The XC-equation, Applying simple mathematics for improving XC flying

Prologue

When there is a scientific approach to a practical issue, there are two important preconditions in order to conclude reliably. The first is to keep accurate data so that we can analyse without large statistical discrepancies in order to identify a significant conclusion. The second, which is probably more valuable, is asking the right questions in order to approach an issue correctly.

Focusing my interest on how to develop XC skills, my intention is to practically translate theory to the language of everyday flying decisions and tactics.

What was missing was the “right question” which came to me from my friend Dionisis from flyfreedom.gr school. What he asked was how to train evolving XC pilots in order to be able to identify a good day and understand those parameters of flight or parameters of piloting that combine to a successful XC flight at a specific day. My eyes flashed from excitement when during this conversation the word “Equation” was used…What if there really was such a mathematic equation that would help us understand the XC potential of the day and guide us to a great Km achievement!!! The idea of the XC Equation was born! The way to understand the day potential and the way to optimize our flying so that we can maximize our performance in a specific day with specific conditions.

Due to the fact that my life free time equation is difficult to solve, I was working the story in my mind for a couple of weeks until I had the opportunity to take a pen and a white paper to sketch & play, and try to imagine how I would finalise a simple conclusion through a complex and chaotic pathway. I finally understood that with the maths I knew I should simplify things, and start from a simple model where geometry of things would set parameters and combine them.

Checking back in my older book for Cross Country and more specifically in the chapter of speed to fly in a soaring flight, I reopened my eyes to see again things under a different perspective. My conclusion, right or wrong, seems to be very close to reality and help someone who is trying hard to understand the parameters of an XC flight and intends to push himself to become a better XC pilot. Knowledge has no value if you do not share it with others; therefore I announce my conclusion to anyone may find it useful to understand things better.

XC-equation

Based on Reichmann, 

Fig 1

Distance between thermals related to Cloud base 

When we fly with a glider over a homogeneous terrain, e.g. a flat ground area, we observe a repetition in the soaring phenomena, such as position of clouds, thermals shape, Cloud streets setting e.g. parallel, or hexagon positions etc.

Every soaring day is different regarding cloud coverage, cloud base and cloud tops, overdevelopment or blue thermals, complete or broken cloud streets, lined up or hexagon patterned Cumulus, Inversion level or levels, different winds in direction and speed at different flight levels, different temperature and Dew point gradient, stronger lift in thermal cores or sink during big glides between clouds. Thermals may not be the same across their way up, and usually there is a specific ideal zone where the difference of lifted air parcels is the highest between the surrounding air and the thermal core that is expanding adiabatically, inside the expanding thermal column. Therefore, to design an XC equation is anything than simple, and at first glance, it seems to be a challenge to conclude on practical rules. 

Returning to Reichmann's theorem, in the homogeneous field we are discussing, we follow the assumption that the thermal cores close to the ground are complex and intersecting like a tree route system, while above the inversion the many cores-branches intersect and become fewer and stronger, resulting a large strong core of warm lifting air that is forming a Cu cloud at a specific altitude where during the adiabatic cooling of the ricing core, the temperature reaches Dew point. The distance between this Cumulus and the next one, is expected to be at a distance S that is related to the Cloud base. The higher the cloud base, the more the distance to the next core. According to Reichmann,

S=2,5 X Cloud base

Thus, at least theoretically, and flying above the inversions across an area with well-organized and repeatable thermals there is a saw shaped kind of flight pattern we are following in order to cover a long distance flight. For people flying across huge flat areas with strong soaring conditions, this pattern may seem familiar.

Now we have the first critical question:

“Following the above repeatable thermal pattern, how much altitude do we need to gain at least, in order to reach the next thermal at the same altitude we started thermalling the previous thermal, therefore maintaining the saw XC pattern and loosing as less as possible time in each thermal?”

If we do the maths, it seems that this depends on many parameters, but in order to simplify things, the answer relates to our achievable glide ratio during the big glide that will follow. More specifically, the calculation concludes that the thermal gain altitude-X we have to gain is at least: 

Χ=2,5Χ(Cloud base/Glide ratio)

This is a useful equation, which helps us to follow a simple rule. If e.g. the cloud base is at 2.000m and under the conditions we can achieve a 10:1 glide ratio between the thermals, it is enough for us to gain at least 500m. If we gain more than this, we may lose time, and if we gain less than 500m then we may not reach the next thermal high enough in the ideal zone (above the inversion and where the lapse rate is ideal).

Next important question for our calculations: “what time do I need to climb?”

If I gain X m of altitude inside an average thermal of θ (θ from the Greek word «θερμικό» which means «thermal»), the needed time T_climb is:

T_climb =X/θ

Where  Χ is the gained altitude, and  θ is the average thermal (most Varios calculate this for us).

For example, if we have an average daily climb of 3 m/s and we have to gain 500m, we need 166 sec. We all understand, that thermalling is a continuous optimizing process where we intend to maximize our climb all the time, staying in the thermal core, or at the ideal altitude zone as with this tactic we can minimize the time loss, a critical strategy to achieve a long XC flight.

Combining the previous equations we can have the following one:

T1=2,5 X Cloud base / (θ Χ Glide Ratio)

 Which means that the stronger the θ and the better Glide ratio we can achieve, the lowest time loss we have. But days with very high cloud base, may not be the ideal ones as the time loss will be more in order to climb higher. Ideally, days with average Cloud bases are days where the thermals are more reliable, and the distance is shorter between them, which may offer the opportunity for dolphin flying conditions where the time gain is even better (ideally XC-ing with a sailplane).

Let’s now move to the next stage of our flight, the big glide!

So far we have minimized our time loses, while climbing and gaining the altitude needed to reach the next thermal. The ideal glide in this saw model is to reach the next thermal, above inversions, in the ideal thermal strength zone, as fast as possible, in the same altitude we entered the previous thermal.  The speed calculation here already exists from Mc Cready theory, which means we increase our speed to the point of our polar curve that is defined from the thermal we expect to find in front of us. The higher the thermal the faster we have to reach it, as shown in the Mc Cready figure.

 While on glide, and according to Reichmann, the distance S we are going to cover until we reach the next thermal is:    

S=2,5 Χ Cloud base

The time we will need for this glide, depends on our airspeed:  

T2=2,5 X Cloud base / (V_horizontal)

Where V_horizontal is the combination of airspeed and wind velocity during our glide. Actually it is the GPS speed during this phase of the flight in our saw model.

Vhorizontal = VLD + Vwind

where VLD is the airspeed that gives us the Glide ratio according to our polar curve.

The total time to complete a « tooth » to our saw type flight is:

Ttooth = T1 + T2

 

Fig 2 – Saw shaped flight

 

Combining the above the total time we need for a single thermal climb and glide is:

T_tooth = 2,5 X Cloud base X { 1 / (θ.LD) + 1 / (VLD + Vwind)}

After all this brain challenge, we are in an interesting point of thoughts:

If the duration of the model conditions last for example 3 hours, how many teeth can we achieve in this specific total XC flight time?

If the flight lasts for time T_{flightduration}, then… 

Potential XC km = S. T_{flightduration} / (T1+Τ2)

 

AND the BIG moment:

The final XC-equation becomes:

Interesting! Because we can make some specific conclusions from this:

For example, case study-1:

Θ=4μ/s

LD=10

T_{flightduration} =5 hours

VLD= 40 kph

Vwind= 10 kph

The equation predicts a maximum potential XC=185Km. It may be high or low, but actually it represents a flight to the razors edge, where we fly to the limit of speed below which we gain more energy than we need and pay with time loss, and above which we are at risk of falling very low, below the inversions, at a not well organized thermal and again lose a lot of time to recover our impatient decision to increase risk.

In the same case, if the thermal average was 2m/s then the max XC potential would be 147 km.

Case Study-2

Comparing the equation results for different assumptions and different performance gliders, we can graphically understand similarities and differences from different category paragliders to high performance sailplanes. As you can clearly seen in the next figure:

during a strong thermal day, the advantage of having a higher glide ratio at a higher speed is higher, while during smooth and weak thermal days the differences are less important.

Another conclusion from the XC-equation, is that we can predict from the model the average XC speed which is a critical predictive performance indicator of the XC flight. Check the next table:

As we can see, if e.g. we fly with a paraglider which has a best LD =10 at 40kph and the average thermal is +3m/s, the average speed we can achieve can be around 33,7kph according to this model, but if we are playing around thermalling anything around +1,5m/s then the average XC speed will fall dramatically to 25,7kph. That is a reason why it is the logical decision to make hard decisions and take the risk of not thermalling every thermal we find, but stop only in the strongest cores. At the same time, we can see that the average XC speed is easier to be achieved when we fly with higher performance wings. At this point it becomes clear that "high performance" is when you have better LD at higher speed in your polar curve, and this explains a lot of things regarding the need for speed and why the 500km XC flights are achieved due to small performance details.

Conclusions

While playing and critically thinking about the XC equation, we can make several logical conclusions that may confirm the reality, as we know it.   

I am really excited after all these flying years since 1988 with airplanes, paragliders, and sailplanes to understand that there is a simple logical calculation  to understand what are the right tactics for a successful long XC flight, and it is mathematically proven that:

•       The relation between a strong thermal day and XC potential is logarithmic

•       The stronger thermals the longer XC

•       The better L/D the longer XC

•       The higher speed of a wing to achieve max L/D the longer XC

•       Pushing winds are a favourite!  

•       Increasing airspeed (V_LD) is positive for more XC but destroys LD, which reduces XC potential. This explains why there is an optimal speed to fly which actually is Mc Cready Speed.

•       The longest day the better XC (see what is happening in Finland)

•       The stronger the thermals, the bigger difference in XC potential between different LD performance. 

•       For specific XC km target, the better LD, the less strong thermals are needed to achieve it.

•       Cloud base is not appearing in the equation! Therefore, we may have a big potential during low cloud base days. 

•       Floater wings can gain more time from thermalling and lose time from gliding, while gliders the opposite. This explains that specific conditions and glider characteristics may have a significant correlation.

•       From the figures we can understand that if we leave weak thermals we can significantly increase our average XC speed. That is why we have to identify the strongest possible achievable day conditions and exploit them at risk.

•       The more complex the terrain and the more changes of the real algorithm in a flying day, make XC a more demanding process in areas with anything but flat lands, and make XC more difficult but also an exciting process

•       As a final conclusion (although I can think of more) I would say that the most important thought that comes out of this equation is tha XC flying is not a battle with Km, Sun, Wind, Thermals, Clouds or gliders. It is a battle with time, where your ultimate goal is to fit in a specific flying time the most possible teeth in a saw shaped flight.

Calculator in EXCEL to download

Finally, if you have reached studying my philosophical approach of XC strategy, please download this excel calculator and play with numbers. Interesting things may appear, especially if you think the opposite way, e.g. what should the conditions be if I want to achieve a specific flight in a specific place, with a specific glider, and what attribute and skills I should train myself on in order to succeed ?

Evangelos Tsoukas