together stuff to present this information. Here's a legend to show where the cross section slices came from:
<span style="font-weight: bold">Slice Locator</span>
Concentric rings of pressure wave are shown at the various slices, you can see how the density change moves out from the bullet as you get farther back from the meplat of the bullet (that is where the shock starts from).
Slice 1
Slice 2
Slice 3
Slice 4
The surfaces wrapping around the bullet are called Isodensity surfaces and they show the region around the bullet that have all the same density. Each different surface shows a different density value which bends the light in it's own way as the bullet proceeds downrange. He had to piece this together from snapshots and I think he did a kickass job with it.
The video shows only local results around the bullet but it propagates away from the bullet and leaves a dissipation region much like a boat leaves a wake as it travels through calm water.
So hopefully this helps to give guys a better concept of what they're actually seeing with the trace, as well as something to look for.
We're thinking about how to turn this into a realistic business venture to make turned solids and optimized bullets available to far more shooters who want to stretch the legs of their rifles, we'll make any other public disclosures regarding a business after talking to LowLight and getting clearance on it. The goal of this was to put out some helpful tools, so let's try to steer to that primarily.
Any in depth questions should be directed to Ply1951Guy, I'll try to answer others as I can.
A few months ago Ply1951Guy and I were talking about being able to simulate bullet flight accurately. Two primary topics of discussion were the dynamic simulation with a 6DOF model that's driven with the aerodynamic solution as well as a highly accurate aerodynamic solution. I'm working on that dynamic solution at this point, but Ply1951Guy has some preliminary results from the Aero solutions that I thought people might be interested in taking a look at.
Bryan Litz's book and Eaglet's calculator help us (as a community) get an estimate on a theoretical bullet calculation and Bryan did a huge amount of work to test a number of actual bullets in flight to get highly accurate BC's for each of the bullets we commonly shoot.
The question/discussion that always comes up is how accurate are the factory prediced BC's? Again back to Bryan's book with the data in the back, there's not much to argue with his presentation of data; the excellent work is a hit with everyone I've talked to about it and I really like it myself.
Ply1951Guy has developed a way to model the flow characteristics around the bullet during flight with a Compressible Computational Fluid Dynamic Simulation (C-CFD). It works on similar principles that Finite Element Analysis is based upon for solid objects where stress, strain, modal behavior, etc. is being investigated, it's simply a different set of partial differential equations to be solved.
The hitch is that it's compressible and highly shock dependent behavior. To our knowledge, nobody else is modeling bullets this way unless it's proprietary or classified research that has yet to be released to the general public. I will freely admit that while I understand the concepts of the model and the conceptualization of Finite Element Theory, if you want details on how the model is built you need to contact him. My specialty is in solid mechanics and mechanisms, not fluids...
The models I'm presenting here are the G7 standard bullet. They have been scaled to a 7mm projectile size and are being shown at Mach 2.76, which is 3000fps at standard conditions. Realistically this would be approximately a 140gr 7mm Boat Tail bullet with a 10 caliber ogive. Since everything we talk about is referenced off of this standard he started his simulations with that.
To validate the accuracy of the model he also built models of some other 7mm bullets that are included in Bryan's book and compared the doppler radar data to the results he obtained via his proprietary method. The predicted model is has less than 2% error compared to the standard table (1.91% if memory serves correctly) and the other bullets ranged between 1.5% and 1.9% lower drag than the radar measured.
We know what the source of this error is, it's in the fact that the bullet is simulated without rotation (not spinning in the air) but the complexity involved with that modeling is not realistic to include for reasonable solution times.
A common question that comes up is new shooters who've never experienced long range shooting, especially with quality optics want to understand what “the trace” is all about. This is where the background information above comes in.
<span style="text-decoration: underline">The Trace</span>
The trace is somewhat difficult to capture on a video and without a video that can be paused, backed up and played again for others to watch with instruction it can be difficult for someone to catch. Additionally I've seen a number of comments asking “why does the trace happen” and “what dictaes how strong the effect is” amongst others. The informed reader knows that this is because of the shock waves and density change around the bullet as it flies at supersonic speeds towards the target. The question arises though: “What am I actually looking at?”
Ply1951Guy put together some screen shots of various orientations and a kick-ass video showing the isosurfaces of constant density around the bullet at various points.
First is a cross sectional slice of the flow field around the bullet that shows the relative velocity of the air flowing around the bullet (keep in mind that the bullet doesn't know it's flying, it just knows what is acting on it from the outside, so the model treats it as a stationary object with the air moving by it instead).
The color gradient shows the local velocity of the air as it flows around the bullet and you can see the well defined shock waves formed from the supersonic flow around the bullet.
Keeping this in mind, now it's time to think about the way air acts as a shock wave moves through it. The density of the air is dependent upon several things, one of them is the pressure. The shockwaves create a pressure discontinuity where a large jump in pressure (and therefore density) happens as you move across the shock wave.
As the pressure increases the density increases. This means little if you have a large region like a "high pressure system" of weather. The change is very slow, the region is large and your eyes see nothing differently.
The bullet is a small, localized pressure difference that makes the light come through "clear" but that buttery ripple of the trace is where the high pressure wake of the bullet bends the light differently because the Refractive Index has changed in the small local region around the bullet as it flies downrange. The governing equations wouldn't add much but confusion to this synopsis, so I'll skip it. Here's the lookup for those who might be interested in the Elden and Updated Elden equations for the Refrative Index of Air
"An Updated Elden Equation for the Refractive Index of Air" by Birch and Downs
Density Plot, Side View
So that might help some, but Ply1951Guy really did quite the job putting together stuff to present this information. Here's a legend to show where the cross section slices came from:
<span style="font-weight: bold">Slice Locator</span>
Concentric rings of pressure wave are shown at the various slices, you can see how the density change moves out from the bullet as you get farther back from the meplat of the bullet (that is where the shock starts from).
Slice 1
Slice 2
Slice 3
Slice 4
The surfaces wrapping around the bullet are called Isodensity surfaces and they show the region around the bullet that have all the same density. Each different surface shows a different density value which bends the light in it's own way as the bullet proceeds downrange. He had to piece this together from snapshots and I think he did a kickass job with it.
The video shows only local results around the bullet but it propagates away from the bullet and leaves a dissipation region much like a boat leaves a wake as it travels through calm water.
So hopefully this helps to give guys a better concept of what they're actually seeing with the trace, as well as something to look for.
We're thinking about how to turn this into a realistic business venture to make turned solids and optimized bullets available to far more shooters who want to stretch the legs of their rifles, we'll make any other public disclosures regarding a business after talking to LowLight and getting clearance on it. The goal of this was to put out some helpful tools, so let's try to steer to that primarily.
Any in depth questions should be directed to Ply1951Guy, I'll try to answer others as I can.
