For the current Boeing models, the flight director maintains V2 to V2 +10 (744/748), V2 + 15 (757/767/777/787), or V2 + 20 (737NG/MAX) if an engine fails. The flight director also limits the bank angle to 15 degrees until V2 + 10 (747/777/787) or V2 + 15 (737NG/MAX, 757, 767) with newer Flight Control Computer (FCC) software. If less than V2, the F/D commands pitch to accelerate and maintain V2 speed. If above V2 + 10/15/12, the F/D commands pitch to slow to V2 + 10/15/20. If the airspeed is between V2 and V2 plus X, the F/D commands pitch to maintain the current speed.

V2 MIN must be no less than 1.13 X VS-1G. However, V2 may be higher than V2 MIN due to V-LOF (such as the 777-300ER), among other factors. In reference to AA191, the DC-10 was certified under the old FAR stall speed, or VS-FAR. Before 1985, all aircraft were certified under VS-FAR, and the V2 minimum was at least 1.2 times the VS-FAR value. Stall speed testing under VS-FAR was variable and primarily dependent on the individual test pilot. Following the implementation of FAR Amendment 25-108 (circa 1985, 767-300 forward) and the change to VS-1G, stall speed testing became much more consistent, resulting in a higher relative stall speed.

However, all takeoff performance is predicated on the V-Speeds, including V2, not V2 + 10, etc. FAR Part 25.121 lists the requirements for each takeoff segment with an engine inoperative. Note that for the second segment climb, it states that the steady gradient climb may not be less than 2.4%, 2.7%, or 3.0% (for two, three, and four engines, respectively) at V2. The minimum climb gradient is what determines the climb limit takeoff weight. Recall that the maximum takeoff weight is the most restrictive of the field limit, obstacle limit, climb limit, brake energy limit, tire speed limit, enroute limit, and landing limit weights. As such, the maximum takeoff climb limit weight is the maximum weight that allows a 3.0% climb gradient (4-engines) during the second segment climb (regardless of whether obstacles are present or not), at V2. It follows that to maximize the second segment climb gradient, V2 should be as close as possible to the best angle speed for the takeoff flap setting. Since the acceleration factor (1 + V/g {dV/dh}) is essentially nil at airport elevation, the best angle is a direct function of T/W - D/L, or the minimum value of D/L. However, with newer high-bypass engines, thrust increases as a function of velocity, and the best angle speed is usually higher than the speed at minimum D/L. Using the 747, V2 is approximately 10-15 knots below the takeoff flap maneuver speed.

Takeoff performance is complex and typically involves a series of trade-offs. Most of the time, the maximum takeoff weight is limited by the takeoff field limit or the obstacle limit weight. However, if the takeoff climb limit weight is the limiting factor, then there is likely excess runway available. In this case, improved climb speeds can be used to increase the climb limit weight. Improved climb speeds increase all the V-speeds, including V2, which, as discussed above, now approaches the best angle speed for the takeoff flap setting. Improved climb speeds can also be used if limited by the takeoff obstacle limit weight, if excess runway is available.

Finally, the Obstacle Assessment Area must be considered. If the splay is straight ahead, then the turn radius does not need to be considered. However, if a turn is required during the second segment, the effect of V2 on the turn radius becomes a factor, as well as the bank angle.

So, is there reliable info on how best angle of climb speed in the takeoff configuration depends on the speed for modern jets?

How is this a serious question? This is basic, and something a student pilot should understand.

If this relates to your original question of V2, then "best angle" is irrelevant.

The links provided, and the explanations provided, covered your questions quite well. What is it that you don't understand?

None of the links or extensive explanations in the posts above answered my original question: "what happens if V2 is significantly exceeded during the second segment? Any mathematical models or real flight tests used to prove it?"
Since people here emphasised (among other things) the best angle of climb for the purpose of obstacle clearance, I narrowed my question down to a more specific one: is there some publication where I can e.g. see a graph or some numbers on how angle of climb depends on speed for a modern jet. That would answer at least in part to my original question.

If it was disscussed here and I missed it, could you please point out to that, so I could see how much climb performance will be lost if instead of climbing at V2+20 any jet of your choice will be climbing at V2+50 during the second segment? I'm interested in numbers, not in "worse" or "somewhat less".

Thanks.

For specific values, you'll need the performance date for your aircraft. You'l need weight, temperature, thrust, and altitude, for a still-air computation, along with aircraft configuration, to include any performance-limiting statements that are true for that particular takeoff (bleeds, flaps, configuration deviation items such as missing inspection covers, gap seals, etc).

Your original post asked about a publication explaining what the climb segments are, and the criteria that defines each segment. Yes, such publications exist, multiple sources were cited, and additional explanations given.

You asked "what happens if V2 is significantly exceeded during the second segment". You were told.

You asked if there were any mathematical formulas or flight tests done to prove it. You were also told: yes, there are. Yes, they were used for the certification of the airplane, and the result of the design of the airplane, and the extensive flight testing that followed, is the operating parameters published in the performance data for that aircraft. Your performance is met when you follow the correct procedure. Your performance will NOT be met if you do not follow that procedure. By how much will you fail to meet the published climb data, if you don't fly on speed? Depends on numerous factors, the combination of which is specific to your aircraft in that location on that day at that power setting at that weight at that temperature at that power setting at that elevation in that configuration and condition.

You appear to be trying to make a case to others that their practice of not flying the published speeds, will have a negative effect. You appear to be saying you want the math to prove it. You have better access to the performance data for your aircraft than the rest of us will especially given that we don't know what aircraft you are flying.

What is correct, regardless of which aircraft you're flying, or any of the particulars, is that unless you fly the established parameters and profile for your departure, you will not get the performance desired, or predicted or required.

With all engines operating, the airplane may well exceed minimum performance gradient by a substantial margin, but don't forget that the minimum climb gradient in climb segments isn't an all-engine proposition, You may well exceed minimum climb gradient criteria at V2+50, with all engines operating. The same may not be true with loss of thrust on one (or more) engines. We conduct the departure with plans for the potential loss of thrust that may accompany an engine failure, or the need to reduce thrust following a problem. We conduct a runway or departure analysis for accelerate and stop distance, and the subsequent climb away from that runway, using a specific set of values. Speeds less or greater than those values will compromise the predicted performance. How much those values are compromised depends on numerous variables specific to your aircraft and environmental considerations. The math has been done by the manufacturer, and flight tests have verified the predicated data, which has then been published for your aircraft. If your aircraft does not include climb data in each segment for V2+50, then you may have to stop and think why. You may lack that information from the manufacturer that did the math and provided the performance numbers you do need, and it falls upon you to use and follow those numbers, if you want the performance predicted.

You could level off and clean up, and you may get a higher climb rate, but the segments of the initial climb are about gradient; distance forward vs altitude gained, not gross climb rate. Therefore, if you're looking for a math formula, you're going to need to know the forward progress of the aircraft over the ground, as well as the altitude increase in time, relative to that forward gain, and the impacts of your takeoff weigtt and CG, as well as ambient temp, elevation, etc. Density altitude matters. If you change any of the published parameters, you push outside those parameters. By climbing at V2+50, you're increasing forward speed, and altering your climb gradient. If you're experiencing sufficient rate of climb at that time and location, with all engines, perhaps you're still well above the desired minimum gradient. Lose thrust on one or more engines and all bets are off. How much are they off? Depends on numerous operational variables, and variables that you've added to the problem. At that point you may have a real problem that you've created, and you may not have published data to account for it. You certainly won't have had that data available to do your takeoff or runway analysis, which means you're guessing at your performance, which is never a good idea (especially given that you already have data guaranteeing your takeoff performance, if followed).

"What happens if I violate the parameters of the guaranteed climb segment performance predictions?" you may ask. You don't get the performance. By how much? Depends. If you choose to attempt to recreate the test program for the aircraft, without access to any of the design and prediction data used for that test and evaluation program, that's all on you. "I guess it will climb better at V2+50" isn't much of a test card, but at that point, the manufacturer data becomes irrelevant, and you're playing test pilot. Save your data.