A Simple Horizontal Velocity Model of a Rising Thermal Instability in the Atmospheric Boundary Layer (In other words, how to take a closer look at downwind thermal drift) Abstract This pilot, as well as others, was taught from the very beginning that rising thermals drift downwind at the same speed of the wind. Brigliadoris [1] talk with more complexity about thermal drift in fairly strong winds. Simply put, they show that the horizontal velocity of the thermal at a given altitude is not necessarily exactly the same as the surrounding wind at the same altitude. This paper describes an example of a simple model that can be used to prove this theory. Finally, and most importantly, from the model results one can gain insight into thermal structure as an aid in efficient thermal usage. Nomenclature y Altitude above ground, ft W(y) Wind profile, mph m Mass of thermal slice (cookie) h Height of thermal cookie R Radius of thermal cookie V v Vertical velocity of thermal cookie v Horizontal velocity of thermal cookie ρ Density of air µ Viscosity of air A Cylindrical reference area L Characteristic length Coefficient of drag C d Introduction The mathematical approach is probably not perfect. Perhaps the fluid dynamics is also wrong in some ways. However, I ve learned in my short time as an engineer that very simple back of the envelope calculations can be valuable. We make assumptions in order to arrive at simple solutions to aid in our understanding of a complex phenomenon. Approach Imagine a simple, cleanly cylindrical thermal column of air in the boundary layer. Take a single horizontal slice of the column. Call this the cookie slice. The slice has height h, diameter R and mass m. The slice is moving vertically at speed V v up through the wind profile W, which is a function of altitude, y. On the ground, the thermal cookie starts B.R.Resor Page 1 of 1 November 1, 7
with horizontal velocity of zero. The goal is to determine the horizontal velocity, v, as a function of height, y, due to the forcing of the wind caused by aerodynamic drag. V v y h R m v W(y) Figure 1. Diagram of analysis In this case I will completely ignore the buoyancy problem of solving for vertical velocity, V v. Most glider pilots would agree that a thermal starts rising from V v = at the ground, accelerates to a fairly steady V v, and at some point stops rising. I will simply define the vertical velocity as such. The goal of this exercise is to look the effects of the horizontal inertia of the cookie as it rises up through the wind gradient. If you think about it, the only way that horizontal thermal speed can equal wind speed is if the thermal has zero inertia or that the wind can exert infinite force on the thermal. However, air has mass and thermals are big. They do have inertia. Also, it is unlikely that wind is able to exert infinite force on another body of air. As we will see, there is a parameter that illustrates this relationship between mass and drag force. This author is a structural dynamic engineer and the first step in the simple approach is to write the equation of motion using Newton ma = F (1) Acceleration is simple the derivative, with respect to time, of the velocity. m v& = F () From aerodynamics, the force on the cookie could be directly proportional to the square of the relative velocity, with dependence on density, reference area, and drag coefficient. ( W ( y v) mv& = d ρ (3) 1 ) C A B.R.Resor Page of 1 November 1, 7
Define the drag proportionality constant as 1 a = C Aρ (4) Then d ( W ( y) v) ( W ( y) v) a v& = sgn (5) m As you can see from the previous equation, the change in the velocity of the cookie is directly related to the drag on the cookie and indirectly related to the mass of the cookie. If the mass of the thermal is higher the thermal will more slowly approach the speed of the wind (it is harder for the wind to pull the thermal along). When more drag is available between wind and thermal the thermal will more quickly approach the speed of the surrounding wind (it is easier for the wind to pull the thermal along). It makes sense. The drag proportionality constant, a, is directly related to coefficient of drag, reference area and air density. The signum function is added to account for situations when v happens to be larger than W. The mass of the cookie can be written as m = ρπr h (6) The reference area can be written as A = Rh (7) Simplifying the a over m term a m Cd Rhρ C = d ρπr h πr 1 = (8) Notice that we have lost dependence on h and ρ. Though, here we see that the ρ s would not cancel out if we took account for the fact that air inside the thermal is a little less dense that air outside the thermal. The net effect might be to increase the ratio a over m in this study. However, I will keep this model simple and ignore that information. An estimate of C d is needed, and depends on Reynold s Number. ρvl Re (8) µ An example is shown below for a cylindrical body. B.R.Resor Page 3 of 1 November 1, 7
Figure. Cd versus Re for a cylinder []. Using basic estimates of viscosity and density of air, and characteristic length equal to diameter of the thermal, Re is about 1. This yields a C d of about 1. Assume a thermal diameter, R, of about 5 feet. Then a/m is approximately.1. Results Assume V v = at y= and then accelerates to a good thermal value of about 4 knots (~4 fpm) Equation (5) can be solved numerically using Newton s Method. Assume v& ( ) =, nonmoving thermal source. Wind speed on the ground is 1 mph, with winds increasing with altitude. This is Case #1. 1 1 thermal strength 1 1 wind thermal 1 1 relative velocity (wind-thermal) altitude, ft 1 3 4 5 vertical velocity, ft/min 5 1 15 horizontal velocity, mph 4 6 8 1 relative horizontal velocity, mph Figure 3. Results of simulation for Case #1, fairly strong winds and fairly strong gradient. Notice that the thermal never does completely catch up with the speed of the surrounding air (until the wind gradient dies down up high). In other words, there is always a little bit of relative velocity between thermal and wind. B.R.Resor Page 4 of 1 November 1, 7
An uncertainty analysis on the parameter a/m shows the calculations for a/m that is higher and lower by a factor of five, for reference, in case there has been an error in the choice of specific values to arrive at a/m. Dotted lines are the two extra cases: 1 1 thermal strength 1 1 wind thermal 1 1 relative velocity (wind-thermal) altitude, ft a/m low x5 a/m high x5 1 3 4 5 vertical velocity, ft/min 5 1 15 horizontal velocity, mph -5 5 1 relative horizontal velocity, mph Figure 4. Uncertainty on Case #1 simulation. How about another case, Case #, where wind is not so violent aloft. Less gradient and less velocity overall: 1 1 thermal strength 1 1 wind thermal 1 1 relative velocity (wind-thermal) altitude, ft 1 3 4 5 vertical velocity, ft/min 4 6 8 1 horizontal velocity, mph 4 6 8 relative horizontal velocity, mph Figure 5. Results of simulation for Case #, mild winds and mild gradient. There is still a noticeable amount of relative horizontal velocity between thermal and surrounding atmosphere. The velocity data in both horizontal and vertical directions can be integrated with respect to time to arrive at a trajectory plot for Case #. The plot shows the trajectory of a massless particle ascending through the wind profile at the same speed as the thermal cookie: B.R.Resor Page 5 of 1 November 1, 7
(axis equal scaling) 1 1 massless neutral balloon thermal packet -.5 1 1.5 horizontal displacement, ft x 1 4 Figure 6. Trajectories for Case #. The axes are scaled equally such that the trajectory seen here is what one would expect for thermal drift in a 7 mph nominal wind and 4 knot climb. Keep in mind that it is likely that the human eyes and brain are not able to truly comprehend depth vs altitude when watching a glider climb, whether it is from the ground or the cockpit. This angle is fairly steep for a day with only 7 mph of wind. A true look at the downwind drift in SeeYou on such a day will prove my point. Implications of model results Implications of thinking about thermal drift in more complex terms other than they drift exactly with the wind include many interesting ideas: Atmospheric mixing The idea makes sense because it is probably a major mechanism by which the boundary layer mixing in the lower atmosphere occurs. Think of the thermals as little bristles in the boundary layer that reach up from the ground, W(y)=, to the upper air, W(y>>)>>, to create resisting forces that eventually get the whole boundary layer to blow as the day becomes breezy. Flow around the thermal Think of what this means for entering and leaving the thermal in a sailplane, and the feedback you experience as you try to locate and center a thermal core. Think also of how at the edge of the thermal/wind interface the thermal can get so shredded up. It is due to relative velocity and resulting shearing! There could be some especially interesting flow convergences, and perhaps turbulence, at the back side of the cylinder. B.R.Resor Page 6 of 1 November 1, 7
Figure 7. Relative flow around the thermal "cookie." Thermal streets Next think about how the air might flow around a pattern of isolated thermals as compared to how it would flow around an organized line of closely spaced thermals. It is easy to see why the rising air organizes into a vertical plate under a cloud street with enough wind. This configuration presents lower drag to the atmosphere than would several smaller columns of thermals. Thus, 1) the relative velocity error could be even higher on the edges of the vertical plate and ) the cloud street is a stable formation because it is the configuration presenting the least resistance to the more quickly moving boundary layer. Figure 8. A cross section of thermals at high altitude (not near the ground): Progression of evenly spaced thermals toward a configuration of thermals that presents a lower resistance to the atmosphere. B.R.Resor Page 7 of 1 November 1, 7
See Brigliadori [1] for intuitive description and demonstration of how lift gets organized in this manner starting from a hexagonal pattern to linear streets. Location of lift underneath cumulus clouds It makes even more sense now why we generally find lift on the upwind side of a large cumulus cloud base. The rising thermal is constantly moving up through the wind profile. However, the condensed air, or cumulus cloud, at the top of the thermal is moreor-less parked at a given altitude. This wind at that altitude has more time to react on that mass of air and moisture and the cloud will tend to drift down wind of the thermal input point (of course we know that in NM there are often several thermal inputs to a single cloud). Some pilots have also seen examples in the past of thermal lift that is found on the downwind side of a cumulus cloud. See the following example simulation. 1 1 thermal strength 1 1 wind thermal 1 1 relative velocity (wind-thermal) altitude, ft 1 3 4 5 vertical velocity, ft/min 5 1 15 horizontal velocity, mph - 4 6 8 relative horizontal velocity, mph Figure 9. Example of wind profile with peak velocities at lower altidtude. Assuming that cloud base is found at 1, ft AGL in this figure, one might imagine in this case that the lift could be found on what appears to be the downwind side of the cloud due to the change in sign of relative velocity of the thermal and wind. Wind calculations based on circling data If what this paper suggests is true, the glide computers that rely on circling to make wind measurements are going to usually be a little low on their estimation of magnitude. Upon close reading of Brigliadori, some people already knew this. This would explain several experiences that I have had over the years. It would also help to explain why a downwind final glide always seems to be a little easier and upwind final glides are a little harder (if you have underestimated the magnitude of your tailwind or headwind!). If you dial in the right wind magnitude and you actually have the polar that you think you do, then both cases should work out equally well. Knowing this, we can plan our glides accordingly. B.R.Resor Page 8 of 1 November 1, 7
Thermal strength As the thermal vertical velocity V v is increased, the effect is to increase the relative difference between wind and thermal horizontal velocity. The effects seen above in the simulations are magnified. Following are example of good thermal, 4 knots, and great thermal, 8 knots. 1 1 1 1 wind thermal 1 1 thermal strength dotted: 8kt thermal solid: 4kt thermal relative velocity (wind-thermal) altitude, ft 4 6 8 1 vertical velocity, ft/min 5 1 15 horizontal velocity, mph 4 6 8 1 relative horizontal velocity, mph Figure 1. Effect of lift strength on relative velocity. Time of day Time of day probably has an effect on how this model could be realized. One might argue that this effect is most pronounced at the beginning of a soaring day when the mixing just begins. Perhaps by evening, when energy input is decreased and the whole boundary layer has reached a more or less steady state it is not the same situation. It might help explain why evening thermals are so nice and smooth compared to the early thermals. Thermal source (fixed or moving) Each pilot needs to consider their beliefs on whether most thermal sources are fixed on the ground, or moving with the wind. Most would argue that the source, or trigger point, is fixed in most cases. However, if the source is moving for some reason it affects the results of this model with initial condition v()>. Choice of speed-to-fly for cross country flight between thermals This paper shows that thermals do not move at the same speed as the wind. Simple assumptions regarding MC theory applied in situations with headwind or tailwind could be inaccurate. Some might argue that it is a minor effect, but Brigliadoris [1] show how the choice of speed to fly between thermals is not the simply-chosen MC setting. One should fly a little faster when penetrating upwind. See their book for more information on this subject. B.R.Resor Page 9 of 1 November 1, 7
Conclusion In the end, of course, thermals are complicated creatures and there is no one description of their true behavior. This simple model helps to demonstrate how the mass inertia of the thermal allows it to resist acceleration by the wind in the atmospheric boundary layer. Actually, after reading Brigliadori very closely, I have come to realize that at least some pilots have knowledge of this phenomenon already and they use it regularly in their soaring decision making. Apparently I ve just been ignorant all this time. Hopefully this little exercise has increased the awareness of the thermal structure and has generated some thinking that will help to understand what might be going on the next time you get low and just can not seem to figure out what s going on with the lift you are chasing. References [1] Brigliadori, Leo and Ricky. Competing in Gliders Winning With Your Mind. [] http://scienceworld.wolfram.com/physics/cylinderdrag.html B.R.Resor Page 1 of 1 November 1, 7