Ballistics Modelling with the McDrag Program

[rsterne]

For a few years now I have been working with two online calculators that are based on Robert McCoy's "McDrag" algorithms.... He developed the program, written in Fortran, in the 1970s while working at the Aberdeen Proving Grounds, and it is one of the best algorithms publically available.... Both the Kolbe Drag Calculator.... http://www.geoffrey-kolbe.com/drag.htm .... and the JBM Drag Calculator.... http://www.jbmballistics.com/cgi-bin/jbmdrag-5.1.cgi .... are based on the McCoy program.... They both have limitations that I have found frustrating, however.... The JBM program won't allow you to input Meplats larger than 30% of the caliber, nor Noses shorter than 1.0 cal.... The Kolbe program plots a nice graph of Cd, but doesn't give the detailed BCs in the Transonic range like the JBM.... So over the last week I created a spreadsheet of my own, using the McDrag Algorithm, and customized it to allow me to change variables one at a time, and see how they change the Cd in the range I am interested, from Mach 0.5 to 1.5.... I checked my program against both the Kolbe and JBM Calculators with a few different bullets, and it seems to track extremely well, so I am confident I copied all the formulas accurately.... With the G1 Model dimensions, my results track with the JBM exactly, to the third decimal.... When I input the dimensions for the G1 and G7 models into my spreadsheet, I get the following results....



The results for the G1 model are a bit "spiky"in the Transonic region, and show a divergence at over Mach 3 from the measured drag (blue).... but the McDrag results for the G7 model are a near perfect match to the experimental results (red).... The claims for the McDrag program is that it is within 3% Supersonic, 11% Transonic, and 6% Subsonic.... Nose lengths less than 1 caliber, or Meplats greater than 50% of the caliber may result in the predicted drag being too high in the Transonic and Supersonic ranges.... We push both of those limits with many of our big bore bullets, which tend to be short and fat.... Boattails longer than 1.5 calibers, or smaller than 65% of the caliber, may produce errors as well, but those limits are outside of practical limits anyway.... For both sets of limits, it doesn't mean the data is useless for close comparisons, just that it may be out of the normal error parameters above....

Here is what I get when I input the dimensions for my latest Bob's Boattail design, the .308 cal 200 gr. BBT Whiteout.... It is intended to have the same ballistics as the .300 AAC Blackout PB round.... Note that I have eliminated all velocities over Mach 1.5, and expanded the horizontal scale to make the charts easier to read....



As you can see, I managed to lower the drag below Mach 1 compared to the G7 Model, which pleased me very much.... I look forward to using this new tool to optimize my bullet designs in the future....

Bob

[Josan]

He Bob. Nice work. Can this in any way be used with airgun pellets?

[rsterne]

Sorry, Josan, these calculators are not capable of handling Diabolo (waisted) pellets.... they are for bullets only....

Now that I have the ability to study single changes in bullets much more readily, and observe the effect of those changes on the drag.... I may as well get on with the first project, which is to study the Meplat diameter.... I input two generic bullets, both in .25 cal., and both with a flat base.... They both use Tangent Ogives, with a constant length nose and overall length.... I vary the Meplat, as a percentage of caliber, between 0% (pointed) and 80%.... The first bullet I studied was 2 calibers long (0.50") and had a nose 1 caliber long (0.25"), half the length of the bullet.... This might be pretty typical of a bullet in the 65 gr. range in .257 cal.... Here are the results.... The dotted blue line is the G1 Model, shown for reference....



The first thing you notice is that the drag below Mach 0.8 (~900 fps) is the same, regardless of the Meplat diameter.... By the time we get to Mach 0.9 (~1000 fps), the drag is in exactly the opposite order to what you might logically expect.... with the bluntest bullets having the least drag.... This trend continues until nearly Mach 1, at which point the bullet with the 80% meplat (the bluntest) starts to drastically increase in drag until it has the most drag about about Mach 1.33.... You might be quite shocked by the trend in the Transonic range, as was I the first time I noticed it.... but once you look at the next chart you may understand better.... Here is the other bullet, which is longer and much more pointed (larger Ogive radius).... It is 3 calibers long (0.75"), with a 2 caliber (0.50") long nose.... This might be a good choice for a long range bullet in the 95 gr. range in .257 cal.... The charts both have the same scales and colours to allow direct visual comparisons....



Once again, the Subsonic drag is the same for all bullets, but the range is extended upwards to Mach 0.85 (~950 fps) before they diverge.... The bluntest bullet from above, with the 80% Meplat, has hardly changed above Mach 1, so you can see that the large flat nose is dominating the drag in the Supersonic range.... Between Mach 0.9-0.95 (1000-1050 fps) all the bullets are still very close in drag.... Above that, first the 80% bullet diverges, and above Mach 1.15 the 60% Meplat bullet also shows an increase in drag relative to the ones with the thinner noses.... Those remain tightly clustered together, but by the time you reach Mach 1.5 the 40% Meplat has the most drag, and the 20% has the least.... In the Supersonic region, all the bullets with the thinner noses have about a third less drag than they did on the bullets with the shorter nose we first looked at.... This tells me that the length of the nose, or more properly the Ogive radius, is very important to the overall drag in the Supersonic region.... Simply stated, pointy bullets create smaller shockwaves when Supersonic....

The really interesting part is the difference in the two bullets between Mach 0.9-1.0.... The two bullets with the thinnest noses (0% & 20%) have MUCH more drag on the shorter blunter bullet.... This can be explained if the Ogive radius in that velocity range is more important than the Meplat diameter.... Think about it, and you will realize that when you hold the length of the nose constant, and reduce the diameter of the Meplat, the Ogive radius gets smaller (blunter), and the edge of the nose where the air first starts flowing around the side of the nose is at a greater angle to the airflow.... That angle creates more drag than the flat nose.... This makes sense if the air is "piling up" on the Meplat and creating a rounded buffer, around which the air is flowing smoothly to the shallower angle on the Ogive, and that is exactly what happens.... it's called the Stagnation Zone.... Once the bullet crosses into the Supersonic region, shockwaves disrupt that smooth flow, and the large Meplat pays a huge drag penalty....

So what conclusions can we draw so far if we are trying to determine the "best" Meplat size?.... Well, below Mach 0.8 it doesn't matter.... If we cared about the Supersonic range (above Mach 1.2), the 80% Meplat is too big.... For short bullets, we can also eliminate the thinner Meplats, as their drag starts to increase sooner.... In the Transonic range, this leaves us with choosing between Meplats of 20-80%.... It appears that for shorter bullets, a Meplat of 60-80% is optimum (thank you, Elmer Keith!), extending the low drag plateau up to Mach 0.9.... while for longer bullets 20-60% looks like the optimum, again extending the low drag plateau to Mach 0.9.... Are you ready for round two?.... Let's set the Meplat at 60% and vary the nose length and see what happens.... and again with a Meplat of 30%....

Bob

[rsterne]

Let's continue with our exploration of this subject.... I input a .25 cal bullet an inch long (so that I was not limited on nose length), with a 60% Meplat (0.150" diameter) and then varied the length of the nose from 0.5 calibers to 3 calibers.... Here is the result....



Then I did the same thing, but with a 30% Meplat (0.075" diameter)....



On first glance, everything seems just what you would expect.... The longer the nose, the less the drag.... This is particularly true for the bullet with the 30% Meplat above Mach 1.... The short nose (0.5 cal., just 0.125") is just plain too blunt to be low drag.... The only time you would use a nose that short is if you were forced to by other considerations, such as the requirement for a compact overall length, or the desire to incorporate a boattail into a short bullet.... Having said that, you will notice that on the bullet with the 60% Meplat (typical for an airgun hunting bullet), there is NO drag penalty under Mach 0.8 (~900 fps), so even though it might be a poor choice for speeds over that, for a short bullet intended to be shot at 900 fps, there is no disadvantage.... There is one small anomaly which you can barely make out on the graphs.... If you look at the black line, for the longest nose, at speeds below Mach 0.9 (where they all appear to have the same drag).... it actually has slightly more drag than the others.... In fact, in that range, the least drag is the 2 caliber long nose, although the 1.5 cal. long is almost as good, and the 1 cal. nose has less drag than the 3 cal. long one.... but you have to drastically expand the vertical scale to see that....

So once again, let's look at that troublesome Transonic region, and in particular between Mach 0.9-1.0.... The least drag is with a long nose (2-3 cal.) with the 30% Meplat.... That is why I chose the dimensions I did for the .308 Whiteout.... If the nose is about 1.5 calibers long, the Meplat size doesn't matter much (compare the green lines on both graphs).... However, if the nose is only 1 caliber long, the 60% Meplat has less drag than the 30%, from Mach 0.85 to Mach 1.1.... and for the very short, 0.5 cal. long nose, the larger Meplat has less drag all the way from Mach 0.7 and up.... Therefore, by doing this second analysis, we have confirmed that there is no single correct answer to the question "What is the best Meplat diameter?".... If you have the luxury of a long bullet with a long nose, then a relatively small Meplat has less drag above 1000 fps.... However, if you have to work with a short, stumpy bullet, you are better off to use a larger Meplat.... It turns out that my first choices of looking at bullets where the noses were 1.0 calibers and 2.0 calibers long were pretty much ideal to determine the answer to that question.... For the short nose, something in the 60-80% range is ideal.... while for the long nose, 20-40% is the area to examine further.... While each specific bullet will have a different answer, it wouldn't hurt to look at a couple of short bullets and a longer one in more detail, concentrating on the Transonic range only, from Mach 0.8-1.2....

Bob

[rsterne]

Going back to the original bullets, I explored the Meplat size in more detail, using smaller steps, and a large scale on the charts for a clearer view of what is happening in the Transonic region.... Here are the charts.... First, the shorter bullet, with a 1 caliber long nose....



I am concentrating on the high Subsonic region, as I have no interest in driving the bullet Supersonic, with the corresponding large increase in drag and therefore wind drift.... It now becomes a matter of eliminating the Meplat sizes that aren't as good, which in the case of the 1 caliber long nose are the smaller ones, at 40% and 50%.... Realistically there is no practical difference in the other three choices, or between them.... so any Meplat between 60-80% will effectively produce great results.... with the absolute optimum for this bullet being about a 70% Meplat.... Let's hear it for Elmer Keith !!! .... If you were going to extend the velocity into the high Transonic range, up to Mach 1.2, a 60% Meplat would be the best choice for this length of nose.... Now what about a longer bullet?....



Again, I am only interested in the high Subsonic region, so we can eliminate the 10-20% Meplat bullets, and concentrate on those from 30-50%.... There is really very little to choose from between the three, with the 40-50% being a fraction better from 0.9-0.95 Mach, but the 30% having a slight edge at Mach 0.95-1.0..... If you were going to extend into the high Transonic, up to Mach 1.2, the 30% would be the best choice for a bullet with a 2 cal. long nose.... With my .308 BBT Whiteout, I wanted to improve the BC as much as possible at around 1050-1060 fps, so I choose the 30% Meplat.... Even though I was working with the JBM Calculator when I made that decision, it is nice to see that decision confirmed using my own spreadsheet....

Anyway, that shows you how you can use a Ballistics Modelling Program to optimize each part of a bullet's design.... I hope you found the process interesting and educational.... In both cases, the correct choice of Meplat allows the bullet to stay on the low-drag plateau all the way to Mach 0.9 (over 1000 fps).... more than worth the effort....

Bob

[rsterne]

Now that we have dealt with determining the optimum nose shape to minimize drag in the Transonic region.... it's time to look at the idea of a Boattail.... First of all, do they work to reduce drag in the Subsonic and Transonic regions?.... Here is the answer to that.... A boattail diameter of 100% of caliber means a flat base bullet (no boattail).... I used a boattail length of 75% of caliber as being middle of the road, and it happens to be what my .308 cal. 200 gr. BBT Whiteout uses, which is the bullet I am basing the charts on....



The yellow line is the flat base bullet.... You can see that below Mach 0.96 (1070 fps) it has more drag than any of the boattails, even one with a 90% base diameter such as you might find in a gas-check or bevel base design.... The smaller you make the boattail, the lower the drag when Subsonic (ie below ~Mach 0.85), but you can see some wild swings in the drag occurring with the 60% and 50% diameter boattails in the Transonic region.... which is why the McDrag program cautions against using a boattail diameter of less than 65% as it introduces errors.... The chart below drops out the 50% boattail, and enlarges the scales....



So, we can conclude that using a boattail is, indeed, very useful in reducing the drag in the subsonic and transonic regions.... and boattails of 80-90% of the caliber can even reduce the drag right at Mach 1.0 compared to a flat base.... If you compare the 90% line to the 100% (flat base) line, you can see what can happen with just a simple reduction in the base diameter from a gas check shank, or a bevel base.... Just those features can reduce the drag.... I think it is worthwhile to examine the 60-80% base diameter in more detail.... but before that, let's have a look at what happens when we change the boattail length....

Bob

[rsterne]

So let's have a look at what happens when we fix the diameter of the boattail, and vary the length.... Using my .308 cal. 200 gr. BBT Whiteout as the base bullet, here are the charts....



You can see the same wild swings in drag with the short boattails.... This leads me to think that it is not the length or diameter of the boattail that causes problems with the calculations in McDrag, but the slope or angle of the boattail.... When it is too steep, the calculations can't cope.... The chart below once again shows more detail in the Transonic region....



You can see a clear trend that at Mach 1 the longer the boattail (ie the shallower the slope), the less drag.... Below Mach 0.85 (~950 fps) there is no difference, and between the two the faster you go, the shallower you want the slope to be.... ie for a given base diameter, the longer the boattail.... Perhaps we should combine the data from this graph and the one in the post above and look at what angles don't work....

We can calculate the angle of the boattail from the following equation (all dimensions in calibers)....
Slope per side = (caliber - base diam.) / 2 / BT length .... This is the tangent of the boattail angle.... Taking the Inverse Tangent of that number gives the angle....

Now we only have a couple of examples to work from above, but it would appear that for a 75% length, a 60% base is too small, and that has a tangent of (1-0.60)/2/0.75 = 0.266 which is 15 degrees.... The other example is a 60% length with a 70% diameter, and that works out to (1-0.70)/2/0.60 = 0.250 which is 14 degrees.... For convenience, let's use a value for the tangent of 0.250 (a slope of 1 in 4), which works out to that 14 degrees, as a maximum slope.... So if the boattail is 14 degrees or steeper per side, the McDrag program probably doesn't work very well in predicting boattail drag.... For our 75% length, that works out to a minimum base diameter of 63%.... and for a 70% boattail diameter the minimum length would be 60%.... I will stay within those limits in the next section....

Bob

[rsterne]

Here is the final analysis for the boattail on my .308 cal 200 gr. BBT Whiteout.... I tested base diameters from 65-80% of the caliber, using the actual boattail length of 76%.... Only the Transonic region is shown....



The drag curves cross at Mach 0.96 (~1070 fps), and below that the smaller the base diameter the less the drag (and above that, the opposite is true).... However, looking at the rapid increase in drag above that for the 65% base, and the fact that the Corbin and Lapua rebated boattail designs use a larger base with a shallower angle, I felt more comfortable using the 70% base diameter.... not to mention that the smaller the base, the harder it is to make a mold.... Corbin use a 73% base, and Lapua between 73-80%, depending on where on the base you measure it, as it has a radius between the taper and the actual base of the bullet.... So, I'm on the small side for the base diameter, but only by a few percent, and that favours lower drag in the velocity range the bullet will spend all of it's time.... so I think it was a good choice.... I use the same 8* boattail angle, from the rebate back, as the Corbin, and the same 0.015" rebate as well.... just a slightly longer boattail that ends a few thou smaller.... For those of you who missed it earlier, here is the predicted drag curve for the .308 cal. 200 gr. BBT Whiteout....



I'm very pleased with how useful my spreadsheet based on the McDrag algorithms is, and how easy it is to use.... I can vary one parameter at a time and see what it does to the drag of the bullet, and so optimize it for the designed task.... I think it will be an invaluable aid to the design process.... and is was very pleasing to see it confirm the design of the Whiteout that I achieved through much more labourious methods....

Bob

[Rizen 1]

That is a great accomplishment Bob!    We look forward to testing the whiteout bullet.

[rsterne]

I decided to try and develop a series of long-range Bob's Boattails, optimizing the drag for each.... I wanted to make generic designs, that are based on several length/diameter ratios, so that they are scalable to different calibers, but I had to start by chosing a caliber to work with, so why not one of the most popular used for long range, the .257.... Here are the four designs I ended up with....



This uses a 60% Meplat and a 3R Tangent Ogive, the BT Base is 75% of Caliber, and it has a Sectional Density of 0.167....



This uses a 50% Meplat and a 4R Tangent Ogive, the BT Base is 72% of Caliber, and it has a Sectional Density of 0.192....



This uses a 40% Meplat and a 5R Tangent Ogive, the BT Base is 70% of Caliber, and it has a Sectional Density of 0.216....



This uses a 30% Meplat and a 6R Tangent Ogive, the BT Base is 70% of Caliber, and it has a Sectional Density of 0.240....

All the designs use the 8* taper, rebated boattail design which I had previously optimized during work on the .308 cal. 200 gr. BBT Whiteout.... The two longer bullets have the same 70% base diameter, but as they get shorter from there it was necessary to truncate the boattail slightly (while keeping the same angle).... For each bullet, I chose an Ogive radius that was as long as possible, but that kept the length of the nose less than half the overall length of the bullet, and then optimized the Meplat diameter to minimize the drag, using the McDrag program to look at all appropriate diameters.... Considering the results from above, it is not surprising that the longer the ogive, the smaller the optimum Meplat in the Transonic region, so I spent my time trying to extend the low drag plateau to as high a velocity as possible.... in all cases I pushed it to about Mach 0.9 (1000 fps) before the Cd starts to increase rapidly....

After all the designs were drawn out, I noticed something interesting.... All the designs had (visually) about the same angle where the Ogive meets the Meplat.... Since I had taken a lot of time to optimize each design, I thought that was unlikely to be a coincidence, so I measured the angle.... It turns out that in all four designs, the point where the Ogive meets the Meplat is within 1/2 degree (plus or minus) of a 20 degree angle to the centerline of the bullet.... I have never read of the importance of this angle anywhere in Ballistics before, but it certainly appears to be important, and it makes perfect sense that Transonic and Supersonic flow in that region would be affected by that angle, and how the air flows around the stagnation zone formed on the Meplat and then over the rest of the nose of the bullet.... According to what I have learned from the McDrag program, at Subsonic speeds there are a large range of angles (and hence Meplat sizes) that work equally well, and at Supersonic speeds, the most critical thing is having a small Meplat (although not zero!).... but in the Transonic region (Mach 0.8-1.2) this 20* angle, or something close to that, seem to be the sweet spot....

It is very interesting that when you look at the shorter designs, your brain tells you that a smaller Meplat should have less drag.... but that when you work through the design carefully to optimize the drag, it turns out to be the angle between the Ogive and the Meplat that is more important in the Transonic region (yet with no increase in drag when Subsonic).... making the blunter design actually better for airgun use if you are limited to how heavy/long a bullet you can use in practice.... You must remember that which of the above designs, if indeed any of them, are suitable for your rifle, will depend on the power level you can develop.... As a rough guide, double the bullet weight, and if your gun can't produce that in FPE in a usable tune (not maxed out), you need a lighter bullet.... That FPE represents 950 fps.... and these bullets are all optimized for muzzle velocities of up to 1000 fps or slightly more.... The twist rates shown are only a rough guide, and should provide a Stability Factor of 1.3 at 1000 fps (and more below that)....

Bob

[rintafile]

I ques these models go on LBT's list...Accurate can't do this caliber,smallest is 7mm and NOE can do but via NOE's forum to get actually anything You need to get at least 10 buyers...

[rsterne]

They will eventually, assuming the Whiteout works out as planned.... The 8 degree Boattail pattern is being done by Veral to use for that, and subsequent, designs.... Veral has Ogive patterns available for the two lighter bullets, but would have to cut new ones for the two heavier designs.... Of course the radius required would change with caliber, as would the weight, Sectional Density, and pressure requirements for the gun to launch them.... At the moment it is a work in progress....

Bob

[Bill G]

Nice designs Bob!  111gr for a .257 is big man. I have been studying your designs and think that I may very well be able to swage them. The issue is that each die will likely produce that size of bullet.  There are silhouette bullets that have been swaged that are the bore riding design. The driving band width is what changes as the weight is changed. so each die would have a very narrow range of weight adjustment. The pro would be that the band width could be adjusted for research and to make it fit the chamber just right.  So that being said, The most cost effective route, IMO, is to see how the molds pan out. If the guns like them, I suppose I may try one. Keeping a sharp transition may be difficult.  Then again, the .250 body with a knurl to take it up to .257 would make friction very low. I have to do some of my own testing and to do that, I am currently working on perfecting chamber reamers for myself. 

[rsterne]

I think your knurling method to upsize the body to form the driving band would be perfect for this style of bore-riding bullet.... Swage them at about 0.251" so that they just touch the lands, and then knurl up the driving band to groove diameter plus a thou.... Should work perfectly.... You could change the parallel mid-body to change the weight in your dies as per usual....

Bob

[Bill G]

Yes.  The current .25cal measures .2508".  The knurl can add .012 repeatable and that makes the groove diameter .2628.  The knurl is pretty fragile and I was thinking that if go to .262, which is way big, the knurl will lay down easy enough.  the average of the lan diameter and the knurled diameter is .2568. It would seem to me, that as long as there is adequate body riding the bore, that the knurl will act as a great seal and fill the throat really well for concentricity.  Another thought for over sizing the knurl would be to fit the groove diameter leaving the surface more smooth.  Effectively extruding the band to fit the bore precisely and minimize friction, while not leaving the dreaded petals at the tail.   The idea is that it would reduce the likelihood of the knurl imparting an added spin drift affect or reducing BC.  All speculation of course.     

[rsterne]

One more addition to the series....



A 65 gr. that is 2.0 calibers long.... It still has the 8 degree boattail, and the Meplat is increased to 70% with a 2R Tangent Ogive to stay close to the 20* angle on the nose.... It still is nearly flat on the drag up to Mach 0.9 (1000 fps).... This is pushing the limits, anything shorter will need a 10 degree boattail and a shorter nose Ogive radius to maintain a decent bearing length.... Both will increase the drag between Mach 0.8-0.9....

Bob

[PakProtector]

One more addition to the series....



A 65 gr. that is 2.0 calibers long.... It still has the 8 degree boattail, and the Meplat is increased to 70% to stay close to the 20* angle on the nose.... It still is nearly flat on the drag up to Mach 0.9 (1000 fps).... This is pushing the limits, anything shorter will need a 10 degree boattail and a shorter nose Ogive radius to maintain a decent bearing length.... Both will increase the drag between Mach 0.8-0.9....

Bob

That one should still be cram-able into a modified 25 cal marauder magazine. Relieve the magazine body a wee bit, and cut the cover as well. It is also of weight suitable for a Marauder built around a Hot valve...
cheers,
Douglas

[rsterne]

You would have to gain about 0.060" in length in the magazine, but likely can do that....

Bob

[Redlock]

How much weight would these loose as hollow point?
Thanks Bob.
Red.

[rsterne]

Depends entirely on the size of the HP cavity.... and that depends on the use of the bullet.... If you want it to blow up, you use a large, deep cavity.... If you want it to mushroom but stay intact, you have to use a shallower cavity.... The actual dimensions depend on the hardness of the lead and the velocity as well.... The HP would normally remove about 3-6 gr. on that 65 gr. bullet.... but if you went really big, could be more....

Bob